Carlos Eduardo Suprinyak
email: csuprinyak@aup.edu

GROTIUS AMONG THE ENGLISH MERCHANTS: MARE LIBERUM AND ANGLO-DUTCH RIVALRY IN THE EARLY SEVENTEENTH CENTURY

Keywords: Hugo Grotius, Mare Liberum, free trade, British Empire, fisheries B11, B17, F52, F54

Hugo Grotius was a widely read and influential figure in seventeenth-century England. Whereas later generations portrayed him as a forefather to modern theories of natural and international law, the publication of Mare Liberum (1609) offered a grounded argument for free trade against the restrictions imposed by the Spanish and Portuguese colonial empires. But Grotius's notion of free trade, of course, was far removed from the later ideal of Richard Cobden. His model of commercial organization was firmly anchored on the chartered mercantile company, and he served as a spokesperson for the Dutch East India Company (VOC) during the Anglo-Dutch Conferences of 1613 and 1615, which sought to appease growing tensions among Dutch and English trading interests in Asia. As the diplomatic and economic relations between England and the Dutch Republic progressively deteriorated during the early decades of the seventeenth century, Grotius's rendition of the law of nations came increasingly to be regarded as a rationalization of Dutch dominance over long-distance trade routes. English commentators, who wished to emulate Dutch success but feared aggression and subordination, had to fashion a different framework of international politics to sustain their vision of an emerging English maritime empire. Tracing the uses of Mare Liberum in the mercantile literature of early Stuart England, the paper will study how this foundational work of modern international thought shaped the political economic discourse of English merchants, as well as their supporters and adversaries in the political arena. Changing attitudes toward Grotius's arguments will help us identify the origins of certain ideas about empire that later came to the fore by the time of the Navigation Acts and the Anglo-Dutch wars. We will thus explore how Grotius's version of the Law of Nationsbefore it was retrospectively converted into a cornerstone of liberal internationalismwas entangled in concrete disputes between rival imperial projects.

Introduction

Hugo Grotius is a thinker with many legacies. At least since the founding of the Grotius Society in 1915, his name occupies a distinguished position in contemporary legal scholarship, denoting an approach to international law that embraces an ethics-laden interpretation of natural law doctrine [START_REF] Lauterpacht | The Grotian Tradition in International Law[END_REF]Jeffery 2006, pp. 85-111). His statement of the principle of the freedom of the seas, likewise, proved very influential over time (Reppy 1950;[START_REF] Butler | Hugo Grotius and the Law of the Sea[END_REF]). Grotius has also been popular among students of political philosophy. Richard Tuck (1979, p. 58) described him as "the most important figure" in the modern history of natural rights theory, an appraisal shared by [START_REF] Haakonssen | Hugo Grotius and the History of Political Thought[END_REF]. Tuck himself later portrayed Grotius as an essential link in the transition between humanist and liberal political vocabularies during the seventeenth century, and even a precursor to the state of nature hypothesis [START_REF] Tuck | Philosophy and Government 1572-1651[END_REF][START_REF] Tuck | The Rights of War and Peace: Political Thought and the International Order from Grotius to Kant[END_REF]. [START_REF]Hugo Grotius and England[END_REF] highlighted yet another facet of Grotius: an avowed disciple of Erasmus and influential contributor to early modern theological arguments, a connection recently reinforced by Marco [START_REF] Barducci | Hugo Grotius and the Century of Revolution, 1613-1718: Transnational Reception in English Political Thought[END_REF].

The Grotian legacy has also been deeply entrenchedthough subject to much controversyin the field of international relations. Distinguished postwar IR scholars like Martin Wight and Hedley Bull built heavily on Grotius' De Jure Belli ac Pacis (1625) in their accounts of international society and international order. [START_REF] Wight | Western Values in International Relations[END_REF][START_REF] Wight | International Theory: The Three Traditions[END_REF] famously popularized the typology of three competing and intertwined 'traditions' of Western international thought: the two extremes of Hobbesian (or Machiavellian) realism and Kantian utopian cosmopolitanism, complemented by the via media of Grotian rational internationalism. This tripartite scheme was further elaborated by [START_REF] Bull | The Anarchical Society: A Study of Order in World Politics[END_REF], who presented Grotius as an early exponent of the notion of an international society of states kept together by tacit respect for a set of rules firmly inscribed in natural law. Through these and related works, Grotius was turned into a canonical figure in the history of international thought. As usually happens when thinkers thus become canonized, the connections between works belonging to a putative 'Grotian tradition' and the writings of Grotius himself grew increasingly murky (Kingsbury and Roberts 1990;Jeffery 2006, pp. 14-26).

But Grotius was not an ivory tower intellectual exclusively concerned with elaborating the abstract principles of international law, politics, and religion. For most of his life, on the contrary, his philosophical and theoretical writings were closely entangled with his personal involvement in government and diplomacy, first in his native Low Countries and then in France. As such, Grotius was an active participant in contemporary arguments about the reconfiguration of the world order following the emergence of modern European empires. This aspect of the Grotian legacy has been discussed by David Armitage (2000, pp. 100-124), who showed how Grotius' Mare Liberum (1609) provided a benchmark against which British claims for an empire of the seas could be formulated. Edward Keene (2002) highlighted how certain lesser-known aspects of Grotius' account of the law of nations "have a striking proximity to the practices of colonialism and imperialism that Europeans adopted in the extra-European world" (p. 3). Peter [START_REF] Borschberg | Hugo Grotius' Theory of Trans Oceanic Trade Regulation: Revisiting Mare Liberum (1609)[END_REF] reconstructed the arguments of Mare Liberum as a rationalization of the practices of the Dutch East India Company in Asia, concluding that Grotius should be counted "among the architects of early modern colonial rule " (p. 46). The same connection between Grotius' early writings and the political agenda of Dutch mercantile elites was explored in detail by [START_REF] Van Ittersum | Profit and Principle: Hugo Grotius, Natural Rights Theories and the Rise of the Dutch Power in the East Indies, 1595-1615[END_REF], who portrayed him as "one of the founding fathers of the First Dutch Empire" (p. xix). Focusing on transnational reception, Erik [START_REF] Thomson | The Dutch Miracle, Modified. Hugo Grotius's Mare Liberum, Commercial Governance and Imperial War in the Early-Seventeenth Century[END_REF] showed how Mare Liberum provided an imperfect blueprint for early modern European powers seeking to reinvent themselves as commercial empires, following the Dutch model. This chapter will follow a similar path to explore how Grotius' articulation of the case for freedom of the seas was received, across the North Sea, by a group directly concerned with Dutch imperial ambitions: the English mercantile community. As we will see, the merchant pamphleteers commonly regarded as typical examples of early modern political economy -Gerard de Malynes, Edward Misselden and Thomas Munengaged extensively with the arguments advanced by Grotius in Mare Liberum. But these contemporary commentators, rather than treating the work as an authoritative account of maritime law, discussed and criticized it as a partisan piece at the service of specific political and ideological agendas. By examining the reception accorded by the English merchants to Grotius' vindication of a trading empire based on naval power, we will gain some insight into competing images about the future of the British Empire itselfone of them based on free open seas and long-distance colonial trade, the other on closed territorial seas and the fishing industry.

Mare Liberum and Anglo-Dutch Commercial Rivalry

During 1607-8, Spain and the Dutch Republic conducted negotiations for a truce to the armed conflict that had raged since the rebellious United Provinces seceded from the Spanish Empire in 1581. One of the matters in dispute was Dutch access to the East Indian spice islands, controlled by the Portuguese since the 16 th century and hence indirectly by Habsburg Spain after the union of the Iberian crowns in 1580. To sustain its commercial incursions in the region, in 1602, the Dutch created the Vereenigde Oostindische Compagnie (VOC), a chartered mercantile company with exclusive trading rights to Asia. The VOC profited from official hostilities with Spain to follow an aggressive course in its commercial exploits, striking diplomatic agreements with native local rulers while engaging in interloping and privateering activities against Portuguese merchants. The truce negotiations thus presented a potential threat to the company's established interests, as the Spanish crown insisted on a recognition of its right to exclude other nations from the East Indian trade as a precondition for any long-term ceasefire (Clark 1934, pp. 49-50;Reppy 1950, pp. 251-8;Roelofsen 1990, pp. 105-6). 1 This was the immediate context for the publication of Mare Liberum in 1609, the year when the Twelve Years' Truce was finally signed. Written in Latin and published anonymously, the small tract presented a case for Dutch rights of navigation and trade to the East Indies against Spanish attempts to restrict access to the region. The choice for anonymity suggests how the work was consciously designed to intervene in ongoing, politically charged arguments. Indeed, Grotius prepared it for print at the instigation of a group of VOC directors, who were engaged in a broad public campaign against the truce (van Ittersum 2006, pp. 283-358). Conversely, the use of Latin signals how the message of Mare Liberum was aimed at an elite international audience. To some commentators, the episode symbolizes the decisive incorporation of commercial affairs into official European political and diplomatic discourse.

As framed by Erik Thomson (2009, p. 109), the debate surrounding the truce "reshaped European political thought by figuring the governance of commerce as crucial to the preservation, conservation and expansion of sovereignty".

But this was only one chapter of Grotius' much broader involvement with the VOC.

In fact, most of the material contained in Mare Liberum had been written a few years earlier, in response to another incident involving the Dutch company's operations in the East Indies.

In 1603, as part of its war of attrition against the Iberian powers, the VOC had captured a Portuguese carrack named Santa Catarina off the coast of Singapore. This otherwise ordinary act of early modern privateering acquired greater significance after the ship's cargo was 1 Portuguese and Spanish claims to exclusive rights in the East Indian trade were based on the bulls of donation issued by Pope Alexander VI in 1493, which divided overseas territories -both newly discovered and yet to be discovered -into two encompassing zones, granted in possession to each of the two kingdoms.

apprehended and brought back to Europe, where it fetched a spectacular sum in auction [START_REF] Borschberg | The Seizure of the Sta. Catarina Revisited: The Portuguese Empire in Asia, VOC Politics and the Origins of the Dutch-Johor Alliance (1602-c.1616)[END_REF][START_REF] Armitage | Introduction[END_REF]). As a private company, the VOC's rights to wage war on the Portuguese and lawfully seize the prize were ambiguous, to say the least. Between 1604 and 1606, Grotius worked on an elaborate legal apology for the company's actions. Though never published during his lifetime, the manuscript was later rediscovered and made publicly available under the title De Jure Praedae. 2 In this work, he justified the VOC's conduct as part of a just war prompted by Portuguese aggression against both Dutch merchants and their East Indian allies, in which the company was legally entitled to seize war prizes because acting on commission by the Dutch government [START_REF] Borschberg | Hugo Grotius, East India Trade and the King of Johor[END_REF]van Ittersum 2006, pp. 1-52).

