Caroline Guibet Lafaye

Global distributions of world resources 1

Article 25: Everyone has the right to a standard of living adequate for the health and well-being of himself and of his family, including food, clothing, housing and medical care.

1-A global difference principle

1-1 An original position global or a global original position

Firstly it is significant that the grounds on which Rawls asserts that a fair equality of opportunity and the difference principle constitute requirements of justice militate against confining these requirements within national borders. An institutional scheme, which preserves a distribution of basic rights, resources and index goods, if constructed along Rawlsian lines, would be developed through a single unified original position global in scope. As we end up, in the framework of a closed society, with a domestic difference principle as a result of the original position, if the framework considered is the whole humanity we should encounter a global difference principle. In the Rawlsian original position global, the partiesrepresenting people of various societies -will, in a first stage (R 1 ) and through an implementation of the maximin, choose a criterion that makes the globally least advantaged the preeminent touchstone of global justice, no matter which societies they may belong to. Global institutions will then be assessed by how well they satisfy Rawls's criterion interpreted globally (Rawls 1971: 377-379). On a second session (R 2 ), the parties are conceived as "representatives of states" (Rawls 1971: 378 and 379). But R 2 entails weaker constraints than R 1 on international socioeconomic inequalities. Then the various nations agree to a second contract, whereby every principle of distributive justice, and especially the difference principle, is decisively rejected at the global level. Therefore the difference principle is applied only at the national level, in democratic-liberal states. In the Rawlsian view a just global basic structure needs merely to ensure that no states are too poor to be able to satisfy Rawls's criterion domestically.

By contrast if we consider not an original position global where the parties represent the people of various nations -as Rawls does -but a global original position -as Pogge does -in which all the members of the mankind are represented as individuals, we end up with a strong claim to a world resources redistribution4 . In fact, some defendants of the liberal-egalitarian approach -as [START_REF] Beitz | Political Theory and International Relations[END_REF], [START_REF] Pogge | An Egalitarian Law of Peoples[END_REF][START_REF] Van Parijs | Sauver la solidarité[END_REF], [START_REF] Chauvier | Justice internationale et solidarité[END_REF] -endeavour to understand and conceive distributive justice on a planetary scale. They aim to determine a principle of global distributive justice closer to the approach developed by Rawls in his own Theory of Justice than what Rawls judges itself appropriate at the global level. Nevertheless and at this stage, the implementation of an "international original position", imagined by Rawls, instead of a "global initial situation", in which peoples would behave as representatives, not of their nations, but of any individual in the world, in other words the application of the difference principle beyond the national borders would generate a strongly levelling redistribution of the world resources.

1-2 Alternatives to the difference principle

If we consider a globalisation of the difference principle, we must examine the moral status of the natural goods (such as mineral resources, fertility, climate, etc.) within the States and in various States -because it is not enough to study the natural and social contingencies on which the international socio-economic inequalities are based or to which they are connected (Pogge 1989: §14.3). Charles Beitz for example proposes a way to use the difference principle at a global level. He suggests, deriving from the parallel between natural endowments and natural assets, a global resource redistribution principle5 . One can also consider that a global difference principle may justify not merely a general adjustment of market prices but a different specification of property rights over natural assets involving, for example, an international tax on (or international ownership and control of) natural assets.

Another solution, regarding a globalized second principle, is to conceive the terms of international cooperation so that the social inequalities due to natural contingencies (the distribution of natural assets) tend to optimize the worst representative individual share (Pogge 1989: §6.4). In this way, a globalized second principle would constrain but not preclude income inequalities that reflect biased exchanges (for example, crude oil versus cotton) arising from the morally arbitrary geographical distribution of natural assets. Finally one may however suppose that a globalized difference principle is less plausible as a constraint on inequalities based upon social contingencies than a globalized difference principle based upon natural contingencies. But in all cases, how a just global institutional scheme would regulate ownership and control over natural assets depends upon the empirical question of which institutional design would optimize the worst social position.

1-3 The difference principle as institutional criterion of justice

Nevertheless some assert that in the world as it is there are special factors, relevant on the global scale but not on the national or domestic level, that make inappropriate to apply Rawls's maximin criterion to a global basic structure6 . In fact, the Rawlsian conception centrally involves the claim that any self-contained social system satisfying the two principles is morally acceptable. Admittedly we are led to conclude that there is no feasible and morally viable avenue of institutional reform toward a global regime fairer and even if a fairer global basic structures are out of reach, it would not show that the Rawlsian criterion is inappropriate on the global plane because it still "can serve as a standard for appraising institutions and for guiding the overall direction of social change" (Rawls 1971: 263), to be carried "as far as circumstances permit" (Rawls 1971: 246). In any case, it seems quite impossible to deny that the position of its least advantaged participants is at least one important measure of the justice of an institutional scheme. All we may ask is that a conception of justice provides a criterion for assessing our global order that would allow us to choose among the feasible and morally accessible paths of institutional change and thus requires from us to improve gradually the justice of this order. However a plausible evaluation of the morally significant consequences of feasible institutional schemes must give a prominent place to the satisfaction of basic social and economic needs. And this is the case even if this claim is often opposed by the assertion that it is more important that social institutions recognize and protect basic civil and political rights and liberties than they ensure that basic social and economic needs are met.

