Vicente Luís

Pedro Lomelino

Fernando Carreira

Francisco M Campos
email: francisco.campos@isel.pt

Mário J G C Mendes

A Luís Osório

J M F Calado
email: joao.calado@isel.pt

Industrial Collaborative Robotics Platform

Keywords: Interoperability, Industrial Collaborative Network, Communication Platform, Systems of Systems

Cyber-physical systems are not designed to integrate an industrial collaborative network when their integration with other systems is required. It is necessary to adopt a strategic model of integration and value creation that allows the interoperability of equipment supported by open technologies, based on the concepts underlying Industry 4.0. To this end, a conceptual information architecture is proposed to manage industrial robotic platforms based on a Cyber Device Bus design. The challenge is to contribute to an open technological framework involving computer systems, cyber-physical systems, and IoT elements in a logic of integration by adaptation without the need for specialized adapters. Adaptation occurs through the adoption of the Informatics System of Systems (ISoS) technological framework, providing an integrated serviceoriented (micro-services) view of technology elements and establishing cooperation between computer and cyber-physical systems under different responsibilities and based on a diversity of technological frameworks.

Introduction

Increasingly, associations seek to increase flexibility and responsiveness to meet the individual requirements of each client. Industry 4.0 has demanded that physical processes interconnect with cyber components, defining cyber-physical systems (CPS), enabling an environment where the physical world merges with the cyber world [START_REF] Wang | A collaborative architecture of the industrial internet platform for manufacturing systems[END_REF]. From the point of view of information and computer science, new information technologies must be gradually applied on the shop floor, integrating CPS framed into architectures based on the Internet of Things (IoT) as a strategy to create a collaborative environment. The IoT represents a technological revolution where the goal is to connect independent devices (each having its computing and communication parts, technologically differentiated and possibly ownership from different responsibilities) Vicente L. et al. (2021) Industrial Collaborative Robotics Platform. In: Camarinha -Matos L.M., Boucher X., Afsarmanesh H. (eds) Smart and Sustainable Collaborative Networks 4.0. PRO-VE 2021. IFIP Advances in Information and Communication Technology, vol 629. Springer, Cham. https://doi.org/10.1007/978-3-030-85969-5_53 Industrial Collaborative Robotics Platform 549 in a network, through the establishment of a global communication system, providing them with the ability to communicate with each other [START_REF]internacionais com vista à identificação das melhores e mais eficientes praticas e processos no âmbito da Indústria 4.0[END_REF][START_REF] Malik | Industrial Internet of Things and its Applications in Industry 4.0: State of The Art[END_REF].

Currently, any process under the concept of smart factories is carried out using several enterprise information systems (EIS), such as supervisory control and data acquisition (SCADA) systems, manufacturing execution systems (MES), enterprise resource planning systems (ERP), human resource management systems (HRM) and customer relationship management systems (CRM), among others. In this context, it is essential to guarantee the cooperation between EIS and their interoperability in order to achieve an effective management structure of corporate systems. Collaboration between different EIS is hampered by the fact that they are often developed by different suppliers, which creates the need to integrate them into a single collaborative platform [START_REF] Wang | A collaborative architecture of the industrial internet platform for manufacturing systems[END_REF].

The ability to interconnect systems within a system of systems framework emphasizes the need for an innovative approach to organize the diversity of technological artifacts that may be under the responsibility of different entities. This approach amounts to providing intelligent and communication capabilities to all devices, thus driving automation and monitoring capacity [START_REF] Vargas | Approaches for integration in system of systems: A systematic review[END_REF]. Currently, the typical EIS framework of organizations is still built on automation islands, as a set of disparate systems for process automation in a given application domain [START_REF] Calado | An adaptive IoT management infrastructure for ecoTransport networks[END_REF].

This paper addresses industrial robotic platforms' management that consists of a collaborative industrial network connecting robots and automated systems. A conceptual information system architecture based on the Informatics System of Systems (ISoS) framework and architectural methodology to achieve that goal is proposed, which guarantees interoperability and allows easy collaboration between different EIS.

This article is organized as follows: section 2 analyses the state of the art of collaborative information architectures; section 3 describes the ISoS methodology; a conceptual information system architecture based on the ISoS approach to achieve a collaborative platform of industrial robots is presented in Section 4. Finally, the conclusions and recommendations for future work are summarized in Section 5.

State of the Art

In the last century, the added value for industries came from the development of the mechanical domain. Today, the computer domain made possible through the evolution of information technologies has become the main driver of the development of organizations [START_REF] Ding | A cyberphysical production monitoring service system for energy-aware collaborative production monitoring in a smart shop floor[END_REF]. The growing mass customization requires that large volumes of data be simultaneously exchanged in real-time between different organizational systems. Thus, the challenge is to achieve direct communication between the Fieldbus and the application layer that promotes interoperability between all software and physical processes. This communication paradigm translates into a communication infrastructure over the internet. Each node represents a computer or CPS within the organization that can communicate with many other nodes (Fig. 1), thereby creating an environment based on collaborative processes. In this interwoven network, machines and applications need to communicate in the same language, coexist in the same infrastructure, and comply with multiple requirements [START_REF] Coito | A Middleware Platform for Intelligent Automation: An Industrial Prototype Implementation[END_REF]. Usually, computers and CPS are not designed to cooperate when the goal is the integration (interoperability) with other systems, potentially from a different business organization, seller, or supplier. When two or more systems under different responsibilities communicate, their managers must agree on the interfaces to use or provide access to a repository (database) through which systems can share data. Despite being a common practice to share information between systems, the access to databases generates strong dependencies between them, reflected in the dependence on changes, which are difficult to manage, e.g., in the face of a change to the database schema [START_REF] Calado | An adaptive IoT management infrastructure for ecoTransport networks[END_REF]. In order to avoid strong dependencies in networks, mediation strategies are adopted involving specialized computer systems to establish "bridges" through the "Hub and Spoke Integration" model [START_REF] Pels | Optimality of the hub-spoke system: A review of the literature, and directions for future research[END_REF] or the organization/company service bus (Enterprise Service Bus -ESB) [START_REF]IBM Cloud Education: What is an ESB (Enterprise Service Bus)?[END_REF][START_REF] Bhadoria | Analyzing the role of interfaces in enterprise service bus: A middleware epitome for service-oriented systems[END_REF]. However, the use of mediation systems, e.g., ESB, contributes to strengthening the technological dependence. Often, these systems centralize dependencies of different technological systems and do not support a more competitive framework of responsibilities, in which exchanging systems or parts of systems would be seamless [START_REF] Osório | Collaborative Networks as Open Informatics System of Systems (ISoS)[END_REF]. From the above, it can be concluded that system diversity hinders traditional systems from achieving the goals of Industry 4.0.