What appeared in print as Mare Liberum was a slightly revised version of Chapter 12 of De Jure Praedae. The text was cleared of references to the Santa Catarina incident, and the resulting argument was pitched at a higher level of generality [START_REF] Armitage | Introduction[END_REF]). Grotius counteracted Iberian claims to exclusive access to the East Indies with an appeal to the principles of natural law and what he termed the 'primary' law of nations, which established as a "certain rule" that "it is lawful for any nation to go to any other and trade with it" [START_REF] Grotius | The Free Sea, trans. Richard Hakluyt, with William Welwod's Critique and Grotius's Reply[END_REF][START_REF] Grotius | The Free Sea, trans. Richard Hakluyt, with William Welwod's Critique and Grotius's Reply[END_REF], p. 10). He justified this principle explaining how divine providence had spread its bounty across the globe "to maintain human friendship by their mutual wants and plenty". This allowed Grotius to assert that even if the Portuguese "had been lords of those countries whither the Hollanders go, yet they should do wrong if they stopped the passage and trade of the Hollanders" (p. 12).

The other essential piece of his case rested on a theory of property rights, elaborated in the lengthy and controversial Chapter 5. "God gave all things not to this man or that but to mankind," Grotius argued, but the profitable use of certain things required they become the property of individuals (p. 22). Certain constraints, however, should be respected: first, "those things which cannot be occupied or were never occupied can be proper to none because all propriety hath its beginning from occupation"; second, property could not be lawfully instituted on "all those things which are so ordained by nature that anyone using them they may nevertheless suffice other whomsoever for the common use" (p. 24). To Grotius, the seas clearly fell under both these categories, being "so infinite that it cannot be possessed and applied to all uses, whether we respect navigation or fishing" (p. 25). This led him to conclude "no part of the sea can be accompted in the territory of any people" (p. 30). Having thus discredited the Portuguese claim to exclude other nations from the sea route to the East Indies, Grotius expanded on his attempt to frame the freedom to trade as part of natural law: Therefore, the liberty of trading is agreeable to the primary law of nations which hath a natural and perpetual cause and therefore cannot be taken away and, if it might, yet could it not but by the consent of all nations, so far off is it that any nation, by any means, may justly hinder two nations that are willing to trade between themselves. [START_REF] Grotius | The Free Sea, trans. Richard Hakluyt, with William Welwod's Critique and Grotius's Reply[END_REF][START_REF] Grotius | The Free Sea, trans. Richard Hakluyt, with William Welwod's Critique and Grotius's Reply[END_REF], p. 51)

In a context of overseas imperial disputes, Grotius' defense of the natural freedom to trade carried certain implications. One of them derived from his conception of the limits imposed by nature on private property: "that all surely might use common things without the damage of all and, for the rest, every man contented with his portion should abstain from another's" (p. 6). When extending this reasoning to the East Indian trade, Grotius asked: "What, therefore, may seem so unjust as that the Spaniards should have the whole world tributary, so that they might neither buy nor sell but at their pleasure?" (p. 56). Iberian attempts to monopolize this trade infringed on the rights of other nations to enjoy the common fruits of divine providence, and thus stood in flagrant breach of natural law. This led to a second notable implication, expounded in the final chapter. Reasserting the hierarchy between natural law and political expediency, Grotius asserted: "we wholly maintain that liberty which we have by nature, whether we have peace, truce or war with the Spaniard" (p. 57). The grievances inflicted by Spanish and Portuguese on the rest of the world should be recognized by any fair judge observing the principles of natural law. But in matters of this kind, recourse to a fair and impartial judge was often unavailable, which led Grotius to proclaim: "that which should be obtained in judgment, where justice could not be had by just war could be revenged" (p. 59).

Moreover, the rights to wage just wars and punish the offenders of natural law were not restricted to sovereign rulers but could be undertaken by private parties as well. "Wherefore he that shall stop the passage and hinder the carrying out of merchandise may be resisted by way of fact," Grotius concluded, "even without expecting any public authority" (p. 60).

While presenting his case for the freedom of trade and navigation and against the monopolistic ambitions of certain early modern overseas empires, Grotius therefore also legitimized some of the aggressive practices increasingly adopted by private actors like the VOC. By doing so, he contributed some compelling ideological groundwork for an emerging mercantile empire where chartered companies wielded significant judicial and political power (van Ittersum 2010; Barducci 2017, pp. 158-61). In the words of Peter Borschberg (2005, p. 43), "Grotius envisions a freedom of the high seas, but in actual fact this translates into a brutal free-for-all struggle, an indivisible trinity of commerce, war and plunder," an outlook that was "happily espoused by his political superiors and by a powerful faction among the merchant community in the early seventeenth century Netherlands". Among his English readers, this aspect of Grotius' argument would not be lost either.

Contrary to some of the VOC's fears, the Twelve Years' Truce significantly enhanced the Dutch position in the East India trade. It also inaugurated an era of heightened Anglo-Dutch commercial rivalry, partly explained by James I's desire to preserve friendlier relations with Spain than his predecessor Elizabeth I, but also reflecting the growing success of both nations in maritime trade (Edmundson 1911, pp. 11-33). Disputes between the English and Dutch East India companies prompted the convening of two diplomatic conferencesone in London in 1613, the other at The Hague in 1615designed to settle colonial claims of both parties and find a workable modus vivendi [START_REF] Borschberg | Hugo Grotius, East India Trade and the King of Johor[END_REF]. In a quick reversal of roles, it was now the Dutch who stood accused of monopolistic practices by restricting the access of English merchants to the spice islands in Asia. English negotiators seized on the same rhetoric previously employed by Grotius against the Iberian powers, arguing the Dutch violated the law of nations when denying liberty of trade and navigation to others in the East Indies, thus repeating what they had formerly condemned in the Portuguese [START_REF] Clark | Grotius's East India Mission to England[END_REF].

Grotius himself was a member of the Dutch delegation sent to London for the 1613 conference, during which, in a much-commented episode, he heard the arguments of Mare Liberum explicitly quoted in support of the English case (Clark 1934, pp. 78-9;Borschberg 1999, pp. 241-8;Armitage 2000, pp. 111-2). His work had already crossed the North Sea, at a time when its authorship was no longer kept under much secrecy. 3 Stepping up once again to defend the VOC's actions, Grotius appealed to treaties that had been signed by the company with East Indian sovereign rulers to justify its rights to exclude others from the trade. Natural law dictated, as an immutable principle, that contracts must be honored, and this overrode the prescription contained in the law of nations in favor of free trade and navigation. The contracts in question had been signed voluntarily by local rulers in return for their liberation from Portuguese yoke, and their continued enforcement was amply justified by the costs incurred by the Dutch to keep them safe from further Iberian aggression (van Ittersum 2006, pp. 359-71).

Predictably, this qualification of the arguments advanced in Mare Liberum proved less than convincing to the English delegates. The episode demonstrated how the discourse of natural law and the law of nations was flexible enough to accommodate different imperial claims and ambitions (van Ittersum 2016, pp. 490-2). It also shaped in important respects the context for the reception of Grotius' work in England.

The Fisheries Dispute

There is another aspect of Anglo-Dutch relations during the early decades of the seventeenth century that significantly affected the way in which Mare Liberum was read and discussed in England. At the time, the North Sea was the site of a thriving fishing industry open to the competition of Dutch and British vessels. The Dutch were regularly allowed to fish off the coasts of England without paying any fees, a privilege confirmed by different treaties over the centuries (Edmundson 1911, pp. 20-1;[START_REF] Benson | England, Holland, and the Fishing Wars[END_REF]. The Scottish crown, however, had been much more zealous of its sovereignty over territorial seas, seeking to protect its fishing grounds from the encroachment of foreigners. When James I acceded to the English throne in 1603, he began extending this Scottish understanding of maritime law to the whole of the British Isles (Armitage 2000, pp. 108-9). This policy dovetailed with entrenched popular convictions about the importance of the herring fisheries to Dutch naval and imperial might, which prompted initiatives by both crown and parliament to support the British fishing industry [START_REF] Benson | England, Holland, and the Fishing Wars[END_REF].

In 1609, the same year Mare Liberum first came out, James issued a proclamation that forbid all foreigners from fishing in British and Irish coasts without previously securing a license. The Dutch soon protested this measure, arguing it violated the ancient privileges they had enjoyed with the acquiescence of countless predecessors of the new English king. The dispute was further pursued during a diplomatic conference held in London in 1610, when the Dutch questioned English rights to close off the seas adjoining their coasts, as these belonged in common to all humankind and could not be brought under the exclusive possession of anyone. Negotiations were concluded without any satisfactory agreement, with only James' promise of postponing the enactment of his proclamation for two years (Edmundson 1911, pp. 26-33). The result presaged a sustained legal and commercial controversy that would flare up on multiple occasions before mid-century.

The North Sea fisheries dispute thus shaped, from the very beginning, the way in which the arguments of Mare Liberum were perceived by British audiences. The most obvious illustration of this pattern is William Welwod's An Abridgement of All Sea-Lawes (1613), one of the earliest written responses to Grotius' tract, which portrayed his defense of Dutch rights of navigation to the East Indies as an elaborate ruse to justify their trespassing on British fishing grounds. This reading found corroboration in certain passages of Mare Liberum, as when Grotius argued: "And the same regard that is to be had of navigation is to be had likewise of fishing, that it may remain common unto all" (Grotius 2014[START_REF] Grotius | The Free Sea, trans. Richard Hakluyt, with William Welwod's Critique and Grotius's Reply[END_REF], p. 28). Welwod conceded Grotius's case with respect to the "great, huge, and main body of the sea", but reaffirmed the rights of sovereigns to establish dominion over the territorial waters lying within the "just and due bounds […] properly pertaining to the nearest lands of every nation" [START_REF] Welwod | Of the Community and Propriety of the Sea[END_REF][START_REF] Welwod | Of the Community and Propriety of the Sea[END_REF], pp. 72-4).