2-The Global Resources Dividend (GRD)

Even if global justice does not imply that the substantial differences of wealth which will appear through the generations are constantly levelled by a supranational institution in charge of maximizing the economic level of the worst off (see Arnsperger and Van Parijs 2003: 83), we could seek and use a distributive model in order to equilibrate the word resources inequalities. In fact who make more extensive use of the resources of our planet (that is roughly the affluent countries) should compensate those who, involuntarily, use very little. This idea does not require that we conceive global resources as the common heritage of humankind nor that they are controlled through a global democratic process and that their full value (determined through resource auctions, for example) be divided equally among all members of the human species. In this perspective T. Pogge suggests establishing a Global Resources Dividend (GRD)7 . This GRD is a tax upon the use of natural resources and aims to a global distribution of the proceeds raised by this natural resources tax.

2-1 Pogge's proposition

The first point to considered is that the status quo can be reformed in a way that we would recognize as an important step in the direction of a just world order due to the fact that the existing radical inequality can be traced to the structure of our global economic order8 . Precisely the GRD is grounded on a kind of difference principle: "it maintains that the most acceptable global difference principle would be the one which would most improve the conditions of the world's neediest people" (Pogge 1998: 511). This reform proposal is conceived as the first step towards implementing Pogge's version of the global difference principle among states 9 .

The GRD proposal envisions that states and their governments shall not have full libertarian property rights with respect to the natural resources in their territory, but can be required to share a small part of the value of any resources they decide to use or sell. This principle supposes a right to a share of natural resources 10 . Moreover the word "dividend" suggests that the global poors own an inalienable stake in all limited natural resources. This stake confers no right to participate in decisions about whether or how natural resources are to be used and therefore does not interfere with national control over resources. But it does entitle its holder to a share of the economic benefits from the use of resources considered, if they are used.

In light of the vast extent of global poverty today, one may think that a massive GRD would be necessary to solve the problem. But this is not Pogge's opinion. Firstly the initial goal could be fully achieved by targeting a limited number of resources and pollutants whose extraction or discharge is reasonably easy to monitor or estimate, in order to ensure that every society is paying its fair share and also in order to assure everyone that this is so. Initially more may be needed, that is perhaps as much as one percent of the world's social product. If radical inequalities have once been eradicated, quite a small GRD may, in the context of a fair and open world market system, be sufficient afterwards to balance those ordinary unequal tendencies of economic markets. In this view proceeds from the GRD are to be used for raising the world's minimum standard of living, in order to ensure that all human beings will be able to meet their own basic needs 11 . The GDR funds would enable governments to eradicate poverty in their territories in whatever way is most effective: by maintaining lowerincluding negative -tax rates, higher tax exemptions and/or higher domestic spending for education, health care, microloans, infrastructure, land reform, etc., than would otherwise be possible 12 . 9 Pogge's declared intention with the GRD scheme is "to make an institutional proposal that virtually any plausible egalitarian conception of global justice would judge to be at least a step in the right direction" (Pogge, 1994: 199), i.e., a step in the direction of his version of the difference principle among states (cf. [START_REF] Pogge | Realizing Rawls[END_REF]. 10 As Jean-Christophe Merle underlines it "the notion of a right to share in all natural resources is accepted both by Pogge and by Rawls, but it can be interpreted differently. Pogge understands it as 'a share of the economic benefits from the use of the resource in question' (Pogge 1998: 511). He interprets it as a right to introduce a tax upon other people's use, but it could just as easily be interpreted simply as a right to full participation. The difference is significant, since the tax on other people's use refers to a product that arises through a combination of natural resources and the work of processing them" (2005: 344). But "if we assume only a right to full participation in the use of natural resources, there is no justification for implementing a different distribution of dividends, i.e., there is no occasion for the GRD scheme. The GRD is only justified if we follow Pogge and assume not just a right to equal use of natural resources, but a right to a share in anyone's use of them" (Merle 2005: 344). 11 "Proceeds from the GRD are to be used toward the emancipation of the present and future global poor: toward assuring that all have access to education, health care, means of production (land) and/or jobs to a sufficient extent to be able to meet their own basic needs with dignity and to represent their rights and interests effectively against the rest of mankind: compatriots and foreigners" (Pogge 1994: 201). Jean-Christophe Merle notes that "this definition does not go quite as far as a genuine difference principle, since it acknowledges a threshold beyond which no redistribution occurs". "It works like a difference principle until that threshold is reached" (2005: 333). 12 Nevertheless the Pogge's proposal exemplifies what J.-C. Merle calls a minimalism conception of distributive justice: "The principled minimalism of redistribution is therefore rounded out by a minimalism of means, according to which the implementation of a different distribution from the existing one would not require a central authority (see for instance [START_REF] Pogge | An Egalitarian Law of Peoples[END_REF]Pogge : 202, 224)" (2005: 330): 330). This type of minimalism -which is false according to Merle -suggests that "distributive justice does not require the intervention of a national or an international legal order. Global distributive justice may be conceivable in the absence of a world government, but it cannot be implemented without one" (Merle 2005: 330).