Communication on an industrial network is performed through a cyber platform commonly called Cyber Device Bus. To ensure communication between different systems, a computer systems architecture based on the design of an open Cyber Device Bus is necessary, with the following requirements: provide an open computational platform, ensure collaboration and ensure interoperability between computational entities and/or cyber-physical systems [START_REF] Panetto | Challenges for the cyberphysical manufacturing enterprises of the future[END_REF]. The design of this platform and the underlying architecture requires a set of non-functional requirements, typical of this type of open and distributed systems, namely: interoperability, transparency in location, fault tolerance, data, and communication security and scalability. In addition to the nonfunctional requirements, further requirements are needed, such as resilience, performance, portability, and substitutability [START_REF] Morel | Historical survey and emerging challenges of manufacturing automation modeling and control: A systems architecting perspective[END_REF].

In recent years, there has been much research into new collaborative reference architectures for Industry 4.0. However, works on conceptual architectures often lack a Industrial Collaborative Robotics Platform 551 physical implementation, and application examples do not typically generalize to other systems [START_REF] Biffl | Introduction to the multi-disciplinary engineering for cyber-physical production systems[END_REF].

There are currently several initiatives aimed at developing standardized system architectures. As an example, the German Reference Architecture Model Industry (RAMI 4.0) (DIN 2016) [START_REF] Bousdekis | A RAMI 4.0 view of predictive maintenance: Software architecture, platform and case study in steel industry[END_REF] proposes a reference model focused on system hierarchy, vertical communication network, and life cycle management, while the American Industrial Internet Reference Architecture (IIRA) [START_REF] Pedone | Model similarity evidence and interoperability affinity in cloudready Industry 4.0 technologies[END_REF] aims to achieve global adaptation and collaborative autonomy in an industrial Internet of Things (IIoT) system.

In the current technological landscape, the trend is towards a concentration of functionalities that, depending on the risk, are only accessible to proprietary technological frameworks (SAP, IBM, ORACLE, Microsoft, Amazon, Google, among others). These approaches, which include the concept of micro-services [START_REF] Sjödin | An agile co-creation process for digital servitization: A micro-service innovation approach[END_REF], do not convey relevant innovation in the context of a Service-Oriented Architecture (SOA) [START_REF] Niknejad | Understanding Service-Oriented Architecture (SOA): A systematic literature review and directions for further investigation[END_REF], so they are unlikely to promote integrated approaches that support processes in the digital age. There are currently several approaches to perform interoperability tasks between CPS, but these are proprietary systems. The diversity of techniques means that to perform changes or upgrades in the system, the customer is very dependent on the proprietary organization. Thus, the existing standards do not guarantee the substitutability of systems or elements of a computer system with potentially more competitive alternatives. In other words, the current standards are not complete in the sense of guaranteeing the exchangeability of systems or computer system elements with potentially more competitive alternatives [START_REF] Osório | On Reliable Collaborative Mobility Services[END_REF]. The lifecycle of a CPS may span decades, depending on its application domain. Thus, it is necessary to ensure those information architectures and their associated support services are available throughout the CPS lifecycle. Current approaches are struggling with the problem of technological dependency, known as the vendor lock-in situation. It is well known that public and private entities experience difficulties developing their innovation processes when they depend on technological devices, from the IoT level to the decision support services [START_REF] Osório | Collaborative Networks as Open Informatics System of Systems (ISoS)[END_REF].

To avoid problems associated with vendor dependency, many companies using CPS have recently chosen to keep the development of their information architecture in-house as a means of avoiding potential risks inherent in commercial architectures. This strategy gave rise to open-source solutions, e. g. Papyrus (2020) [START_REF]Eclipse Papyrus ™ Modeling environment[END_REF] and Capella (2020) [START_REF]Model Based Systems Engineering | Capella MBSE Tool[END_REF], which allow companies to develop their own customized information architectures. Several open-source initiatives have also been launched, as evidenced by the efforts of NASA, the Japanese Science and Technology Agency, and several consortia of industry and academia in Europe in creating open-source tools to assist in the development of collaborative information architectures [START_REF] Vara | Assurance and certification of cyber-physical systems: The AMASS open source ecosystem[END_REF].

More cooperative EIS environments based on information/knowledge are needed to meet current business demands. The cooperative EIS need contrasts with today's information architectures in most applications, which simply connect devices and can hardly support collaborative optimization for production systems. Other than that, information architectures should promote a relationship chain where all computers and CPSs provide support for knowledge construction, process monitoring, decision support, requirements management, and control [START_REF] Wang | A collaborative architecture of the industrial internet platform for manufacturing systems[END_REF].

There are currently numerous technological approaches that can be used to build a collaborative information architecture ensuring interoperability between various EIS. However, most of these are proprietary approaches that are difficult to budget and are not based on previously validated models. Most technological artifacts are not based on standard or reference models, which implies that the market does not moderate their cost. Thus, any change to the system, such as an integration requirement, is dependent on the involvement of those responsible for it, at a price that has no reference to be supported [START_REF] Osório | Adaptive Integration of IoT with Informatics Systems for Collaborative Industry: The SITL-IoT Case[END_REF]. The fact that state-of-art relies on unique computer systems developed as custom software without a formal framework and open modularity foundations has been a barrier to sustainable innovation.