The topic was frequently addressed in the English pamphlet literature over the following years. While not engaging the legal argument about rights of dominion over the seas, Tobias Gentleman's England's Way to Win Wealth, and to employ Ships and Marriners (1614) painted a vivid portrait of the benefits enjoyed by the Dutch from fishing off the British coasts.

The author highlighted the "inestimable summes of money taken yearely for fish and herrings out of his Majesties Seas by strangers," through which they had "not onely maintained their warres many years against the Spaniard," but also "growne exceeding rich and strong in fortified Townes and beautifull Buildings, in plenty of money and gold, in trade and trafficke with all other Nations," besides having "so increased and multiplied their shipping and Marriners, that all other Nations and Countries in the world doe admire them" (Gentleman 1614, pp. 4-5). The pamphlet played upon feelings of Anglo-Dutch rivalry when describing how English fishermen were "daily skorned by these Hollands, for being so negligent of our profite, and carelesse of our fishing" (p. 44). Building on Gentleman, Robert Kayll's The Trades Increase (1615) advanced a more inflammatory argument in favor of stricter control of British territorial seas. Kayll stressed how the Dutch derived their wealth and might from fishing "in our owne Seas, […] in his Maiesties dominions, on the coast of England, Scotland, Ireland" (Kayll 1615, p. 36). While recognizing Dutch claims of ancient privileges in this trade, he cautioned: "howsoever it pleaseth his Maiesty to allow of his royall Predecessours bounty, in tollerating the neighbour Nations to fish in his streames, yet other Princes take more straighter courses" (p. 37).

Kayll's pamphlet is also revealing for his antagonistic attitude toward long-distance colonial trade. Though this was "the best Trade in appearance", yet closer inspection revealed "how remote it is, and with what cost of purse, and losse of people followed, yet without such satisfaction" (p. 40). The herring fisheries, by contrast, offered a much safer and reliable foundations for a maritime empire:

Wee thinke the West Indie gold to be the cause of the pride and presumption of the Spaniards; we may assure ourselves that our North Indies countervaile that treasure, and are the onely confidence of the Hollander even by breeding sea-men, and increasing of shipping in that aboundance, as that hereby they both swarme every where, and France, Spaine and the East Countries are full of their shipping. (Kayll 1615, p. 47) Sir Dudley Digges came to the rescue of the East India Company with The Defence of Trade (1615), where he commended the goal of reviving the British fishing industry but warned against the danger of leaving the colonial trade in the hands of the Dutch, despite the latter being England's "best assured friends" both "in reason of state, and through band of Religion" (Digges 1615, p. 4). A similar message was advanced in an anonymous pamphlet titled Britaines Busse (1615), which further promoted the case of the North Sea herring fisheries while striving to portray them as complementary, rather than opposed to the East India trade.

The prospect of recapturing for Britain part of the extraordinary gains derived by the Dutch from the fisheries moved James to action once again in 1617, imposing a toll on Dutch vessels fishing off the coast of Scotland. This quickly turned into a sour diplomatic incident after one Scottish official was captured and taken as prisoner to Holland by Dutch captains who refused to pay the toll (Edmundson 1911, pp. 50-5;Benson 2015, pp. 450-1). Meanwhile, tensions between the two mercantile companies in the East Indies escalated once again, prompting the dispatch of yet another Dutch embassy to London, in 1618, to discuss a peaceful settlement of disputes. By now, Grotius had fallen from grace after recent political upheavals in the United Provinces culminating in the execution of Johan van Oldenbarneveldt, his longtime patron. As the Dutch envoys insisted on the rights of their people, under the law of nations, to fish on British coasts, James told them the misfortunes of Grotius "should be a warning to others not to accept his theories" (van Ittersum 2010, p. 390). These charged negotiations provided the occasion for the first drafting of John Selden's celebrated rejoinder to Grotius, Mare Clausum (1635), which defended British rights of dominion and sovereignty over the Northern seas.

As suggested by David Armitage (2000, pp. 103-5), during the early decades of the seventeenth century, Britain was struggling to square the circle between her ambitions for unimpeded commercial expansion overseas and the desire to establish more secure and stable bases for a maritime empire at home. This translated into a readiness to embrace the principles of mare liberum when applied to the open seas, while sticking closely to a policy of mare clausum on the territorial seas adjoining the British and Irish coasts (Loth 1995, pp. 717-22).

In both cases, British imperial ideology was developed against the background of growing Anglo-Dutch rivalrybetween chartered mercantile companies in Asia, or fishing vessels in the North Sea. But the relative importance of these two planes of commercial and maritime expansion for the strength and opulence of the British empire was a matter of dispute.

Mare Liberum vs Mare Clausum in the English Mercantile Literature

During the first half of the 1620s, England was hit by a severe economic depression caused by disruptions in the European cloth trade, an early reflection of the seismic movements brought about by the Thirty Years' War. The depression prompted much public discussion about the economic maladies afflicting the kingdom and the remedies appropriate for relieving them. Some of the written interventions in this debate later came to be considered landmark works in the canon of early modern political economy. The pamphlets of Thomas [START_REF] Mun | A Discourse of Trade, From England unto the East-Indies[END_REF][START_REF] Mun | England's Treasure by Forraign Trade. Or, The Ballance of our Forraign Trade is the Rule of Our Treasure[END_REF], Gerard de Malynes (1622[START_REF] Malynes | The Center of the Circle of Commerce[END_REF] and Edward [START_REF] Misselden | Free Trade, or the Meanes to Make Trade Florish[END_REF][START_REF] Misselden | The Circle of Commerce, or the Balance of Trade, in defence of free Trade[END_REF], to mention the most conspicuous examples, were drafted to engage ongoing political arguments about the dire conditions of the English economy, long before they became symbols of 'mercantilist' doctrine (Supple 1964, pp. 197-224;Magnusson 1994, pp. 60-93;[START_REF] Suprinyak | Trade, Money, and the Grievances of the Commonwealth: Economic Debates in England During the Commercial Crisis of the Early 1620's[END_REF]). As such, they echoed many of the background conditions and arguments mentioned in the previous sections.

What may be more surprising to current readers are the frequent references to Mare Liberum found in the writings of these merchant pamphleteers. Even if Richard Hakluyt had prepared an English translation of Grotius' tract already by the mid-1610s, this only circulated in manuscript form at the time, so it remains unclear how widely accessible it was to average readers. 4 But one should not be too astounded by the prospect of an early modern English merchant engaging a Latin text on the principles of maritime law, as much as this may conflict with pre-conceived images of these pamphleteers as vulgar pleaders for vested interests. As shown by Andrea [START_REF] Finkelstein | Gerard de Malynes and Edward Misselden: The Learned Library of the Seventeenth-Century Merchant[END_REF], early modern merchants typically belonged to an urban elite that displayed remarkable levels of literacy and erudition. Both Malynes and Misselden regularly engaged ancient classics and contemporary humanist scholarship, quoting from sources in Greek, Latin and Hebrew. While less prone to learned discourse, Mun (1664, p. 8) nonetheless suggested when describing the ideal qualities of a perfect merchant: "although there be no necessity that [he] should be a great Scholar; yet it is (at least) required, that in his youth he learn the Latine tongue, which will better enable him in all the rest of his endeavours".

Mun's first pamphlet, A Discourse of Trade, From England unto the East Indies (1621), does not contain any explicit allusions to Mare Liberum. But as an apologia for the activities of the English East India Company against its domestic detractors, it is steeped in the language of imperial expansion and Anglo-Dutch rivalry. Mun portrays the opening of the sea route to the East Indies via the Cape of Good Hope as a victory for Christendom, as it diverted the European trade in Asian commodities away from the Ottoman empire (Mun 1621, pp. 4-9). But even if all of Europe ultimately stood to gain from the new arrangements, this did not preclude the emergence of disputes between different national interests, as witnessed by recent altercations between the English and Dutch companies (pp. 37-8). Mun expressed some cautious optimism that merchants of both nations could find the means for a peaceful coexistence, following the more recent rounds of diplomatic negotiations. He then proceeded to identify the true causes of England's economic troubles, only to run again into another instance of commercial strife with the Dutch Republic, this time much closer to home. After listing the fishing industry as one possible channel for boosting English trade and employment, Mun observed: "The Dutch in particular, are said to reape such an infinite wealth yearely by this fishing Trade, that without more certain knowledge thereof I dare not set the sum, it seemeth so incredible" (p. 57).