Finally, the GRD is supposed to satisfy two conditions which grant its implementation: a constraint of feasibility and a constraint of reality. (1) Firstly the GRD seems to be feasible: Pogge believes that this institutional alternative should be able of gaining the approval and support (especially in the wealthy countries) that is necessary to implement it. (2) Secondly the GRD should be practicable, that is able of sustaining itself (by eliciting enough cooperation and support from its participants) in the world as we know it 13 . It stays reasonably close to the current institutional arrangements. Finally it is obviously responsive to the moral concerns implicit in the maximization of the economic level of the worst off and in the concern for compensating or leveling unequal developments 14 .

2-2 Conditions of the implementation of the GDR

Nevertheless instituting a GRD would therefore require significant further institutional reforms, including central enforcement mechanisms. The Pogge's reform proposal accepts the existing state system and, in particular, leaves each national government in control of the persons and natural resources of its territory. But, according to Pogge, the entire GRD scheme would require no central bureaucracy and certainly nothing like a world government, as governments could simply transfer the GRD amounts to one another, through some facilitating organization, such as an appropriately reorganized World Bank, for example 15 . Nevertheless supranational institutions and organizations will be required to limit the sovereignty rights of states more severely than it is actually the case. In fact, the global reduction of inequalities suggested by Pogge "would have to be accompanied by modifications in the political and social system" (Merle 2005: 341).

3-Institutional and political conditions for a globalisation of equity

It seems thus that a radical redistribution of the world resources could not be implemented without a global government (a world government) or without a highly improbable consensus between states in order to apply such provisions. Just as Rawls requires domestic institutions to distribute whatever burdens remain in ways satisfying the second principle, the difference principle or any international redistributive mechanism should thus require international institutions to distribute such burdens analogously.

Currently the interpersonal redistribution capacity is located at the national level of the State. But this capacity could be transferred to a world State. In fact and in order to satisfy the requirements of distributive justice, the institution of a juridical and political community is absolutely necessary. For example, if there were a global state and if we consider with David 13 In fact its amount is rather small for the rest of us. 14 The pragmatic advantage Pogge claims for his reform proposal is that it should overcome Rawls's objections to global distributive justice, which mainly have to do with "inadequate administrative capabilities and the dangers of a world government " (1994: 199).

Copp the basic needs principle 16 , this global state would have a duty to ensure the ability of each subordinate state to enable its members to meet their basic needs 17 . We can imagine, as Copp suggests, different ways to extend the basic needs principle to the case of a global federation. (1) Firstly, we could think of the global state as having the same duty with respect to enabling people to meet their needs that the individual subordinate states would have with respect to their members, if the global state didn't exist. On this view, the subordinate states are morally transparent. The global state will deal directly with the needs of individual people, just as if the subordinate states did not exist. This can be called the "transparency view". (2) Secondly, we might instead conceive that the individual subordinate states have the primary responsibility to ensure that their residents are able to meet their needs. In this view the global state is required only to ensure that the subordinate states have sufficient resources to be able to meet this primary responsibility. This can be called the "divided responsibility view" (Copp 2004: §6, 15). In both cases, this juridical community and this world state would establish a closed juridical unity, i.e. a legal community (see Merle 1997: 237) 18 . The divided responsibility view may be the more sensible and immediate in our world, but in different circumstances it might not be. By this way, it is possible to conceive that the global state has a duty in justice to organize a fair system for the transfer of resources from the more wealthy countries to the less well-off countries to ensure that every country comes to be in favourable circumstances (Copp 2004: §7, 16). Therefore the subordinate countries have a duty to do their parts in implementing this system.

Although the idea of a distributive justice model on a global scale seems to involve a certain type of state organization, namely a centralized world government, this shows that we can imagine other model of world institution than a world government. First we have to avoid the confusion between sovereignty and centralization (Merle 1997: 231) 19 . We suppose that humanity, by becoming a single juridical community, should constitute a single state. But we can, as well, conceive a world government as a very decentralized federation. In the economic field for example, J.-C. Merle suggests considering a decentralized application of the decentralized federation model. By this way he avoids the objection of a necessary centralized authority against a world government. In a same prospect, we can conceive a world organization without a global state. The global society might be organized in a way that 15 The decisions regarding the GRD funds distribution could be made, as Pogge suggests, not only through local governments, but also -or even only -through "facilitating organization" under the supervision of "an international group of economists and international lawyers". 16 The "basic needs principle" requires a society to ensure that its members have access to high quality medical care, a sound basic education, decent housing, clean water, a nutritious diet, and so on. 17 D. Copp shows also that "in the present situation, existing states have a duty (ceteris paribus) to work to create a global state or system of institutions that could discharge the global society's duty to enable people to meet their basic needs" (2004: 2). 18 Merle suggests that the major objections with this model of world legal community is grounded on their applicability, i.e. on the possible transition starting from the current situation, and more precisely on the pragmatic and ethical requirements which this transition should respect. 19 For example, Pogge underlines that his GDR's model accepts that "sanctions could be decentralized: Once the agency facilitating the flow of GRD payments reported that a country has not met its obligations under the scheme, all other countries would be required to impose duties on imports from, and perhaps also similar levies on exports to, this country in order to raise funds equivalent to its GRD obligations plus the cost of these enforcement measures" (Pogge 1998: 510, 517f). Nevertheless J.-C. Merle's point is that "the GRD scheme makes it able of joint actions. There could be a quasi-state. Rawls, for example, takes into account the idea of a "federative union" of states (Rawls 1999: 70). Similarly, the arguments developed by D. Copp require only the viability of a global federation or system of institutions with the capability of discharging duties regarding basic needs.