Informatics System of Systems Approach

Today's industrial challenges have made production structures more complex and require new management and organizational strategies that offer greater flexibility, agility, efficiency, and effectiveness. In order to meet this challenge, remote and intelligent control and monitoring systems must be developed that deal with the distinctive functionalities of sensor and actuator networks, monitor hardware and software, and connect with production management and planning. These smart systems require high automation and information processing levels, allowing for rapid response to shop floor changes. An outcome of coupling smart supervisory with real-time decision support systems is the reduction of lead-times and costs on the shop floor, achieved through flexibility improvement of the production processes and by enhancing management control, which is supported by new decision support tools based on self-learning approaches and smart analytics. Hence, this paper addresses the implementation of a service-oriented Cooperation Enabled Services (CES) as a modularity abstraction framework to reduce the technology dependencies and empower the modeling and intelligence capabilities.

The Informatics System of Systems (ISoS) [START_REF] Osório | Collaborative Networks as Open Informatics System of Systems (ISoS)[END_REF] founds an open technological framework and incorporates the concept of Open Adaptive Coupling Infrastructure (OACI) to facilitate interoperability. This framework has been applied in several systems to structure technological artifacts of a company/organization, e.g., the Brisa tolls, speed control network (ANSR/SINCRO), and the automatic payment control in service area networks [START_REF] Osório | Adaptive Integration of IoT with Informatics Systems for Collaborative Industry: The SITL-IoT Case[END_REF] . In the ISoS framework, a computational or cyber-physical element of an Informatics System (Isystem) is structured as the composition of one or more CES. A CES models a composite of Service elements. An innovative aspect of the ISoS model is that a CES is made up of services (computational parts of CPS or computer systems) that can be developed under any technological framework. The ISoS model allows any service belonging to an Isystem to be instantiated as an element of a computer or cyber-physical system. ISoS makes it possible for an organization/company to cooperate with any other Service of a CES from another Isystem without the need for specialized adapters.

The aim of developing ISoS is to contribute to an open technological framework that encompasses computer systems, CPS, and IoT elements and promotes adaptation without special adapters. Instead, adaptation takes place by obtaining meta-data on the service elements and through cooperation between computer systems and CPS under Industrial Collaborative Robotics Platform 553 different responsibilities and on the basis of a diversity of technological frameworks [START_REF] Osório | Collaborative Networks as Open Informatics System of Systems (ISoS)[END_REF][START_REF] Calado | Dynamic integration of mould industry analytics and design forecasting[END_REF].

The proposed OACI as a loose coupling strategy based on the CES and Isystems concepts unifies the factory of the future computational infrastructure allowing the separation (decouple) of intelligent processes sharing a common distributed computational infrastructure (coordination, security, scalability, fault tolerance, quality of services reliability, and costs moderation through a multi-supplier framework). The capability of establishing hybrid computational execution environments on-premises and on the cloud, aligned with the microservices dynamics, makes the proposed approach scalable (elasticity) and prone (open) for emerging intelligent services and facilitates business intelligence applications integration.

Information System Architecture for Collaborative Networking

In the industry 4.0 approach, robot control and supervision system can be considered as a CPS, henceforth referred to as a robotic cyber-physical system (CPS-R). It is essential to reorganize them on a computational platform based on the internet to achieve a global collaborative network that allows covering all EIS, including CPS. The exchange of information between systems creates a collaborative, productive environment. In this work it is defined a system architecture based on the open technological framework ISoS, addressing the use of CPS-R to perform the collaboration and interoperability between CPS and EIS in an industrial plant with robots. Fig. 2 shows the context of the adoption of the ISoS framework and the concept of a Cyber Device Bus, forming a computational platform that integrates computational and/or cyber-physical entities. Notice that the presented robot is illustrative only. The information system architecture can be applied to any robot configuration. In this approach, the CPS-R constitutes a CES that in turn is composed of Services that are constituted as computational elements that guarantee its operation and the availability of functionalities to be accessed by other services, the cyber-physical system, or other computer systems. The CPS-RCES occurs as an element of the ISystem MCPS (A computerized management system for cyber-physical systems). This ISystem may be composed of other CES elements associated with other CPS of industrial robots, which exist or may be added to the organization.

This approach allows you to integrate The CPS with the organization's computer systems, e.g., an ERP computer system can obtain data from a CPS. Each computer system, according to the ISoS approach, will constitute an ISystem, which is in turn composed of CES. In fact, the CES elements are a composition of independent computational elements to which we call the Service. Thus, with this approach, the access of an ERP system modelled by an ISystem is considered as an ISystem to a service of the CPS, as shown in Fig. 2.

Communication in the proposed approach can be carried out between two services, from any CES, even if they are in conformity with different systems. A Service uses the Isytem0 computer meta-system in its management/coordination function to locate each computational element, that is, the information that allows access to a particular service. Additionally, with this architecture may be developed user interfaces (UI), for example, a web interface, from which a person can have access to information from the various services and in specific relevant aspects of the desired CPS.

Usually, processes are performed by several robots that work collaboratively and share data between them or control and supervision systems. Thus, this approach allows to be extended with the creation of new services associated with existing Isystems or new Isystems to take into account the several CPS-R, i.e., with the CPS-R1, CPS-R2, Industrial Collaborative Robotics Platform 555 …, CPS-Rn, being n the number of robotic systems that may come to incorporate into the organization (Fig. 3). This computational platform allows responding to the new service-oriented functionality management paradigm, providing an industrial collaborative network that guarantees interoperability, security, and scalability in accessing and exchanging data. With this computational architecture, an infinite number of computers and CPS can be integrated. At any time, a new service, computer system, or cyber-physical system can be added, integrating it into the organization's collaborative network.