Whereas Mun only obliquely touched upon the maritime claims in dispute between

England and the United Provinces, Edward Misselden's Free Trade, or the Meanes to Make Trade Flourish (1622) addressed the problem head-on. Andrea Finkelstein (2000, p. 9) called attention to how Misselden reproduces a passage from Grotius on commercial intercourse and the law of nature, without acknowledging the source. But there are plenty of other occasions where the pamphlet explicitly engages the arguments of Mare Liberum. Misselden's critical attitude, perhaps surprising at first given his choice of title, is perfectly in line with the reception accorded to Grotius' tract among English readers at the time. When discussing the inconveniences brought about by the "encroaching of strangers, in fishing upon our coasts," Misselden (1622, p. 35) described how "not onely the bread is taken out of the subiects mouth, but that infinite wealth, which God hath made proper and peculiar unto Us, is become common unto them." He then continued: I am not ignorant that a learned man of that side, pressing hard on a treatise entitled Mare liberum, the Community and freedome of the Sea against the Portugall trade unto the East Indies: doth cunningly and obliquely […] defend and maintaine, in the fift chapter thereof, their Fishing upon our Coasts. (Misselden 1622, p. 36) The pamphlet next invoked Welwod's De Dominio Maris (1615)an elaboration of the critique contained in his previous An Abridgment of All Sea-Lawes -to support England's claims for sovereignty over territorial seas:

To part of that Treatise, there is an answere entitled, De Dominio Maris, to which I refer those that desire further satisfaction in this matter. But in my judgment, […] the author of Mare liberum, though otherwise very learned, strayneth his Arguments for that purpose beyond their strength. For Ius is said to be Scriptum, or Non Scriptum. And Non Scriptum is Consuetudo. And Consuetudo non minus est species iuris, quam ius scriptum. And by both these, the proprieties of the Seas may be proved, to belong to those Princes and Countries, to which they are next adiacent. (Misselden 1622, p. 37) Misselden was careful not to challenge Grotius' arguments for Dutch rights in the East India trade, only denying their applicability to the fisheries dispute. Later in the pamphlet, however, he singled out both the colonial and the fishing trade as sources of abuse inflicted by strangers that caused damage to the king's honor, to public revenue, and the wellbeing of the commonwealth. Such abuses came from either friends or foes. Even if Misselden classified the Dutch among the former, he did not hesitate to highlight how, "by Usurpation, those Friends of ours, deprive us of our East India Trade, and Fishing" (p. 98). But his envisioned solution was distinct for each of these grievances. Regarding the East Indian trade, Misselden hoped the two companies could devise a way to operate on a strongly collaborative manner to present a unified front in their dealings with local rulers and traders (pp. 112-5). When it came to the fishing business, conversely, he was much less magnanimous:

For the Fishing, the infinite treasure that Stangers search out of our Seas, the variety of Trade that thereby they purchase, the multitude of Mariners they breed, the Fleets of Shipping they maintaine, me thinkes should every of them apart, or all of them together, be unto us as some many provocations to rouze us up to the exercise thereof: Whereby his Maiestie might receive such a Tolle of Custome of them, as other Princes doe in like case, and be once againe Lorde and Master of the Seas, for all the dispute of the author of Mare liberum: and the Native subiect encouraged by some Immunity or Privilege, to lay hold on that benefit, which God and Nature hath brought home to our doors. (Misselden 1622, pp. 125-6) Misselden thus seemed much more concerned with establishing the English case for mare clausum in the North Sea fisheries than using the principle of mare liberum to safeguard English access to the East Indies. A rather similar attitude can be discerned in Gerard de Malynes' critical rejoinder, The Maintenance of Free Trade (1622). While antagonizing Misselden in many of his central claims, Malynes (1622, p. 27) agreed the most desirable course of action in the East Indies involved some form of close cooperation between the English and Dutch companies: one could only expect larger and growing gains "when the said parties shall be united in True Love," and thus better equipped to deal with local agents and interests. Likewise, he concurred the English had every right to charge duties of foreigners who fished on their coasts, and should do so both to enrich the kingdom's coffers and to stimulate the domestic fishing industry (pp. 42-4). Malynes then listed the precedents established by several other European maritime kingdoms to conclude:

These exemplary Actions have long determined the question of Mare liberum, touching the Communitie of freedom of the Seas, which is acknowledged to be so, for Navigation, without that the same doth any manner of way prejudice the Distinct Dominion of the Seas of all Princes concerning fishing; that is to say, the fishing Trade. So that it is superfluous to aleage the opinions of Orators and Poets about the fishing heretofore in the Mediterranean Seas, neither doth it belong to this place to cite the determinations of the learned Civilians which are mentioned in the treatise De Dominio Maris. (Malynes 1622, p. 45) Once again, we see a clear hierarchy of priorities when dealing with the arguments about maritime law in dispute at the time. Malynes conceded Grotius' case with respect to navigation, and subscribed to the same natural law argument according to which "God caused nature to distribute her benefites, or his blessings to severall Climates, supplying the barrennesse of some things in one countrey, with the fruitfulnesse and store of other countries, to the end that interchangeably one Common-weale should live with an other" (pp. 58-9). But these considerations came in second when addressing the matter of fishing rights on territorial seas.

That same year, Malynes also published the first edition of his Consuetudo, vel Lex Mercatoria, or the Ancient Law-Merchant (1622), a compendium on mercantile law destined to become a landmark work in the field (de Ruysscher 2020). Even if the work contained no explicit references to Mare Liberum, Malynes (1622b, p. 3) made it clear where he stood already in the opening pages, claiming the law of nations exhorted sovereigns to preserve "a communitie of the Seas for navigation," while also entitling them to "a distinct dominion of the seas adioyning to the territories and iurisdiction of their countries," which comprised the right to impose "duties for the fishing in their Seas, Streams, and Dominions". In later chapters addressing the principles of community and dominion of the seas, Malynes reinforced how, "according to the common right of mankind, Iure Gentium, the navigation through all the world, is no lesse free and open to everie one, than the use of the ayre" (p. 182). This was then followed by a lengthy demonstration, based on natural law and historical precedent, of English rights to dominion over the seas "in places of fishing" (pp. 185-92). Malynes drove his point home in a chapter dedicated to the fishing trade, where he built on Gentleman's England's Way to Win Wealth to describe the herring fisheries as the "chiefest trade and principall Gold-mine" of the Dutch, "whereby many thousands of their people of trades and occupations are set on worke, maintained, and doe prosper" (p. 242).

A Fishing Empire?

In another pamphlet published a year later, titled The Center of the Circle of Commerce (1623), Malynes followed in the footsteps of Robert Kayll to question the profits derived by England from the East Indian trade. The fishing trade, he argued, "will effectually improve our exportations, and afford much more benefit than any Persian or East India Trade can doe, if his Maiestie were pleased to encourage them with previledges and immunities, as the worthiness of the action deserveth" (Malynes 1623, p. 127). As a member of the East India Company's Court of Directors, Thomas Mun naturally held a much more approving opinion of its business. In England's Treasure by Foraign Tradeposthumously published in 1664 but widely accepted to have been written during the 1620s [START_REF] Gould | The Date of England's Treasure by Forraign Trade[END_REF][START_REF] Muchmore | A Note on Thomas Mun's "England's Treasure by Forraign Trade[END_REF])he praised both staple and long-distance trade, on which rested the prosperity of mercantile republics like Venice, Genoa, and the Low Countries (Mun 1664, pp. 23-4). But even for a seasoned colonial merchant like himself, the North Sea fisheries still retained their allure. After arguing the luxury habits of the nobility were not necessarily damaging to the commonwealth if channeled to goods of domestic manufacture, Mun added: But if any man say, that when the people want work, then the Fishing-trade would be a better employment, and far more profitable; I subscribe willingly. For in that great business there is means enough to employ both rich and poor, whereof there hath been much said and written. (Mun 1664, p. 149-50) Chapter XIX of England's Treasure consists in a long elucidation of the different effects proceeding from 'natural' and 'artificial' riches, structured around a comparison between England and the Dutch Republic. Just as the English were rendered "vicious and excessive" by the abundance of natural riches, shamefully neglecting such valuable trades as "the Fishing in his Majesty's Seas of England, Scotland, and Ireland," the Dutch stood out as a prime example of people who, "having little or nothing in their own Territories, do notwithstanding purchase great wealth and strength by their industrious commerce with strangers" (pp. 178-83). But rather than see this almost exclusive reliance on artificial riches as a mark of unfailing success, Mun implied this constituted their signal weakness. The Dutch waged war on the mighty Spanish and controlled the markets of all Europe, but to do so "they have little foundation besides the Fishing, which is permitted them in his Majesties Seas, being indeed the means of an incredible wealth and strength, both by Sea and Land" (p. 186). Without this foundation, he continued, "it is apparent that they cannot long subsist in Soveraignty," since "the whole building of their wealth and strength both by Sea and Land must fall" (p. 187).

Having thus shown how "the glory and power of these Netherlanders consisteth in this fishing of Herrings, Ling and Cod in His Majesties Seas," Mun then proceeded to inquire about "what right or title they have thereunto, and how they are able to possess and keep the same against all other Nations" (p. 188). While Grotius had appealed to precedent and the law of nations, Mun was clearly of a different opinion:

[…] it is not the Netherlandish author of Mare Liberum, that can intitle them to Fish in his Majesties Seas. For besides the justice of the cause, and examples of other Countreys, which might be alleged, I will only say, that such titles would be sooner decided by swords, than with words (Mun 1664, p. 188) The English might have reasons to condone Dutch fishing in their coasts on grounds of political expediency. So long as the Dutch remained "in perfect league with England, and in war with Spain," this act of magnanimity would make them "ever bound to acknowledge their strong alliance with England, above all other Nations, for there is none that hath the like good means to lend them such a powerful maintenance" (pp. 189-90). But as Dutch maritime power thus rested on precarious foundations, the English had no reason to stand in awe of their North Sea rivals: "The United Provinces (we know) are like a fair bird suited with goodly borrowed plumes; but if every Fowl should take his feather, this bird would rest neer naked" (p. 196).

Thomas Mun, the East India merchant, was thus ready to subscribe to a vision of commercial and maritime empire where the command of territorial possessions close to home, rather than long-distance trade routes, was the foundation of power and opulence. Others were more willing to bet on colonial trade as the key to imperial success. In his Englands Exchequer, or a Discourse of the Sea and Navigation (1625), John Hagthorpe stressed how "the glory and sovereignty at sea, hath at this day three Competitors, the English, Dutch, and Spanish Nations, between whom, though there were no open hostilitie, yet is there a politique secret warre, by striving to undermine and beat each other, out of their trade" (Hagthorpe 1625, p. 7). To fight this 'secret war', he added, "it seemes not onely profitable, but right necessarie to continue the East Indian trade, & if it be possible to settle also a West Indian like the Hollanders have begunne" (p. 20). Dutch expansion in the colonial trade, in other words, had "imposed upon us a meere necessity of doing something in the same kinde" (p. 21).

More often, however, the herring fisheries and the colonies were treated as complementary foundations of a solid maritime empire. Moreover, concern with the North Sea fishing trade persisted even as England expanded its overseas colonial empire. Two decades later, on the eve of the Civil Wars, Henry Robinson's Englands Safety, in Trades Encrease (1641) once again exhorted his English compatriots to "show our selves sole Soveraigne of the Sea, and with our Trident Scepter give laws […] to all nations" (Robinson 1641, p. 2). After claiming the English fisheries to be "a treasure equall to that of both Indies," he concluded in capital letters: "unless the fishing imployment and East India traffique be followed and enlarged, other Nations will gaine upon us, our trade infallibly decline daily, and the whole State with the same speede and paces post on to poverty and utter ruine" (p. 49). The image of a thriving maritime empire firmly grounded in the British Isles and their fishing trade remained powerful among the English merchants.