The explanatory importance of social institutions has been powerfully illustrated by numerous domestic regime changes during the history of the XX th century. Institutional choices (e.g. between socialism and capitalism) can have an impact on the distribution of income and wealth, education and health care, rights and liberties, and hence quality of life. But being familiar with the idea on the national level has made it harder to appreciate its application on the global plane. Dramatic differences in standard of living across countries draw our attention to local factors (national institutions, cultures, resources, climates, environments, levels of development and so on). But any attempt to introduce a fairer economic system can be limited to a particular country, or a particular group of countries, because of the extent of the world inequalities and because of the context of globalisation. At the international level, it already exists certain institutions, as the World Trade Organization (WTO), the International Monetary Funds (IFM) and the World Bank, which could be used to promote economic equality and temperate the exclusive pursuit of productivity and efficiency by a greater economic safety. These organisations could thus adopt rules and implement policies, which support the equality while establishing, for example, a global minimum wage and global provisions for health and safety. Another idea, as we have seen, is to limit the economic authority of individual states by, for example, a GDR or by collecting a tax on natural resources globally and without the assent of affected countries, as Pogge suggests it. Indeed and although the realization of such changes seems currently remote, the whole economic activity is funded on norms which can be changed. As long as the development of administrative capacities and of a worldwide democracy is not encouraged and undertaken, we are condemned to satisfy ourselves with a low resources redistributive capacity at the planetary level.

Will be publised in J.-C. Merle (dir.), Social and Economic Justice, Dordrecht (Pays-Bas), Springer,

(à paraître). 2 caroline.guibet-lafaye@ens.fr 3 Centre Maurice Halbwachs (CNRS) -48, bd Jourdan, F-75014 Paris (France).

Rawls recognizes only persons as "self-originating sources of valid claims" (1980:

543). Then he commits himself to base his criterion of social justice exclusively upon data about individual shares.

He wants the criterion of global justice to govern not merely the terms on which states can develop and market their natural assets but also rights over these assets themselves. But Beitz's parallel between natural assets and natural endowments, for several reasons, is mistaken (seePogge 1989: 252).

Bertrand Guillarme underlines that the principle of difference is reasonable when it is applied to the internal justice of the democratic societies but it does not constitute a realistic means to solve the general problem of the adverse conditions that certain societies know (1996: 84).

Pogge uses word dividend partly in order to avoid the connotations of "tax" and "fee".

Pogge suggests that an intelligent targeting of GRD liabilities makes it possible -without major changes to our global economic system -to raise a revenue stream sufficient to eradicate world hunger within a few years[START_REF] Pogge | A Global Resources Dividend[END_REF].


Vincent Acary
email: vincent.acary@inria.fr

Paul Armand
email: paul.armand@unilim.fr

Minh Hoang

Nguyen

High-accuracy computation of rolling friction contact problems

Keywords: optimization, Jordan algebra, second-order cone optimization, Nesterov and Todd scaling, rolling friction contact

Our aim is to perform numerical solutions of an optimization model derived from the problem of unilateral contact between solid bodies with rolling friction. The model is an optimization problem with a strictly convex quadratic objective function and a second-order cone of constraints that is not self-dual. The solver is an implementation of a primal-dual interior-point algorithm with the predictor-corrector scheme of Mehrotra extended to second-order cone problem. We focused on the analysis of the limits of numerical computation and propose some treatments to achieve optimal solutions with ten significant digits precision.

I. INTRODUCTION

Acary and Bourrier presented in [START_REF] Acary | Coulomb friction with rolling resistance as a cone complementarity problem[END_REF] a general model for the problem of unilateral contact between solid bodies with rolling friction. Let n, m ∈ N be the number of contact points and degrees of freedom. Let d = 5 be the dimension of the velocity and reaction vectors at a contact point. In this paper, we consider only a convex relaxation of this model as

M v + f = H r, Hv + w = u, F * u ⊥ r ∈ F, (1)
where M ∈ R m×m is symmetric and positive-definite, f ∈ R m , H ∈ R nd×m , w ∈ R nd . The vector r ∈ R nd is the reaction force at contact, u ∈ R nd is the velocity of the contact point and v ∈ R m is the velocity of the system. The rolling friction cone is defined by

F = n i=1 Ri :=    ri =   ri 0 ri ri   ∈ R d : ri 0 ≥ max{ ri , ri }    .
The dual cone is then