Through the ability to exchange information between services, machine-to-machine communication is promoted, allowing two or more industrial robots to exchange information autonomously with each other, without the need for human intervention, using this computational platform. Because this architecture is internet-based, this allows the integration of all CPS of an organization regardless of their location. In this way, even in different facilities, the CPS can communicate with each other and with all computer systems, thus ensuring a global collaborative network.

Conclusion

In this article, a conceptual information architecture is proposed to manage industrial robotic platforms based on the design of a Cyber Device Bus, using the ISoS technological framework. The structure of the proposed architecture was built to create a collaborative industrial network through the reorganization of all systems that incorporate an organization, allowing interoperability between systems. This architecture translates into a computational platform that covers an infinite number of EIS, as well as an infinite number of CPS, promoting a collaborative environment. The computational platform presented allows users to have an overview of the entire organization, allowing access to all information from both computer systems and CPS through the various UI. By adopting the architecture presented throughout the organization, a service-oriented colouring mechanism is provided, allowing customers, manufacturing processes, and suppliers to be integrated, making it possible to respond to the high rate of customization required by the market. Future work should focus on the implementation of the architecture presented with a view to its validation in the construction of a collaborative industrial network.

Fig. 1 .

 1 Fig. 1. Collaborative information network.

Fig. 2 .

 2 Fig. 2. Collaborative network based in the ISoS framework using CPS-R elements.

Fig. 3 .

 3 Fig. 3. Isystem architectures to perform interoperability with several CPS-Rs.

Acknowledgments. This work was supported by the project POCI-01-0247-FEDER-039764 under the Program PORTUGAL2020.


D T V An

A Jourani

Keywords: Parametric convex programming, marginal function, subdifferential, singular subdifferential, distance function AMS Subject Classifications: 49J53, 49Q12, 90C25, 90C31

The aim of the present work is to use a metric intersection formula to estimate the subdifferential of the marginal function in the convex setting. This intersection formula includes many interesting situations in parametric convex programming, including the polyhedral one. It is expressed in terms of the objective function and the constrained multivalued mapping which govern the parametric program.

Introduction

Intersection formulas play an important role in optimization and variational and nonsmooth analysis. For example it allows to obtain the relationship between normal cone to set intersections and normal cone to each set in the intersection, to get chain rules for subdifferentials, and to estimate the subdifferential of the marginal function (see [START_REF] Jourani | Intersection formulae and the marginal function in Banach spaces[END_REF]).

To be concrete, we say that two closed sets C and D of a normed space X satisfy the intersection formula at x 0 ∈ C ∩ D if there exist two real numbers a > 0 and r > 0 such that

d(x, C ∩ D) ≤ a(d(x, C) + d(x, D)) ∀x ∈ B(x 0 , r), (1)
where x → d(x, B) denotes the distance function to a set B ⊂ X.

Normal cones to the intersection. Condition (1) ensures the following inclusion

N (x 0 ; C ∩ D) ⊂ N (x 0 ; C) + N (x 0 ; D),
where N denotes either the Fenchel cone in the sense of convex analysis with C and D being closed convex sets or the limiting Fréchet normal cone (see [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation[END_REF]) whenever X is a Asplund space or the nucleus normal cone (see [START_REF] Ioffe | Approximate subdifferentials and applications III: The metric theory[END_REF]) whenever X is any Banach space.

Unfortunately, the intersection formula [START_REF] An | Further results on differential stability of convex optimization problems[END_REF] is not sufficient to guarantee the same chain rule for Clarke's normal cone. However, for the latter one, we obtain

N (x 0 ; C ∩ D) ⊂ cl * [N (x 0 ; C) + N (x 0 ; D)],
where cl * (A) denotes the weak-star closedness of A ⊂ X * and X * is the dual space of X.

Subdifferential calculus. Consider the sets

S 1 = {(x, α, β) ∈ X × R × R : f 1 (x) ≤ α},
and

S 2 = {(x, α, β) ∈ X × R × R : f 2 (x) ≤ β},
for extended real-valued functions f 1 and f 2 on X. We say that f 1 and f 2 satisfy the intersection formula at x 0 if there exist r > 0 and a > 0 such that d((x, α, β), S 1 ∩ S 2 ) ≤ a[d((x, α, β), S 1 ) + d((x, α, β), S 2 )]

for all x ∈ x 0 + rB X , α ∈ f 1 (x 0 ) + rB R and β ∈ f 2 (x 0 ) + rB R . Condition (2) leads to the following chain rule

∂(f 1 + f 2 )(x 0 ) ⊂ ∂f 1 (x 0 ) + ∂f 2 (x 0 ),
where ∂ denotes either the Fenchel subdifferential with f 1 and f 2 being convex functions or the limiting Fréchet subdifferential (see [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation[END_REF]) whenever X is a Asplund space or the nucleus subdifferential (see [START_REF] Ioffe | Approximate subdifferentials and applications III: The metric theory[END_REF]) whenever X is any Banach space.

The study of the behavior of the marginal (or value) function is known to be very important in variational analysis and optimization theory; see, e.g., [START_REF] An | Further results on differential stability of convex optimization problems[END_REF][START_REF] An | Differential stability of convex optimization problems under inclusion constraints[END_REF][START_REF] Aubin | Optima and Equilibria, An Introduction to Nonlinear Analysis[END_REF][START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF][START_REF] Gauvin | Differential properties of the marginal function in mathematical programming[END_REF][START_REF] Gollan | On the marginal function in nonlinear programming[END_REF][START_REF] Jourani | Intersection formulae and the marginal function in Banach spaces[END_REF][START_REF] Mordukhovich | Subgradients of marginal functions in parametric mathematical programming[END_REF][START_REF] Thibault | On subdifferentials of optimal value functions[END_REF][START_REF] Zȃnlinescu | Convex Analysis in General Vector Spaces[END_REF] and the references therein.