The importance of the fisheries dispute in Anglo-Dutch relations during the early decades of the seventeenth century determined the context for the contemporary reception of Mare Liberum in England. Grotius was read and appropriated by English merchants, but not as the authoritative source on maritime and international law that later generations would see.

On the contrary, his tract was interpreted as a partisan piece at the service of specific political and ideological agendas. As the English hesitated between the fishing industry and the colonial trade as the core of their imperial project, Mare Liberum became entangled in arguments about rights of dominion over territorial seasrather than serving as a manifesto for the freedom of trade and navigation as principles inscribed in divine and natural law. To paraphrase Peter Borschberg (2005, p. 47), the fate of Mare Liberum in the hands of English merchants makes it clear that free trade and politics have never been neatly separated in practice, reinforcing how "issues of trade and commerce have been, and may always remain, inseparably coupled with the tenets of political expediency".

The title is usually translated in English as Commentary on the Law of Prize and Booty. Even if the work has been commonly known by this name since its reappearance in the 19 th century, Grotius himself referred to the manuscript as De Rebus Indicis[START_REF] Armitage | Introduction[END_REF]).

Grotius himself had never been concerned with preserving anonymity, a precaution due to the publishing house(van Ittersum 2006, pp. 341-2). In 1614, a new edition of Mare Liberum was published already indicating his authorship.

This translation, possibly commissioned by the English East India Company itself, only survived as a manuscript until very recently, when it became the basis for the Liberty Fund edition of The Free Sea[START_REF] Armitage | Introduction[END_REF]).


Kusol Pimapunsri
email: kusol.p@eng.kmutnb.ac.th

Darawan Weeranant

Andreas Riel

GENETIC ALGORITHMS FOR THE RESOURCE-CONSTRAINED PROJECT SCHEDULING PROBLEM IN AIRCRAFT HEAVY MAINTENANCE

Keywords: Genetic algorithm, RCPSP, aircraft maintenance

Due to complex sets of interrelated activities in aircraft heavy maintenance (AHM), many airlines have to deal with substantial aircraft maintenance downtime. The scheduling problem in AHM is regarded as an NP-hard problem. Using exact algorithms can be time-consuming or even infeasible. This article proposes genetic algorithms for solving the resource-constrained project scheduling problem (RCPSP) in AHM. The objective of the study was to minimise the makespan of the maintenance plan. The proposed algorithms applied five heuristic dispatching rules to generate an initial population based on activity list formation. Resource allocation methods for RCPSPearliest start time (EST) and workgroup and earliest start time (WEST) -were used to evaluate the fitness value. The elitist and roulette wheel methods were applied in the selection process. The sequences of the activity lists were then iteratively improved by crossover and mutation operations. The results show that the proposed algorithms perform efficiently compared to the existing solutions in terms of computational time and resource allocation.

Introduction

The demand for commercial air transport has increased over the last few decades due to short travel times and affordable prices. The annual growth in passenger traffic from 2015 to 2034 is estimated to be 4.6-4.9% [START_REF] Schmidt | A review of aircraft turnaround operations and simulations[END_REF]. This rapid growth has imposed many challenges for planning, scheduling, and operations in the aviation industry.

Aircraft maintenance, repair, and overhaul (MRO) activities are critical for aircraft safety, and periodic maintenance checks need to be carried out on each aircraft after a specified number of flying hours. The cost of MRO is about 9% of the annual operating costs of airlines, and it is the third highest after fuel and labour costs [START_REF] Qin | A mathematical model and algorithms for the aircraft hangar maintenance scheduling problem[END_REF]. Furthermore, airlines cannot generate income during maintenance periods. Therefore, they wish to minimise maintenance lead-time while assuring that all operations are in compliance with the quality standards and regulatory requirements of Airworthiness Authorities, such as the Federal Aviation Administration (FAA), European Aviation Safety Agency (EASA), and International Civil Aviation Organization (ICAO).

Aircraft maintenance scheduling is a complicated task involving the synthesis of a range of economic, political, legal and technical factors. The demands on service, aircraft use and operational costs of the aircraft are the principal drivers of this problem [START_REF] Sriram | An optimization model for aircraft maintenance scheduling and re-assignment[END_REF]. In particular, aircraft heavy maintenance (AHM), also called a heavy maintenance visit, makes the maintenance scheduling problem even more difficult to solve. AHM consists of complex sets of interrelated activities that must be performed within a given period of time. They usually require keeping the aircraft in a hangar for several weeks to carry out inspection and maintenance activities that are compliant with specific maintenance programmes. Due to the enormous number of interrelated activities in AHM, it is challenging for the planner to minimise the makespan of the maintenance plan with scarce resources, while ensuring that the aircraft is delivered on time or with the shortest possible delay. This problem, the so-called resource-constrained project scheduling problem (RCPSP), is regarded as a nondeterministic polynomial (NP)-hard problem in AHM. Using exact algorithms for solving it leads to extremely long computation times for obtaining the optimal solution, and may even be intractable.

In this paper, we present genetic algorithms for solving the RCPSP in AHM. The scope of the problem we dealt with in this study was a heavy maintenance visit (D check) of an aircraft cabin, which is composed of five main workgroups, i.e. cockpit, door, galley, interior, and lavatory. Only scheduled maintenance tasks were considered, while unexpected or unscheduled maintenance requirements were not taken into account. The objective of this study was to develop genetic algorithms to evaluate solutions with a minimum makespan of the maintenance plan in reasonable computational time.

Aircraft Heavy Maintenance (AHM)

Civil aircraft maintenance can be subdivided into four major types of investigations, i.e. from A check through to D check. The type of check depends on the number of hours the aircraft has flown since its last check, the age of the aircraft, and the number of take-off and landing cycles it carried out. A and B checks are lighter checks, while C and D checks are considered heavier checks which must be performed inside a hangar using specialized equipment and highly trained personnel [START_REF] Friend | Aircraft Maintenance Management[END_REF]. The C and D checks are considered heavy maintenance. The C check typically occurs every two years and requires 10,000-30,000 man-hours (2 to 4 weeks) to inspect and exhaustively overhaul the entire aircraft. The D check, the most comprehensive and occurring approximately every six years, involves taking the entire airplane apart for inspection and overhaul. It needs up to 50,000 man-hours to complete (UK Department for Business, Innovation and Skills, 2016). However, it may require a different number of resources depending on the size and complexity of the aircraft, the guidelines of aircraft manufacturers, and the requirements of the airline. Our previous study [START_REF] Weeranant | Application of DSM and simulation technique to improve aircraft maintenance master plan[END_REF] examined the complexity of AHM by applying a design structure matrix (DSM; [START_REF] Yassine | An introduction to modeling and analyzing complex product development processes using the design structure matrix (DSM) method[END_REF][START_REF] Eppinger | Design Structure Matrix Methods and Applications[END_REF] and simulation technique to ascertain the proper sequence for the aircraft maintenance plan.

The Resource-Constrained Project Scheduling Problem

The RCPSP is a classical problem that has received the attention of many researchers for several decades. The objective of RCPSP is to properly schedule dependent activities over time such that the makespan of the project is minimised while precedence and resource constraints are met [START_REF] Wang | Optimization and simulation of resource constrained scheduling problem using genetic algorithm[END_REF][START_REF] Kolisch | Heuristic algorithms for solving the resource-constrained project scheduling problem: Classification and computational analysis[END_REF]. The project scheduling problem is identified as determining the time required to implement the activities of a project to achieve a certain objective [START_REF] Habibi | Resourceconstrained project scheduling problem: review of past and recent developments[END_REF].

Heuristic scheduling is one of the traditional methods for seeking solutions close to the optimal with acceptable computational cost and usually requiring less time. The heuristics are often defined as scheduling rules with dispatch rules. Some heuristic algorithms [START_REF] Tormos | A competitive heuristic solution technique for resource-constrained project scheduling[END_REF][START_REF] Sriram | An optimization model for aircraft maintenance scheduling and re-assignment[END_REF][START_REF] Rosales | Analysing uncertainty and delays in aircraft heavy maintenance[END_REF][START_REF] Kolisch | Heuristic algorithms for solving the resource-constrained project scheduling problem: Classification and computational analysis[END_REF] have been developed to solve the RCPSP, as well as a simulation approach [START_REF] Pimapunsri | Solving complexity and resource-constrained project scheduling problem in aircraft heavy maintenance[END_REF] for examining uncertainty in activities and delays in AHM. However, there are still no promising methods that guarantee optimal solutions and computational feasibility. Although the heuristic methods provide good solutions in reasonable computation time, they are typically limited to a specific set of constraints or problem formulation. In addition, development of new heuristics is difficult.

In general, a large-scale scheduling problem is NP-hard. Algorithms exist for exactly solving some forms of the problem, but they typically take too long when the size of the problem grows or additional constraints are added. As a result, most research has been devoted to either simplifying the scheduling problem to the point where some algorithms can find solutions, or to devising efficient heuristics for finding good solutions.

Genetic Algorithms

Genetic algorithms (GAs) form a subset of stochastic search optimisation methods introduced in the mid-1970s by Holland [START_REF] Holland | Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence[END_REF]. Based on the principles of natural evolutionary processes, GAs create new solutions by employing heuristic search methods, such as selection, crossover, and mutation, to find better solutions [START_REF] Goldberg | Genetic Algorithms in Search, Optimization, and Machine Learning[END_REF][START_REF] Michalewicz | Genetic Algorithms + Data Structures = Evolution Programs[END_REF]. GAs are widely used in various engineering applications, e.g. production scheduling [START_REF] Goncalves | A hybrid genetic algorithm for the job shop scheduling problem[END_REF][START_REF] Goncalves | A genetic algorithm for the resource constrained multi-project scheduling problem[END_REF] and design optimisation [START_REF] Gokdag | Structural damage detection using modal parameters and particle swarm optimization[END_REF][START_REF] Kiani | A comparative study of nontraditional methods for vehicle crashworthiness and NVH optimization[END_REF], and in solving RCPSP [START_REF] Wall | A genetic algorithm for resource-constrained scheduling[END_REF][START_REF] Franco | A genetic algorithm for the resource constrained project scheduling problem (RSPSP)[END_REF][START_REF] Kadam | Solving resourceconstrained project scheduling problem by genetic algorithm[END_REF]. RCPSP consists of several candidate solutions. In GAs, each solution is called an individual (chromosome) which has a set of properties (its genes) that can be mutated to generate the next-generation population of solutions. A fitness function is used in the evaluation to select the best individuals. As the GA is a heuristic search method based on the survival of the fittest on optimisation, it is an effective method for solving RCPSP. The procedures and the pseudo-code of the proposed algorithm are shown in Figures 1 and2.