F * = n i=1 R * i :=    ui =   ui 0 ūi ũi   ∈ R d : ui 0 ≥ ūi + ũi    .
This cone is not self-dual since F = F * , a significant difference with the frictional contact (no rolling) framework [START_REF] Acary | FCLIB: a collection of discrete 3D Frictional Contact problems[END_REF]. The system [START_REF] Acary | Coulomb friction with rolling resistance as a cone complementarity problem[END_REF] coincides with the first order optimality conditions of the primaldual pair of convex problems:

min v,u 1
2 v M v + f v max v,r -1 2 v M v -w r s.t. Hv + w = u, s.t. H r -M v = f, u ∈ F * , r ∈ F. (2)
After the introduction of artificial variables and a reformulation of the constraints, we will show that these problems fall in the class of second-order cone optimization (SOCO) problems. Thus, a primal-dual interior-point method (IPM) is an appropriate choice to efficiently solve them. This paper proposes an adapted IPM for the solution of [START_REF] Alizadeh | Second-order cone programming[END_REF], in order to accurately solve [START_REF] Acary | Coulomb friction with rolling resistance as a cone complementarity problem[END_REF]. We do it by means of a C/C++ implementation of the algorithm in SICONOS 1 and perform tests on FCLIB [START_REF] Acary | FCLIB: a collection of discrete 3D Frictional Contact problems[END_REF], a collection of discrete rolling frictional contact problems 2 . The paper is organized as follows. Section II shows the rolling friction contact model and the system of equations solved by means of the interior point algorithm described in Section III. The numerical difficulties are analyzed in Section IV. The robustness of our implementation is highlighted by the numerical results presented in Section V. The paper ends with a conclusion in Section VI and some basics on Euclidean Jordan algebra as well as the notation in Appendix.

II. ROLLING FRICTION CONTACT MODEL

The cone appearing in a standard SOCO problem, usually the Lorentz cone, is self-dual. This allows to take advantage of the Jordan algebra. But for the rolling friction problems, this is not the case, the cone of constraints is not self-dual. There is no Jordan product such that R * i is a cone of squares, an important property to extend IPM from linear optimization to SOCO [START_REF] Alizadeh | Second-order cone programming[END_REF]. To deal with this difficulty, we propose to transfom the conical constraints of Problem (2) into the product of Lorentz cones by means of artificial variables. Knowing that, for a triplet (a, b, c) of real numbers, a ≥ b + c if and only if there exists t, t ∈ R such that t ≥ b, t ≥ c and a = t + t . We apply this statement by setting ui,0 = ti + t i for all i ∈ {1, . . . , n}. Then, the primal-dual Problem (2) can be reformulated as

min v,z 1 2 v M v + f v max v,r -1 2 v M v -w r s.t. Hv + w = Jz, s.t. H r -M v = f z ∈ L 2n J r ∈ L 2n , (3)
where z ∈ R n×(d+1) is a vector with n components of the form (ti; ūi; t i ; ũi), i ∈ {1, . . . , n} and

J = n i=1 , with  =   1 1 I I   ∈ R d×(d+1) .
By virtue of [2, Lemma 15], the orthogonality condition of (1) becomes z • J r = 0, with z ∈ (L 2n ) * = L 2n and J r ∈ L 2n . This representation of (1) provides a square system of equalities.

It is assumed that the Slater hypothesis holds, i.e., there exists v ∈ R m such that Hv + w ∈ int(F * ). Under this assumption, (r, u) ∈ R 2nd is a primal-dual optimal solution of Problem (2) if and only if there exists (t, t ) ∈ R 2n such that the Karush-Kuhn-Tucker (KKT) are satisfied:

M v + f = H r, Hv + w = Jz, z • J r = 0 n(d+1) , (z, J r) ∈ L 4n . (4)
In practice, the Newton method is applied to a perturbation of (4), such that the solution is maintained in the interior of the cone of constraints. The perturbation is to replace the complementarity conditions by z • J r = 2µe, where the perturbation parameter µ is the duality measure defined by

µ = z J r n .
The perturbed KKT system associated to Problem (3) is then

M v + f = H r, Hv + w = Jz, z • J r = 2µe, (z, J r) ∈ int(L 4n ). (5)
It can be shown that the system (5) has a unique solution (z(µ), J r(µ)) ∈ int(L 4n ) for all µ > 0. The corresponding curve, parametrized by µ, is called the central path. It can be shown that under the Slater hypothesis, when µ ↓ 0, the central path converges to a relative interior point of the set of primal-dual solutions of (3).