In [START_REF] Jourani | Intersection formulae and the marginal function in Banach spaces[END_REF], the author has proposed the method by using intersection formulas to evaluate the G-subdifferentials of the marginal functions of parametric programming problems under inclusion constraints in Banach spaces. Recently, by using the Moreau-Rockafellar theorem and appropriate regularity conditions, An and Yen [START_REF] An | Differential stability of convex optimization problems under inclusion constraints[END_REF], An and Yao [START_REF] An | Further results on differential stability of convex optimization problems[END_REF] have obtained formulas for computing subdifferentials of the marginal functions for convex optimization problems under inclusion constraints on infinite dimensional spaces.

The primary goal of this paper is to use intersection formulas for estimating subdifferentials of the marginal functions of convex parametric programming problems under inclusion constraints in Banach spaces. Thus, on one hand, our results have the origin in the study of [START_REF] Jourani | Intersection formulae and the marginal function in Banach spaces[END_REF]. On the other hand, they are the results of deepening that study for the case of convex programming problems.

The contents of the remaining sections are as follows. Section 2 recalls several definitions from [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF] and [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]. Intersection formulas are analyzed in Section 3. Section 4 presents formulas for computing the subdifferential and the singular subdifferential of the marginal functions of convex optimization problems under inclusion constraints on Banach spaces.

Section 5 compares our results with some known results. Several conclusions are given in the final section.

Preliminaries

Let X and Y be Banach spaces whose duals are denoted, respectively, by X * and Y * . We

dom ϕ := {x ∈ X : ϕ(x) < +∞}
is nonempty, and if ϕ(x) > -∞ for all x ∈ X. The function ϕ is convex if and only if its

epigraph epi ϕ := {(x, α) ∈ X × R : α ≥ ϕ(x)} is a convex subset of X × R. If epi ϕ is a closed subset of X × R, ϕ is said to be a closed function.
Denoting the set of all the neighborhoods of x by N (x), one says that ϕ is lower semicontinuous (l.s.c.) at x ∈ X if for every ε > 0 there exists U ∈ N (x) such that ϕ(x ) ≥ ϕ(x) -ε for any x ∈ U. If ϕ is l.s.c. at every x ∈ X, ϕ is said to be l.s.c. on X.

It is convenient to denote the set of all proper l.s.c. convex functions on X by Γ 0 (X).

Let ϕ be an extended-real valued convex function on X and x ∈ dom ϕ. The subdifferential of ϕ at x is the set

∂ϕ(x) = {x * ∈ X * : x * , x -x ≤ ϕ(x) -ϕ(x), ∀x ∈ X}.
If f ∈ Γ 0 (X), then ∂ϕ(x) is nonempty for every x ∈ int(dom) ϕ (see [START_REF] Ioffe | Theory of Extremal Problems[END_REF]).

For a convex set Ω ⊂ X, the normal cone of Ω at x ∈ Ω is given by

N (x; Ω) = {x * ∈ X * : x * , x -x ≤ 0, ∀x ∈ Ω}.
The singular subdifferential of a convex function ϕ at x ∈ dom ϕ is defined by

∂ ∞ ϕ(x) = {x * ∈ X * : (x * , 0) ∈ N ((x, ϕ(x)); epi ϕ)}.
For any x / ∈ dom ϕ, we put ∂ϕ(x) = ∅ and

∂ ∞ ϕ(x) = ∅.
It is not difficult to show that x * ∈ ∂ϕ(x) if and only if x * , x -x -(α -ϕ(x)) ≤ 0 for all (x, α) ∈ epi ϕ or, equivalently, (x * , -1) ∈ N ((x, ϕ(x)); epi ϕ). Interestingly, for any convex function ϕ, one has ∂ ∞ ϕ(x) = N (x; dom ϕ); see e.g., [START_REF] An | Differential stability of convex optimization problems under inclusion constraints[END_REF].

Let F : X ⇒ Y be a multivalued function. The graph and the domain of F are defined by

Gr F := {(x, y) ∈ X × Y : y ∈ F (x)} and dom F := {x ∈ X : F (x) = ∅}.
One says that a multivalued function F is closed (resp., convex) if Gr F is closed (resp., convex). In addition, if dom F = ∅, F is called proper. The product space X × Y will be endowed with the norm ||(x, y)|| = ||x|| + ||y|| for any (x, y) ∈ X × Y .

The coderivative D * F (x, ȳ) : Y * ⇒ X * of a convex multifunction F between X and Y at (x, ȳ) ∈ Gr F is the multivalued function defined by

D * F (x, ȳ)(y * ) := {x * ∈ X * : (x * , -y * ) ∈ N ((x, ȳ); Gr F )} , ∀y * ∈ Y * . If (x, ȳ) / ∈ Gr F , one puts D * F (x, ȳ)(y * ) = ∅ for any y * ∈ Y * .
Let C be a nonempty subset of X. The distance function of C is defined by

d(x, C) = inf u∈C ||x -u||.
Clearly, the distance function d(•, C) is Lipschitz continuous with modulus one, and convex provided that C is convex. It is obvious that for any convex set Ω ⊂ X containing x, we have

N (x; Ω) = N (x; clΩ) and ∂d(x, Ω) = ∂d(x, clΩ). (3)
We will need some properties of the subdifferential of d(•, C). Assertion (i) of the following proposition has been observed in different places, e.g., [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF]Example 2.130] and [8, Theorem 1]. Assertion (ii) is well known when C is convex as it corresponds to [START_REF] Clarke | A new approach to Lagrange multipliers[END_REF]Proposition 11]. We present a direct proof for the reader's convenience.