Research Methodology Chromosome Structure

Usually, several hundred activities have to be completed to accomplish the heavy maintenance of an aircraft cabin. In this study, we defined an activity list as a chromosome. A chromosome is composed of genes; the number given to each gene corresponds to the ID number and its information (the duration and resources required). Figure 3 provides an example of a network of ten activities. This network can create 10! chromosomes, without taking into account precedence constraints. For example, a chromosome of this network could be [5-6-4-10-9-1-3-7-2-8].

Initial Population

The population of solutions is based on activity list formation. An activity list is composed of activities according to their precedence in the relationship. A highly constrained scheduling problem regularly has a small feasible search space. Purely random generation of activity sequences results in a large number of infeasible solutions. In this study, we used a permutation-based simulation procedure to produce an initial population of precedence-feasible chromosomes [START_REF] Hartmann | Project scheduling with multiple modes: A comparison of exact algorithms[END_REF]. The permutation encoding method creates an initial population based on its size. The objective of this step is to determine a set of activity lists (initial population) by using five dispatching rules: SPT (shortest processing time); LPT (longest processing time); critical path and SPT; critical path and LPT; and random rules. The number of activity lists generated depends on the required population size. For example, if the population size is set to 10, this method creates two equal chromosomes (activity lists) for each dispatching rule. However, only chromosomes that meet precedence constraints are considered. Otherwise, they need to be repaired before being evaluated. According to the chromosome, [5-6-4-10-9-1-3-7-2-8] in the previous step is not a proper sequence for the precedence constraints in Figure 3. Then, a repair process is carried out by exchanging the position of activities (genes) of the same priority and a new chromosome is formed in the population, for example [5-6-4-2-8-1-7-9-3-10].

Evaluation of Fitness Value

In each generation, all chromosomes in the population are evaluated by the fitness function. The chromosomes with better fitness values are included in the mating pool to form new offspring (child chromosomes). The fitness function (Fi) of the proposed GA is defined in Equation ( 1). The fitness value represents the makespan (C max ). The objective function is to minimise the makespan. Then, the chromosome that has the lowest value is considered the best current solution. Selection of chromosomes with better fitness value enables the GA to reach solutions in a shorter period of time.

F i = Minimize C max (1)
A chromosome is an activity list formation which is a representation of the schedule. In this step, RCPSP is used to schedule the resource demand of each activity for the sequences and resource allocation methods. To evaluate the fitness value of each chromosome, we applied the earliest start time (EST) algorithm since it is simple and efficient and is widely used to minimise the makespan. Furthermore, the workgroup of the activity was considered for scheduling resources. Therefore, the workgroup and earliest start time (WEST) algorithm was also applied to evaluate the fitness value in this study.

EST is one of the greedy algorithms that are used to solve scheduling problems. According to the sequences of each activity list (chromosome) generated from the previous step, EST first considers the resource which prompts the activity. In case the number of available resources is higher than demand, EST takes the resource prior to the lowest identification number. This process is repeated until all the activities are accomplished, then the duration (makespan) of each chromosome is calculated. The schedule of the chromosome [5-6-4-2-8-1-7-9-3-10] can be represented as in Figure 4.

The WEST algorithm, according to EST, takes into account the workgroup to which the activity under consideration belongs. Aircraft maintenance (assembly and disassembly) activities are mostly operated in the hangar. Therefore, the resources required are moved to the hangar. When the activity considered is in the same workgroup and requires the same resource as the previous activity, the resource that operates the previous activity is assigned first. The reason is to reduce the moving time of resources and equipment between workgroups that are usually in different locations. For example, the schedule of the same chromosome [5-6-4-2-8-1-7-9-3-10] would be represented in a different way as shown in Figure 5.

Selection Operation

The objective of this step is to select parent chromosomes (sequences of activity lists) obtained from the previous step. This study applied two selection methods: called elitist and roulette wheel [START_REF] Pasala | A study of roulette wheel and elite selection on GA to solve job shop scheduling[END_REF]. The elitist method selects the sequence of activity lists that offers the minimum fitness value, while the roulette wheel method is a probabilistic selection. The reciprocal of the fitness value and the probability of each sequence of activity lists are calculated by Equations ( 2) and (3), respectively. Then, the parent sequences of activity lists are selected based on the probability interval of each sequence of activity lists by using random numbers. This method uses the strength of each chromosome to create area proportions (probability) on the wheel. Let h be the sequence index of the activity list starting from 1 to Ps (population size) and Rh and Ph be the reciprocal of the fitness value and the probability of sequence h:

R h = 1 F h
(2)

P h = R h ∑ R h Ps h=1
(3)

Crossover Operation

Two individuals, denoted as parent chromosomes, are selected from the previous step. To produce a new generation, their genes are exchanged in a certain order to obtain offspring (new child) chromosomes. There are several crossover operators in the literature for creating new child chromosomes. In this study, we applied two well-known crossover operators, i.e. partially mapped crossover (PMX) and position-based crossover (PBX), since they always offer good offspring for permutation encoding [START_REF] Sukkerd | Hybrid genetic algorithm and tabu search for finite capacity material requirement planning system in flexible flow shop with assembly operations[END_REF]. Note that the crossover process is allowed when a random number is less than the crossover probability (Pc).

PMX is one of the most popular and effective crossovers for order-based GA to deal with combinatorial optimisation problems [START_REF] Ting | Multi-parent extension of partially mapped crossover for combinatorial optimization problems[END_REF]. Duplicate numbers in the offspring chromosomes are not allowed. To address the issue of the legality of an order, PMX uses a mapping relationship to legalise the offspring that have duplicate numbers. Figure 6 illustrates how PMX legalises offspring. First, two points on Parent 1 and Parent 2 that will create three substrings are marked as shown in Step-1. Substring 2 (4-2-8-1-7) of Parent 1 and (6-2-5-3-7) of Parent 2 are then exchanged to produce proto-offspring in Step-2. However, this results in duplication of the genes (5, 6, 3) that appear twice in proto-offspring 1 as well as the genes (1, 4, 8) in proto-offspring 2. To fix the illegal offspring, Step-3 partially maps the genes in substring 2, e.g. '6' to '4', '5' to '8', and '3' to '1'. Then, the duplicated genes in substring 1 and substring 3 are replaced with corresponding genes in the mapping relationship as shown in Step-4.

PBX was proposed by [START_REF] Syswerda | Schedule optimization using genetic algorithms[END_REF]. First, a random set of positions is selected in the parent strings, for example, a pair of parents as shown in Figure 7. Suppose that the second, fifth and eighth positions are selected as shown in Step-1. The first offspring's positions are filled by selecting the same positions of Parent 1. Then deselected positions of Parent 2 are taken in order from left to right without duplicate number. The same procedure is applied to the second offspring. The results are shown in Step-2.

Mutation Operation

The mutation operator is applied to prevent the offspring sequences from falling into a local optimum area. Two mutation methods, SWAP and INSERT, were applied in this study since they are very efficient for permutation encoding. These random operators enable individuals to make a new combination. Similar to the crossover process, the mutation process is allowed when a random number is less than the mutation probability (Pm; [START_REF] Chen | Effect of solution representations on Tabu search in scheduling applications[END_REF]. However, similar to the initial step, the sequences of the new combination must respect the precedence constraints. Otherwise, they need to be repaired before the fitness value is evaluated. Then, the existing population is replaced by a new generation obtained by this step. This is followed by checking the stopping criterion. If it is not satisfied, a new loop from the selection operation process is started to find a new generation of the population. For example, suppose that mutation probability (Pm) is 0.01 and the random numbers of the two offspring in Figure 6, [8-4-6-2-5-3-7-9-1-10] and [3-6-4-2-8-1-7-5-9-10], are 0.5 and 0.004. Since the random number of the second offspring is less than Pm, the sequence of this offspring is mutated. In the SWAP mutation method, two genes on the chromosome are selected at random and exchange

Experiment and Analysis

The experiments described in this section used the MATLAB programme on a desktop computer with an Intel® Core TM i5 Duo Processor 3 GHz and 4 GB RAM. The AHM case study was the same as in our previous work [START_REF] Weeranant | Application of DSM and simulation technique to improve aircraft maintenance master plan[END_REF]. The network information was rearranged following the precedence constraints using the design structure matrix (DSM) method [START_REF] Yassine | Complex concurrent engineering and the design structure matrix method[END_REF][START_REF] Yassine | An introduction to modeling and analyzing complex product development processes using the design structure matrix (DSM) method[END_REF]. The characteristics of the case study were as follows: the number of activities was 317; the number of workgroups was 5; and the number of resource groups was 12. This experiment was conducted based on a factorial design. The stopping criterion was computational time which was 120 minutes maximum. There were five parameters with two algorithms, in which the total runs were (5×2×3×2×3)×2 algorithms = 360 runs which required 720 h. The parameters used in this experiment are shown in Table 1.

The response variable of the experiment was the makespan obtained from each run. Table 2 illustrates the results of the minimum makespan obtained from the EST and WEST algorithms. Although both algorithms offered the same minimum makespan of 32.995 days, the EST algorithm consumed less time than the WEST algorithm in the overall results. To evaluate the performance of both algorithms in terms of computational time, the convergence curves of the EST and WEST algorithms are shown in Figures 10 and11, respectively. The best parameter setting of the algorithms that found the minimum makespan with the least time, 4.90 min for EST and 34.31 minutes for WEST, are shaded in Table 2.