III. PRIMAL-DUAL INTERIOR-POINT APPROACH

Basically, an interior point algorithm consists of successive linearizations of (5), while driving µ to zero and keeping the iterates z and J r in the interior of the Lorentz cones. The linearization of equations ( 5) leads to the following linear system

  M -H 0 -H 0 J 0 ZJ R     d v d r d z   = -   M v + f -H r Jz -Hv -w z • J r -2µe   , (6)
where Z = Arw(z) and R = Arw(J r). Unfortunately, this system can be singular, even if the matrices Z and R are positive definite, see [START_REF] Terlaky | Self-regularity: A New Paradigm for Primal-dual Interior-point Algorithms[END_REF]. This is due to the fact that the arrow matrices do not commute, except if they share a common set of eigenvectors. To overcome this drawback, the idea is to make a local change of variables, which keeps invariant the Lorentz cone as well as its interior and so that, in the new space, the arrow matrices commute. An extensive literature has discussed several choices of variable changes and it is commonly agreed that the best one for solving a SOCO problem is the Nesterov and Todd (NT) scaling scheme [START_REF] Tütüncü | Solving semidefinitequadratic-linear programs using SDPT3[END_REF]. It is defined as follows. Let p, p ∈ R nd be the NT directions defined with n conic components of the following form: for i ∈ {1, . . . , n},

pi = Q (t i ;ū i ) 1/2 Q (t i ;ū i ) 1/2 (ri 0 ; ri) -1/2 -1/2 , pi = Q (t i ;ũ i ) 1/2 Q (t i ;ũ i ) 1/2 (ri 0 ; ĩi) -1/2 -1/2 . ( 7
)
Let us define the block-diagonal matrix Qp = n i=1 (Qp i ⊕ Qp i ). The change of variables is then ẑ := Qpz and y := Q p -1 J r.

A fundamental property of the NT scaling is that ẑ = y, see [2, p.42], which implies that the corresponding arrow matrices commute.

After the change of variables, we obtain the corresponding primaldual problems

min v,ẑ 1 2 v M v + f v max v,y - 1 2 v M v -w KQp y s.t. Hv + w = JQ p -1 ẑ, s.t. H KQp y -M v = f, ẑ ∈ L 2n , y ∈ L 2n , (8)
where

K = n i=1 k, with k =   1 0 I I   ∈ R d×(d+1) .
Note that KJ = I. After some obvious simplifications, the new pertubed KKT system associated to ( 8) is given by

M v + f = H r, Hv + w = Jz, ẑ • y = 2µe, (ẑ, y) ∈ int(L 4n ). (9)
The linearization with respect to the variables v, r and z of this new system leads to the following linear system:

  M -H 0 -H 0 J 0 ẐQ p -1 J Y Qp     d v d r d z   = -   M v + f -H r Jz -Hv -w ẑ • y -2µe   , ( 10
)
where Ẑ := Arw(ẑ) = Arw(y) =: Y . By left-multiplying the third equation by Qp Ẑ-1 and after some simplications, we get the following equivalent symmetric linear system:

  M -H 0 -H 0 J 0 J Q p 2     d v d r d z   = -   M v + f -H r Jz -Hv -w y -2µz -1   . ( 11
)
Algorithm 1 is the implemented algorithm that was used to perform our numerical tests. This is the primal-dual interior point method of Mehrotra [START_REF] Mehrotra | On the implementation of a primal-dual interior point method[END_REF], extended to the solution of a SOCO problem [START_REF] Tütüncü | Solving semidefinitequadratic-linear programs using SDPT3[END_REF]. At each iteration of Algorithm 1, two linear systems are solved, with two different right-hand-sides (RHS). The first one corresponds to a pure Newton direction, called the affine scaling direction and denoted by (d v a , d r a , d z a ). It satisfies the linear system (6) (resp. [START_REF] Dolan | Benchmarking optimization software with performance profiles[END_REF]) with µ = 0. The second one corresponds to the computation of a linear combination of the affine scaling direction and a correction step, that is used to force the iterates to get close to the central path. See [5, §14.2] for details. As a result, when solving (6), the third equation is replaced by the following one:

ZJ d r + Rd z = -z • y -d z a (J d r a ) + 2σµe, (12)
where σ ∈ (0, 1) is the centralization parameter, i.e., the desired reduction factor of the duality measure. When solving [START_REF] Dolan | Benchmarking optimization software with performance profiles[END_REF], this equation becomes

J d r + Q p 2 d z = -y -Qp(ẑ -1 • ( dz a • ďy a )) + 2σµz -1 , ( 13
)
where dz a = Qpd z a and ďy a = Q p -1 J d r a . Once a linear system is solved, a line search is performed so that the next iterate remains within the cone of constraints. Let x ∈ L 4n be the current iterate and let d ∈ L 4n be the corresponding solution of the linear system that was solved. The function get_step_length(x, d) returns the largest α ≥ 0 such that x + αd ∈ L 4n . The step length along the affine scaling direction is the largest value α ∈ (0, 1] such that x + αd ∈ L 4n . This value is used to compute the centralization parameter σ. Next, the step length along the full direction is the largest value α ∈ (0, 1] such that x + αd ∈ (1 -γ)x + L 4n . See the detailed formulas for this computation in [START_REF] Tütüncü | Solving semidefinitequadratic-linear programs using SDPT3[END_REF]. At the end of the iteration, the variables are updated at Step 26.

Algorithm 1 IPM with predictor-corrector scheme 1: Choose a tolerance > 0. Set γ = 0.9.