Proposition 2.1. Let C be a nonempty convex subset of X. Then for every x ∈ C, one has

(i) ∂d(x, C) = B X * ∩ N (x; C), where B X * is the closed unit ball in X * ; (ii) N (x; C) = λ>0 λ∂d(x, C).
Proof. We only need to prove assertion (i) beacause (ii) immediately follows from the first one. Indeed, by the definition of the subdifferential, one has x * ∈ ∂d(x, C) if and only if

x * , u -x ≤ d(u, C) -d(x, C), ∀u ∈ X. (4)
Since d(•, C) is Lipschitz continuous modulus one, from (4) one has

x * , u -x ≤ d(u, C) -d(x, C) ≤ ||u -x||, ∀u ∈ X.
It follows that ||x * || ≤ 1. Hence ∂d(x, C) ⊂ B X * . Moreover, for all u ∈ C, from (4) we get

x * , u -x ≤ 0. In other words,

x * ∈ N (x; C). Therefore x * ∈ B X * ∩ N (x; C).
Conversely, suppose that x * belongs to B X * ∩ N (x; C). For any u ∈ X and ε > 0, we can

find u ∈ C such that d(u, C) ≥ ||u -u || -ε. As ||x * || ≤ 1 and x * ∈ N (x; C), the latter implies that d(u, C) ≥ x * , u -u -ε = x * , u -x + x * , x -u -ε ≥ x * , u -x -ε. (5)
Since ε > 0 is arbitrary and x ∈ C, we obtain

d(u, C) ≥ x * , u -x = x * , u -x + d(x, C) ∀u ∈ X
which shows that x * ∈ ∂d(x, C).

Intersection formulas

Let C and D be subsets of X with x 0 ∈ C ∩ D. One says that C and D satisfy the intersection formula at x 0 if there exist two real numbers a > 0 and r > 0 such that inequality (1) is satisfied. In the case where C and D are closed and convex, we obtain (see [START_REF] Jourani | Error bound characterizations of Guignard's constraint qualification in convex programming[END_REF]) the following characterization.

u ε ∈ C ∩ D, x * ε ∈ X * and b * ε ∈ B * such that i) u ε -u ≤ d(u, C ∩ D) + ε 2 , ii) x * ε + εb * ε ∈ (1 + ε)∂d(u ε , C ∩ D), iii) x * ε , u -u ε = u ε -u . Note that u ε ∈ C ∩ D ∩ B(x 0 , s)
. By 2. and ii), we have

x * ε + εb * ε ∈ a(1 + ε) ∂d(u ε , C) + ∂d(u ε , D) = a(1 + ε)∂(d(•, C) + d(•, D))(u ε )
and hence

x * ε + εb * ε , u -u ε ≤ a(1 + ε)(d(u, C) + d(u, D)).
Using iii), we get

u -u ε (1 -ε) ≤ a(1 + ε)(d(u, C) + d(u, D))(because b * ε ≤ 1)
which in turns implies that

d(u, C ∩ D)(1 -ε) ≤ a(1 + ε)(d(u, C) + d(u, D))
and as ε is arbitrary, we obtain the desired result by putting r = s 2 .

Following Borwein and Strojwas [START_REF] Borwein | Tangential approximations[END_REF], we call a set C ⊂ X compactly epi-Lipschitzian (CEL) at x 0 ∈ C if there exist a norm-compact set K and a scalar r > 0 such that

C ∩ (x 0 + rB X ) + τ rB X ⊂ C -τ K ∀τ ∈ [0, r].
This class includes all finite dimensional sets. It is worth to note that in infinite dimensional spaces, there is no relationship between compact sets and CEL sets (because these later are never compact in infinite dimensional spaces). This also implies that the CEL sets are useful only in infinite dimensional spaces; see ( [START_REF] Borwein | Compactly epi-Lipschitzian convex sets and functions in normed spaces[END_REF], [START_REF] Jourani | On a class of compactly epi-Lipschitzian sets[END_REF]).

In [START_REF] Henrion | Subdifferential conditions for calmness of convex constraints[END_REF], it is shown that a boundary condition ensures the intersection formula. More

Then there exist r > 0 and a > 0 satisfying relation [START_REF] An | Further results on differential stability of convex optimization problems[END_REF]. Here "bd (A)" denotes the boundary of the set A ⊂ X * with respect to the norm on X * .

Remark 3.1. Note that the reverse implication is not true. We mean that relation [START_REF] An | Further results on differential stability of convex optimization problems[END_REF] does not implies [START_REF] Borwein | Compactly epi-Lipschitzian convex sets and functions in normed spaces[END_REF]. To see this, take

C := {(x 1 , x 2 ) ∈ R 2 : x 1 ≤ 0, x 2 ≤ 0, x 2 ≤ x 1 }, D = R 2
+ and x 0 = (0, 0). It can be verified that relation [START_REF] An | Further results on differential stability of convex optimization problems[END_REF] is valid for a = 1 and for any r > 0, meanwhile bd N (x 0 ; C) ∩ -bd N (x 0 ; D) = R + × {0}.

Main results

We define the marginal function (also called value function) m : X → R by

m(x) := inf{f (x, y) : y ∈ F (x)}, ( 7
)
where f is an extended-real-valued convex lower semicontinuous function on X × Y and F is a multivalued function from X to Y with closed graph. By the convention inf ∅ = +∞, we have m(x) = +∞ for all x / ∈ dom F. Corresponding to each data pair F, f we have one optimization problem depending on a parameter x:

min {f (x, y) : y ∈ F (x)}. ( 8
)
For every x ∈ dom m, we consider the set of minimizers

M (x) = {y ∈ F (x) : m(x) = f (x, y)}. (9)
We are looking for formulas to estimate/compute subdifferentials of m.

The following proposition gives the continuity of m under some conditions. Proof. We consider the unconstrained problem min {ϕ(x, y) :

y ∈ Y }, (10)
with the objective function ϕ(x, y) being ϕ(x, y) = f (x, y) + δ Gr F (x, y). Obviously, the marginal function of problem ( 10) is also the function m given in [START_REF] Borwein | Tangential approximations[END_REF] Put

C 1 = epi f and C 2 = Gr F × R.
We need some auxiliary results.