Table 3 shows the results for minimum makespan and resource demand. In the case of resource allocation, the WEST algorithm required 115 units, while the EST algorithm required 130 units. Compared to the plan estimated by the priority rule-based heuristic in [START_REF] Weeranant | Application of DSM and simulation technique to improve aircraft maintenance master plan[END_REF] and an experienced planner, the makespan of the proposed algorithms was significantly improved by 25.7-29.8%, while the peak resource demand was improved by 4.4-15.4%.

Conclusions

This study proposed genetic algorithms to solve RCPSP in AHM. Heuristic dispatching rules were applied to determine a set of initial sequences of activity lists as an initial population. The fitness value was calculated by two algorithms of resource scheduling, EST and WEST, to obtain the makespan of each chromosome. The roulette wheel method was applied to select the parent chromosomes which subsequently evolved by crossover and mutation processes to produce a new generation of the population. The performance of the proposed algorithms was controlled by the parameters of GA: population size, operators and their probability of crossover, and mutation processes. The results of the experiment indicate that the proposed algorithms are more efficient than the existing plans for determining project makespan and resource allocation. Due to the probabilistic nature of GA, the proposed algorithms do not guarantee optimality. However, the results obtained from the proposed algorithms are clearly shown to be a significant improvement over master plan and priority rulebased heuristic methods. Future research will focus on other types of checks and heavy maintenance visits of other aircraft which usually have more or less distinct characteristics, e.g. number of activities, number of resources and resource groups, number of workgroups, etc. This future analysis is necessary to make the algorithm applicable to a wider range of case studies.

Figure 1 .

 1 Figure 1. Flowchart of the proposed genetic algorithm

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. An example of a network information

FigureFigure 7 .

 7 Figure 6. Example of partially mapped crossover (PMX)

Figure

  Figure 8. SWAP mutation operator

Table 1 . Parameters of the experiment

 1

	Algorithm	Ps	Cross	Pc	Mutate	Pm
	EST	5, 10, 30, 60 100	PMX PBX	0.7, 0.8 0.9	SWAP INSERT	0.05, 0.01 0.1

WEST Ps: Population size; Cross: Crossover operator; Pc: Crossover probability; Mutate: Mutation operator; Pm: Mutation probability.

Table 2 . Best parameter settings for each algorithm EST Algorithm WEST Algorithm

 2 Population size; Pc: Crossover probability; Cross: Crossover operator; Pm: Mutation probability; Mutate: Mutation operator.

	Ps	Pc	Cross	Pm	Mutate	Time	Ps	Pc	Cross	Pm	Mutate	Time
	5	0.7	PBX	0.05	SWAP	50.02	5	0.7	PMX	0.1	SWAP	88.10
	5	0.8	PBX	0.05	SWAP	33.15	5	0.9	PBX	0.1	SWAP	113.66
	5	0.9	PMX	0.05	SWAP	61.21	10	0.7	PMX	0.01	SWAP	117.68
	10	0.7	PMX	0.1	SWAP	4.90	10	0.8	PMX	0.1	SWAP	60.04
	10	0.7	PBX	0.05	SWAP	33.39	30	0.7	PMX	0.05	SWAP	78.55
	10	0.7	PBX	0.1	SWAP	43.02	30	0.7	PBX	0.1	SWAP	107.04
	10	0.9	PMX	0.05	SWAP	18.82	30	0.8	PMX	0.01	SWAP	119.74
	30	0.7	PMX	0.05	SWAP	38.90	30	0.8	PMX	0.1	SWAP	34.31
	30	0.7	PBX	0.1	SWAP	50.46
	30	0.8	PBX	0.05	SWAP	53.00
	30	0.9	PMX	0.05	SWAP	20.55
	60	0.7	PMX	0.1	SWAP	26.16
	60	0.8	PMX	0.01	SWAP	29.75
	60	0.8	PMX	0.1	INSERT	74.81
	60	0.9	PMX	0.05 INSERT	23.87
	100	0.8	PMX	0.1	INSERT	38.11
	100	0.9	PMX	0.1	INSERT	84.37
	Ps:

Table 3 . Results of makespan and resource demand analysis

 3

	Algorithm	Makespan (days)	Max. of Resource Demand (units)
	Master plan	47.000	136
	Priority rule-based heuristic	44.420	136
	Genetic algorithm: EST	32.995	130
	Genetic algorithm: WEST	32.995

Acknowledgments

This research was funded by King Mongkut's University of Technology North Bangkok (KMUTNB), contract no. KMUTNB-62-DRIVE-25. The authors are grateful to KMUTNB for the financial support received to execute this work.


M Kowalski
email: kowalski@cmi.univ-mrs.fr

B Torrésani
email: bruno.torresani@cmi.univ-mrs.fr

Structured Sparsity: from Mixed Norms to Structured Shrinkage

Sparse and structured signal expansions on dictionaries can be obtained through explicit modeling in the coefficient domain. The originality of the present contribution lies in the construction and the study of generalized shrinkage operators, whose goal is to identify structured significance maps. These generalize Group LASSO and the previously introduced Elitist LASSO by introducing more flexibility in the coefficient domain modeling. We study experimentally the performances of corresponding shrinkage operators in terms of significance map estimation in the orthogonal basis case. We also study their performance in the overcomplete situation, using iterative thresholding.

I. INTRODUCTION

Standard variational approaches to the sparse regression problem, such as LASSO/Basis Pursuit Denoising, generally rely on the implicit assumption that the regression coefficients are i.i.d.. Departure from independence can also be implemented into the problem by suitable choices of the regularization term. Examples of these are given by the joint sparsity approaches [START_REF] Fornasier | Recovery algorithm for vector-valued data with joint sparsity constraints[END_REF], the Elastic net [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF] type methods, Group LASSO [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF] or Elitist LASSO [START_REF] Kowalski | Random models for sparse signals expansion on unions of basis with application to audio signals[END_REF].

We study in this paper some generalizations of the mixednorm based dependent coefficient sparse regression methods, designed to overcome some limitations of some of G-LASSO and E-LASSO.

In the latter, described in Section II (together with the corresponding generalized shrinkage operators), dependences are introduced through a prior rigid hierarchy (defining groups and members). Variants were proposed in [START_REF] Kowalski | Random models for sparse signals expansion on unions of basis with application to audio signals[END_REF] that avoid such a rigidity and allow for adaptive groupings of coefficients. After recalling in Section III the construction of these variants (termed Windowed Group LASSO, and persistent Elitist LASSO), in a somewhat simplified way, we describe corresponding iterative shrinkage algorithms. We describe a numerical simulation setup designed to study experimentally their performances in regression problems. Numerical results, displayed and discussed in Section IV, show that even though the proposed methods do not compare very well with more classical LASSO in terms of SNR, they drastically outperform the latter in terms of significance map estimation.

II. STRUCTURED REGRESSION USING MIXED NORMS

We start here from the popular regression problem known as the Basis Pursuit Denoising (BPDN for short) [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF], or LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]. Given an observation y ∈ C M , BPDN aims to identify a sparse expansion into an overcomplete dictionary {ϕ k } N k=1 (with M ≤ N ), by minimizing the functional

Ψ(x) = y - k x k ϕ k 2 2 + λ x 1 . (1)
The ℓ 1 penalty on the synthesis coefficients x k yields sparsity. In a Bayesian interpretation, this corresponds to a prior with i.i.d. synthesis coefficients, distributed according to a Laplacian distribution. This remark is well illustrated when the dictionary is an orthonormal basis: denoting by y k = y, ϕ k the analysis coefficients, the minimizer of the functional is obtained through the well known soft-thresholding operator, in which all analysis coefficients y k are compared to a global threshold λ:

x k = arg(y k ) |y k | -λ + . (2)
In many cases, the synthesis coefficients can be labelled with two indices. Various example of such two levels indexing can be naturally found in signal processing, for example in time-frequency or time-scale decomposition, or multichannel signals. Using these two indices, a hierarchy may be introduced in the synthesis coefficients, in such a way that they can be gathered into groups. Such a hierarchy can be exploited by a mixed norm penalty [START_REF] Kowalski | Sparse regression using mixed norms[END_REF], [START_REF] Kowalski | Sparsity and persistence: mixed norms provide simple signals models with dependent coefficients[END_REF], the functional to minimize taking the form:

Ψ(x) = y - g m x g,m ϕ g,m 2 2 + λ g m |x g,m | p q/p = y - g m x g,m ϕ g,m 2 2 + λ x q p,q . (3)
The index g is the "group" index, and m is the "member" index. The relationship between a hierarchical model and mixed norms is well illustrated in [START_REF] Szafranski | Hierarchical penalization[END_REF]. For simplicity, let us shall limit ourselves to the cases p, q ∈ {1, 2}. When the atoms ϕ g,m are orthonormal, the minimizer of the functional is obtained through "generalized" soft thresholding operators. Before recalling them, let us remark that generalized soft thresholding (with threshold τ g,m ) can also be written in the general form (supposing that y g,m = 0)

x g,m = arg(y g,m ) |y g,m | -τ g,m + = y g,m 1 - τ g,m |y g,m | + = y g,m 1 -ν g,m (y) + . (4)
In the following, we will use the notation y g = {y g,1 , . . . , y g,m , . . .} to represent the vector of coefficients that belong to the group g. If the ϕ k are orthonormal, the minimizers of Ψ are given by the following shrinkage coefficients ν g,m

• if p = 2 and q = 1, ν g,m (y) = λ y g 2 . ( 5
)
• If p = 1 and q = 2, denoting by {y g,m } the coefficients |y g,m | ordered by descending order within each group g, and choosing M g so that

yg,Mg+1 ≤ λ Mg+1 m=1 (y g,m -yg,Mg+1 ) and yg,Mg > λ Mg(λ) m=1 (y g,m -yg,Mg ) ,
the shrinkage coefficient is given by

ν g,m (y) = λ 1 + M g λ yg,m:Mg 1 |y g,m | . (6)
Actually, these shrinkage operators correspond to the proximity operators, used by Combettes et al. [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF], associated with the mixed norms under consideration.