2: Set a starting point (v, r, z) ∈ R m × R nd × R n(d+1) 3: while 1 do 4: pinfeas = Jz -Hv -w max{ Jz , Hv , w } 5: dinfeas = M v + f -H r max{ M v , f , H r } 6: µ = 1 n z J r 7:
if max{pinfeas, dinfeas, µ} ≤ then Set γ = 0.9 + 0.09α

26:

Set (v, r, z) = (v, r, z) + α(d v , d r , d z ) 27: end while IV. COMPUTATIONAL ISSUES Although the system (6) may be singular, we have performed numerical experiments with it. The singularity of the matrix is a rather rare event in practice. These first experiments serve as a kind of reference for future experiments, both in terms of number of iterations and CPU time. Since the matrix in ( 6) is not symmetric, an LU factorization is performed. The advantage of this "no-scaling" strategy is that it is simple to implement and computationally fast, because there is not a bunch of calculations due to the scaling. However, since an LU factorization is more expensive than a symmetric matrix factorization, this advantage should be lowered for large problems.

Our first experiments with the NT scaling scheme have been disastrous. This is due to the ill-conditioning of the matrix Qp near an optimal solution. This behavior is well known in the community of SOCO and previous experiments with the software SDPT3 [START_REF] Tütüncü | Solving semidefinitequadratic-linear programs using SDPT3[END_REF] warned us about it. We have been searching for a long time for a formulation of the linear system that allows us to obtain suitable results. The form (11) seems to us not too bad, but not enough robust. We propose an alternative equivalent 3 × 3 symmetric linear system, by left-multiplying the last equation of ( 11) by Q p -1 and setting dz = Qpd z :

  M -H 0 -H 0 JQ p -1 0 Q p -1 J I     d v d r dz   = -   M v + f -H r Jz -Hv -w y -2µẑ -1   . ( 14
)
The matrix of ( 14) is less ill-conditioned than the one of [START_REF] Dolan | Benchmarking optimization software with performance profiles[END_REF].

Although the ill-conditioned matrix Qp is introduced in the second equation, which leads to a loss of accuracy in the primal feasibility, the use of ( 14) allowed to solve 97% of the problems with a tolerance = 10 -10 .

Since the matrices in ( 11) and ( 14) are symmetric, an LDL factorization is done by means of the Harwell subroutine MA57 [START_REF] Duff | MA57-a code for the solution of sparse symmetric definite and indefinite systems[END_REF], an efficient code for the solution of sparse symmetric systems. Furthermore, in order to avoid the propagation of computational errors due to the ill-conditioning of Qp, all calculations of the form Qxy are performed without explicitly building the vector x and the matrix Qx. This is done, for example, to compute the vectors y and ẑ in the RHS of ( 14). We do it by using the formula Qxy = 2(x y)x -(det x)Ry and the spectral decompositions of the vectors x and y. In addition, to improve the accuracy, all these calculations are done with the floating-point data type long double instead of double. Indeed, a difficulty due to the computational limits of processors led us to use this type of data. In experiments with tight tolerances, when the algorithm returns an error, it is usually stopped by a computation involving a NaN (Not a Number). This comes from an invalid arithmetic operation, like a division by zero or the calculation of the square root of a negative number. Despite the fact that z and J r are kept in the interior of the Lorentz cones, when approaching the boundary of the constraints, the second eigenvalue (λ2) of some conic components becomes almost zero or even equal to zero. This further justifies our choice to use the data type long double. However, the use of the long double data type may produce a negative λ2. This is due to the fact that the algorithm is executed on 64-bit processors whose epsilon machine is in double precision, i.e., 16 digit precision [START_REF] Rajaraman | IEEE standard for floating point numbers[END_REF].

We also tried some experiments with a solution of the linear system in a reduced form, which might have been preferred because of its smaller dimension. By eliminating the variable d z in the system [START_REF] Dolan | Benchmarking optimization software with performance profiles[END_REF], a reduced version is given by

-M H H JQ p -2 J d v d r = M v -f -H r -Hv -w + 2µJy -1 . (15)
But, like as [START_REF] Dolan | Benchmarking optimization software with performance profiles[END_REF], the matrix of ( 15) is very ill-conditioned. Another reduction with a smaller condition number is given by

-M Ĥ Ĥ I d v d r = M v + f -H r P -1 (-Hv -w + 2µJy -1 ) , (16)
where Ĥ = P -1 H and d r = P d r . The matrix P is such that JQ p -2 J = P P . This factorization can be done in several ways, such as Cholesky, LDU, etc. Whatever the formulation used, in addition to the numerical computation problems mentioned above, we again encounter numerous computations to perform at each iteration, which leads to propagating too many errors. Any reduction to a 2 × 2 form by eliminating the variable d z always returned worse results.