Lemma 4.1. Let y 0 ∈ M (x 0 ). Then, we have (x * , β) ∈ N ((x 0 , m(x 0 )); epi m) if and only if

(x * , 0, β) ∈ N ((x 0 , y 0 , f (x 0 , y 0 )); C 1 ∩ C 2 ).
Proof. Let (x * , β) ∈ N ((x 0 , m(x 0 )), epi m). Then,

x * , x -x 0 + β(r -m(x 0 )) ≤ 0, ∀(x, r) ∈ epi m, (11)
because (x 0 , λ + m(x 0 )) ∈ epi m for every λ ≥ 0. So β must be negative number. Keeping in mind that y 0 ∈ M (x 0 ), ( 11) one has

x * , x -x 0 ≤ -β(r -m(x 0 )), ∀(x, r) ∈ epi m, ≤ -β(m(x) -f (x 0 , y 0 )), ∀x ∈ dom m
≤ -β(f (x, y) -f (x 0 , y 0 )), ∀(x, y) ∈ dom f and (x, y) ∈ Gr F ≤ -β(r -f (x 0 , y 0 )), ∀(x, y, r) ∈ epi f and (x, y) ∈ Gr F, or, equivalently, (x * , 0), (x, y) -(x 0 , y 0 ) + β(r -f (x 0 , y 0 )) ≤ 0, ∀(x, y, r) ∈ epi f and (x, y) ∈ Gr F.

So (x * , 0, β) ∈ N ((x 0 , y 0 , f (x 0 , y 0 )); C 1 ∩ C 2 ), with C 1 = epi f and C 2 = Gr F × R.
Conversely, take (x * , 0, β) ∈ N ((x 0 , y 0 , f (x 0 , y 0 ));

C 1 ∩ C 2 )
. By the definition of the normal cone, we have

x * , x -x 0 + β(r -f (x 0 , y 0 )) ≤ 0, ∀(x, y, r) ∈ C 1 ∩ C 2 . (12)
Similarly, one has β ≤ 0. By the definition of the infimum, for all ε ≥ 0, we can find

y ∈ F (x) such that m(x) ≥ f (x, y) -ε. Put r = ε + m(x), we have (x, y, r) ∈ C 1 ∩ C 2 .
From [START_REF] Gollan | On the marginal function in nonlinear programming[END_REF], we obtain

x * , x -x 0 ≤ -β(ε + m(x) -f (x 0 , y 0 )) = -β(ε + m(x) -m(x 0 )), ∀x ∈ dom m.
The last inequality implies that

x * , x -x 0 ≤ -β(m(x) -m(x 0 )), ∀x ∈ dom m ≤ -β(r -m(x 0 )), ∀(x, r) ∈ epi m.
In other words, (x * , β) ∈ N ((x 0 , m(x 0 )); epi m). 2

Lemma 4.2. Let y 0 ∈ M (x 0 ). Then one has

(x * , β) ∈ ∂d((x 0 , m(x 0 )), epi m)
if and only if

(x * , 0, β) ∈ ∂d((x 0 , y 0 , f (x 0 , y 0 )), C 1 ∩ C 2 ).
Proof. According to Propostion 2.1(i), one has

∂d((x 0 , m(x 0 )), epi m) = B X * ×R ∩ N ((x 0 , m(x 0 )); epi m).
Then, (x * , β) ∈ ∂d((x 0 , m(x 0 )); epi m) if and only if ||(x * , β)|| ≤ 1 and (x * , β) ∈ N ((x 0 , m(x 0 )); epi m).

By Lemma 4.1, (x * , β) ∈ N ((x 0 , m(x 0 )); epi m) if and only if

(x * , 0, β) ∈ N ((x 0 , y 0 , f (x 0 , y 0 )); C 1 ∩ C 2 ).
Combining with ||(x * , 0, β)|| ≤ 1, we obtain (x * , β) ∈ ∂d((x 0 , m(x 0 )), epi m) if and only if

(x * , 0, β) ∈ ∂d((x 0 , y 0 , f (x 0 , y 0 )), C 1 ∩ C 2 ). 2
The first main result of this section reads as follows.

Theorem 4.1. Suppose that C 1 and C 2 satisfy the intersection formula at (x 0 , y 0 , f (x 0 , y 0 )),

with y 0 ∈ M (x 0 ). Then ∂m(x 0 ) = (x * ,y * )∈∂f (x 0 ,y 0 ) {x * + D * F (x 0 , y 0 )(y * )} . ( 13
)
Proof. Pick u * from the right hand side of ( 13), then there exists (x * , y * ) ∈ ∂f (x 0 , y 0 ) such that

u * ∈ x * + D * F (x 0 , y 0 )(y * ).
It gives us (u * -x * , -y * ) ∈ N ((x 0 , y 0 ); Gr F ). By the definition of the normal cone, we have

u * -x * , x -x 0 -y * , y -y 0 ≤ 0, ∀ (x, y) ∈ Gr F,
which shows that whenever y ∈ F (x), we have

u * , x -x 0 ≤ x * , x -x 0 + y * , y -y 0 ≤ f (x, y) -f (x 0 , y 0 ) ≤ f (x, y) -m(x 0 ).
The latter implies

u * , x -x 0 ≤ inf y∈F (x) f (x, y) -m(x 0 ) = m(x) -m(x 0 ), ∀y ∈ F (x).
Therefore u * ∈ ∂m(x 0 ).