The regression problem with ℓ 2,1 penalty is known as the Group-LASSO (G-LASSO) regression, or regression with multiple measurement vectors. Observing the shrinkage coefficient in [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF], one can see that the selected groups are the ones with the biggest ℓ 2 norms. Then, when using this penalty, one keeps entire groups of coefficients, namely the most energetic groups. At the opposite, operator (6) selects coefficients with the biggest modulus within each group. Therefore, the regression based upon the ℓ 1,2 penalty was called the Elitist-LASSO (E-LASSO) regression to illustrate the fact that only the "best" coefficients are chosen for each group.

These operators are used to minimise the convex, but nondifferentiable, functional (3), when the atoms ϕ g,m form an overcomplete dictionary, inside iterative thresholding algorithms such as the thresholded Landweber iteration [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF] or proximal algorithms [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF].

III. SELECTION BY SHRINKAGE OPERATORS

The mixed norms allow one to introduce structure (organized in terms of groups and members) in regression problems, however groups are defined once for all. One main shortcoming is the independence of the groups: a given coefficient cannot belong to two different groups. However, in some tasks, one cannot design such independent groups, but would like to gather a coefficient and its neighborhood. Conversely, even when independent groups can be constructed, one may want to introduce persistence accross the groups. Motivated by these remarks, we present generalizations of the two above defined operators (G-LASSO and E-LASSO). We introduce two new shrinkage operators constructed directly on the analysis coefficients with respect to an orthonormal basis. The shrinkage will be defined by the corresponding ν g,m (y) in equation [START_REF] Kowalski | Random models for sparse signals expansion on unions of basis with application to audio signals[END_REF].

A. Windowed Group-LASSO

In G-LASSO the hierarchy is fixed once for all, and a given member cannot belong to several groups. To relax this constraint we use here a single indice k to label the analysis coefficients y k . Then we associate with any index k a family of neighborhood indices g(k) = {m ; m ∈ Neighborood of k}, and use the G-LASSO shrinkage [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF]. While the neighborhood system still has to be defined in advance, a given index can now belong to the neighborhood of several other indices. Figure 1 gives an illustration of such a neighborhood.

The corresponding shrinkage coefficient (WG-LASSO) takes the form

ν k (y) = λ m∈g(k) |y m | 2 = λ y g(k) 2 . (7)
When the neighborhoods are disjoint (ie. there is no overlap between the groups), WG-LASSO reduces to G-LASSO. We stress that in the iteration, the shrinkage operates only on the original analysis coefficients {y k } to give the corresponding synthesis coefficients {x k }.

B. Persistent Elitist-LASSO

The Persistent Elitist-LASSO (PE-LASSO) is constructed in order to introduce persistence between groups inside E-LASSO. In this case, neighborhood systems have to be defined as in section II. Then, for a given group g, we denote by N (g) the index set of the groups which are defined as close to this group g. Then, we define the following coefficients

z g,m =   g ′ ∈N (g) |y g ′ ,m | 2   1/2 .
For each member of a group, we consider the energy of this member and its neighbors in the close groups N (g). Such a coupling is illustrated on Figure 2. Then, the PE-LASSO shrinkage is defined by (ž g,mžg,Mg ) , the {ž g,m } being the coefficients z g,m ordered by descending order within each group g.

ν g,m (y) = λ 1 + λM g žg,1:Mg 1 z g,m , (8)
Here again, if the groups are independent, the PE-LASSO shrinkage coincides with the E-LASSO shrinkage. Moreover, this operator is also a generalization of WG-LASSO: if the neighborhood are only between the groups for a given member, then M g = 0 and by setting žg,1:Mg 1 = 1, the two shrinkage operators are the same. We stress again that, as for the WG-LASSO, the shrinkages operate only on the analysis coefficients.

Remark 1: The shrinkage ( 8) is in fact related to the proximity operator of the mixed norm defined on triply labelled coefficients x i,j,k (i.e. a two levels hierarchy -see [START_REF] Gramfort | Improving m/eeg source localization with an inter-condition sparse prior[END_REF]):

x 2 2,1,2 = i   j k |x i,j,k | 2   2 ,
and can be viewed as a version of this operator with overlaped groups.

C. Algorithms and simulations

The above defined shrinkage operators have been used heuristically in iterative thresholding strategies. It is worth noticing that unlike G-LASSO and E-LASSO, for which convergence to a fixed point can be proven, the shrinkage operations in [START_REF] Kowalski | Sparse regression using mixed norms[END_REF] and ( 8) are not associated with a simple (convex) functional such as (3). The groups are defined by the black rectangle. Then, for each member of a group, we are considering the left neighbor ant the right neighbor. We compute then the energy of each red groups, and we apply a "E-LASSO like" shrinkage on these coefficients. We do the same for the next coefficients ans its neighborhood defined by blue groups.

Denoting by y → S(y) the shrinkage operator defined coordinatewise in the previous section, the iterative algorithm we use is the following Algorithm 1:

• Let x (0) = 0, γ < 1 Φ * Φ and t max ∈ N. • For t = 0 to t max x (t+1/2) = x (t) + γΦ * (y -Φx (t) ) x (t+1) = S(x (t+1/2) )

End For

One can remark that while the shrinkage operator S is the (proximity) operator ( 5) or (6) defined in section II, algorithm 1 is the iterative thresholded Landweber algorithm [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF] applied to mixed norms [START_REF] Kowalski | Sparsity and persistence: mixed norms provide simple signals models with dependent coefficients[END_REF], or proximal algorithm [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF], and converges to a solution of (3). However, with the shrinkage operators ( 7) and ( 8), nothing is known about the convergence of this algorithm: neither its convergence, nor its limit when it converges. But, as we will see in section IV-B, the numerical results we obtain are nevertheless quite interesting.

In the numerical experiments presented below, we use a time-frequency dictionary {ϕ (t,f ) }, we consider signals of the form y =

(t,f )∈∆ x (t,f ) ϕ (t,f ) + b ,
where b is an additive Gaussian noise, and ∆ is the (structured sparse) significance map. The latter is generated using fixed frequency Markov chains as introduced in [START_REF] Molla | An hybrid audio scheme using hidden Markov models of waveforms[END_REF], and the synthesis coefficients x (t,f ) , (t, f ) ∈ ∆ are generated from a standard normal distribution. An example of such a map is displayed in Figure 3. b is a white Gaussian noise (tuned so as to obtain a signal to noise ratio of about 5 -6 dB). Used as such, the shrinkage operators introduced above turn out to perform quite poorly for denoising task compared to usual soft thresholding, except in some very specific situations. However, they also turn out to outperform very significantly standard soft thresholding if one limits oneself to another problem, namely the significance map estimation. We analyze below the results for this problem in terms of type one and two errors:

Type 1:

π 1 = P{(t, f ) / ∈ ∆ | (t, f ) ∈ ∆} ; Type 2: π 2 = P{(t, f ) ∈ ∆ | (t, f ) /
∈ ∆}. with ∆ the true significance map, and ∆ the estimated one.

IV. EXPERIMENTAL RESULTS

A. The orthogonal basis case

First, assume that {ϕ (t,f ) } is an orthogonal basis such as a MDCT (Modified Discret Cosine Transform) basis. The group index is defined as the frequency index, and the member index is therefore the time index. We compared systematically all the previous shrinkage operators, with various values of the parameter λ (the bigger of λ, the sparser the ∆). The time-persistent groups were constructed as follows: N (g) = {gp, gp + 1, . . . , g, g + 1, . . . , g + p} with p = 1 or 2. We used the same neighborhood system for WG-LASSO. The results are shown in figure 4, where type one and two errors are represented as functions of the estimated significance map size. PE-LASSO and WG-LASSO obviously outperform LASSO and E-LASSO, in particular for type 1 error, but also for type 2 error. The PE-LASSO performs better than WG-LASSO. In addition, results are better for long persistence (p = 2) than short persistence (p = 1) as could be expected. All this clearly shows that taking into account explicitely the persistence properties of significance maps improves significantly their estimation.

B. The frame case

We also used the shrinkage operators inside algorithm 1. In this experiment, the dictionary {ϕ (t,f ) } is a Gabor frame, and the significance map is generated as before. Results shown in Figure 5 are qualitatively similar to the previous ones. Regarding the convergence of the algorithm, we observed that the thresholded Landweber iteration seems to "oscillate" after a number of iterations for both PE-LASSO and WG-LASSO. However, this is not sufficient to undermine once for all the convergence of algorithm 1. However, both outperform significantly simple shrinkage strategies.

V. CONCLUSIONS, PERSPECTIVES

The iterative shrinkage schemes described in this paper provide a flexible way for performing sparse approximation while maintaining some dependence relationships between the regression coefficients.

Numerical results show that such approaches perform extremely well (in comparison with more standard sparse regression) if one limits oneself to the estimation of significance maps, i.e. the locations of significant coefficients. This suggests to develop a different, "two-stage" regression approach, in which significance map estimation would be followed by a more standard regression, limited to the identified atoms. This new approach will be discussed in a forthcoming contribution. The convergence of algorithm 1 with shrinkage operations such as the WG-LASSO or PE-LASSO ones should also be studied. The difficulty comes from that these shrinkages are not the solution of the variational equation associated with the minimization of a simple functional.

Fig. 1 .

 1 Fig.1. Windowed Group-LASSO: two overlaping groups. The neighborhood of the coefficient k 1 is given by the red window, and the neighborhood of the coefficient k 2 by the blue one. These two neighbordhoods share one coefficient.

Fig. 2 .

 2 Fig.2. Persistent Elitist-LASSO: persitence is introduced between the groups. The groups are defined by the black rectangle. Then, for each member of a group, we are considering the left neighbor ant the right neighbor. We compute then the energy of each red groups, and we apply a "E-LASSO like" shrinkage on these coefficients. We do the same for the next coefficients ans its neighborhood defined by blue groups.

Fig. 3 .

 3 Fig. 3. A structured sparse significance map generated with fixed frequency Markov chains.

Fig. 4 .

 4 Fig. 4. Top: type 1 error. Bottom: type 2 error.

Fig. 5 .

 5 Fig. 5. Top: type 1 error. Bottom: type 2 error.