V. NUMERICAL RESULTS

Introduced in [START_REF] Acary | FCLIB: a collection of discrete 3D Frictional Contact problems[END_REF], three collections of 523 rolling friction problem tests have been carried out by Algorithm 1. Detailed descriptions of these problems are given in Table I. The performance profiles were introduced by Dolan and Moré [START_REF] Dolan | Benchmarking optimization software with performance profiles[END_REF] to get the benchmarking optimization solvers through a cumulative distribution function ρs(τ ). It is the probability that the performance ratio of the solver s to the best solver does not exceed a factor 2 τ . The higher the probability is, the better the solver is. Figures 1, 2 and3 show the performance profiles of Algorithm 1, with a linear system under the form (6) "No scaling", (11) "Qp2" and ( 14) "JQinv", with tolerances = 10 -8 , = 10 -9 and = 10 -10 respectively. In Figure 1, the linear systems have been solved for the tolerance = 10 -8 . Within this tolerance, the three solvers are able to solve all problems successfully. The "no-scaling" system (6), represented by the blue curve, has shown it to be faster (better CPU time) but less efficient (larger number of iterations) than the other two. The curves in red and green, corresponding to [START_REF] Dolan | Benchmarking optimization software with performance profiles[END_REF] and (14), do not differ significantly in this assessment. However, when the tolerance is decreased to 10 -9 and 10 -10 , the difference becomes more significant between these curves. In Figure 2 and Figure 3, despite still being the fastest one, the blue curve has shown the worst efficiency as the number of favorable results is only approximately 82% and 74% of the total number of experiments. Meanwhile, the red curves in Figure 2 and Figure 3 show that (11) has started to suffer from problems of NaN, so its result has stopped at 98% and 85%. Finally, through the greens, we can see that (14) still retains its robustness. There is no failure with the tolerances 10 -8 , 10 -9 .

When the tolerance is 10 -10 , the algorithm is stopped by NaN for only 14/523 tests, thus 97% performance.

VI. CONCLUSION

The reformation of the non self-dual cone of constraints of the original problem into a self-dual one of the SOCO problem is an important contribution to the proposed interior-point algorithm. First, a scaling strategy by NT directions to produce a symmetric linear system is performed to overcome the disadvantages that come from the asymmetric linear system. Then, because of the considerable computational volume of this strategy, it is essential to choose suitable alternative formulas, for example the calculation of Qxy. This not only creates efficiency in terms of execution time, but also is to achieve better accuracy of optimal solutions. Finally, the experiments indicate that Algorithm 1 is efficient and robust for solving the rolling friction problem to obtain the optimal solution of 10 digit precision.

APPENDIX

We will take a glance at the Euclidean Jordan algebra which is a well-known framework associated with SOCO. See the reference paper [START_REF] Alizadeh | Second-order cone programming[END_REF] for a detailed presentation. We consider column vectors of the form (x0; x) ∈ R × R N -1 . For each such vector x ∈ R N , we define an Arrow-shaped matrix given by For any arbitrary power p ∈ Q, we define

x p = λ p 1 c1 + λ p 2 c2. For example, a useful formula for the gradient of the logarithm of the determinant is given by ∇x log det(x) = 2x -1 = 2(λ -1

1 c1 + λ -1 2 c2
). The quadratic-representation associated to x is defined by Qx = 2Arw 2 (x) -Arw(x 2 ) = 2xx -det(x)R.

For x ∈ R N , x nonsingular (i.e., λ2 = 0), y ∈ R N and t ∈ Z, we have the following properties [2, Theorem 8] Qxy = 2(x y)x -det(x)Ry,

Q x t = Q t
x , (Qxy) t = Q x t y t if x and y commute.

For vectors x, y of the form (v1; . . . ; v ), such that vi ∈ R n i belongs to the second order cone of dimension ni, we define

σ

  resp. (11)) with µ = 0 to obtain (d v a , d r a , d z a ) 12: α = get_step_length((z, J r), (d z a , J d r a )) = min{1, (µa/µ) e } 22: Solve (6) (resp. (11) with the modified RHS (12) (rep.(13)) to obtain (d v , d r , d z ) 23: α = get_step_length((z, J r), (d z , J d r ))

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Performance profile for tol = 10 -8 .

  Note that x ∈ L := {v = (v0; v) ∈ R × R N -1 : v ≤ v0} (known as Lorentz cone) if and only if Arw(x) is positive definite. In SOCO, it is useful to introduce the identity vector eN = (1; 0) ∈ R × R N -1 , the reflection matrixRN = 1 0 0 -I ∈ R N ×N ,the direct sum of matricesA ⊕ B := A 0 0 B and p i=1 Ai = A1 ⊕ . . . ⊕ Ap.The Jordan product of two vectors x, y is given byx • y := x y x0 ȳ + y0 x = Arw(x)y = Arw(x)Arw(y)e.The Jordan product is commutative, but not associative. The spectral decomposition of any element of this Jordan algebra is given byx = λ1c1 + λ2c2,where λ1,2 = x0 ± x and c1,2 = and eigenvectors of x. The trace and determinant of x are defined by tr(x) = λ1 + λ2 and det(x) = λ1λ2.

x

  • y = (x1 • y1; . . . ; x • y ), e = (en 1 ; . . . ; en ), Arw(x) = i=1 Arw(xi), Qx = i=1 Qx i and R = i=1 Rn i .

https://nonsmooth.gricad-pages.univ-grenoble-alpes.fr/siconos

https://github.com/FrictionalContactLibrary/fclib-library