To establish the opposite inclusion, we note that

u * ∈ ∂m(x 0 ) ⇔ (u * , -1) ∈ N ((x 0 , m(x 0 )); epi m).
According to Proposition 2.1(ii), one gets

N ((x 0 , m(x 0 )); epi m) = λ≥0 λ∂d((x 0 , m(x 0 )); epi m).
In other words, there exist λ > 0 and (x * , β) ∈ ∂d((x 0 , m(x 0 )); epi m) such that u * = λx * and λβ = -1. By Lemma 4.2, the latter is equivalent to

(x * , 0, β) ∈ ∂d((x 0 , y 0 , f (x 0 , y 0 )), C 1 ∩ C 2 ). (14)
Our assumptions guarantee the convexity and the closedness of the sets C 1 = epi f and

C 2 = Gr F × R.
Since C 1 and C 2 satisfy the intersection formula at (x 0 , y 0 , f (x 0 , y 0 )), Proposition 3.1 ensures the existence of a > 0 satisfying ∂d((x 0 , y 0 , f (x 0 , y 0 )),

C 1 ∩ C 2 ) ⊂ a[∂d((x 0 , y 0 , f (x 0 , y 0 )), C 1 ) + ∂d((x 0 , y 0 , f (x 0 , y 0 )), C 2 )]. (15)
Therefore, from ( 14) and ( 15), we can find a vector (z * , y * , γ) ∈ ∂d((x 0 , y 0 , f (x 0 , y 0 )), epi f ), and (v * , p * ) ∈ ∂d((x 0 , y 0 ), Gr F ) satisfying

x * = a(z * + v * ), 0 = a(y * + p * ),
and β = aγ. Hence

u * = λx * = λa(z * + v * ), -1 = λβ = aλγ,
and λay * = -λap * . Thus (λaz * , λay * ) ∈ ∂f (x 0 , y 0 ), and u * ∈ λaz * + DF (x 0 , y 0 )(λay * ).

2 By Proposition 3.2, the next corollary follows directly from Theorem 4.1.

Corollary 4.1. Let y 0 ∈ M (x 0 ). Suppose that one of the following conditions is satisfied

(i) epi f is CEL at (x 0 , y 0 , f (x 0 , y 0 )) and N ((x 0 , y 0 , f (x 0 , y 0 )); C 1 ) ∩ [-N ((x 0 , y 0 , f (x 0 , y 0 )); C 2 )] = {(0 X * , 0 Y * , 0 R )}; (16)
(ii) Gr F × R is CEL at (x 0 , y 0 , f (x 0 , y 0 )) and ( 16) holds.

Then the conclusion of Theorem 4.1 is valid.

The singular subdifferential of m can be computed as follows.

Theorem 4.2. Let y 0 ∈ M (x 0 ). Suppose that one of the following conditions is satisfied (i) dom f is CEL at (x 0 , y 0 , f (x 0 , y 0 )) and

N ((x 0 , y 0 ); dom f ) ∩ [-N ((x 0 , y 0 ); Gr F )] = {(0 X * , 0 Y * )}. (17)
(ii) Gr F × R is CEL at (x 0 , y 0 , f (x 0 , y 0 )) and (17) holds. Then

∂ ∞ m(x 0 ) = (x * ,y * )∈∂ ∞ f (x 0 ,y 0 ) {x * + D * F (x 0 , y 0 )(y * )} . (18)
Proof. We first observe that x ∈ dom m if and only if

m(x) = inf{f (x, y) : y ∈ F (x)} < +∞.
As the strict inequality holds if and only if there exists y ∈ F (x) with (x, y) ∈ dom F , we have

δ dom m (x) = inf {δ dom f (x, y) : y ∈ F (x)}.
Recall that δ C is the indicator function of the set C, i.e., δ C (x) = 0 if x ∈ C and + ∞, otherwise. On one hand, epi δ dom f = dom f × [0, +∞). If the condition (i) is satisfied, then epi δ dom f is CEL at (x 0 , y 0 , f (x 0 , y 0 )). On the other hand, [START_REF] Jourani | Intersection formulae and the marginal function in Banach spaces[END_REF] yields

N ((x 0 , y 0 , δ dom f (x 0 , y 0 )); epi δ dom f ) ∩ [-N ((x 0 , y 0 , δ dom f (x 0 , y 0 ));Gr F × R)] = {0 X * , 0 Y * , 0 R }.
Since f is convex, it follows that dom f is convex. Moreover, as f is lower semicontinuous, it follows that epi f and dom f are closed. Thus δ dom f (.) is a lower semincontinuous function. So all assumptions of Corollary 4.1 are satisfied. Thanks to Corollary 4.1, we have

∂δ dom m (x 0 ) = (x * ,y * )∈∂δ dom f (x 0 ,y 0 ) {x * + D * F (x 0 , y 0 )(y * )} . (19)
As ∂δ dom m (x 0 ) = ∂ ∞ m(x 0 ) and ∂δ dom f (x 0 , y 0 ) = ∂ ∞ f (x 0 , y 0 ). So, we obtain ( 18) from [START_REF] Mordukhovich | Subgradients of marginal functions in parametric mathematical programming[END_REF].

5 Comparisons with some known results

By a different approach, the authors in [START_REF] An | Further results on differential stability of convex optimization problems[END_REF] and [START_REF] An | Differential stability of convex optimization problems under inclusion constraints[END_REF] have studied a problem similar to that considered in this paper. Namely, in our notation, by using some suitable sum rules for convex subdifferential, the authors have proved the following theorems. x * + D * F (x, ȳ)(y * ) and

∂ ∞ m(x) = (x * ,y * )∈∂ ∞ f (x,ȳ) x * + D * F (x, ȳ)(y * ) ,
where m(•) and M (•) are given respectively by [START_REF] Borwein | Tangential approximations[END_REF] and (9).

(0, 0) ∈ int (dom f -Gr F ) (20)
is satisfied, then for every x ∈ dom m with m(x) = -∞, and for any ȳ ∈ M (x), we have

∂m(x) = (x * ,y * )∈∂f (x,ȳ) x * + D * F (x, ȳ)(y * ) .
Suppose further that the set dom f is closed. Then

∂ ∞ m(x) = (x * ,y * )∈∂ ∞ f (x,ȳ)
x * + D * F (x, ȳ)(y * ) .

In [START_REF] An | Further results on differential stability of convex optimization problems[END_REF], the authors have shown the relationship between regularity conditions. Namely, the condition ( 20) is really weaker than both regularity assumptions (a) and (b) in Theorem 5.1

(see [START_REF] An | Further results on differential stability of convex optimization problems[END_REF]Section 4]).

We now give an analysis of regularity conditions. Let X be a finite dimensional space. It is a convenient way to consider the following problem

min {f (x) : x ∈ C}, (21)
where f : X → R ∪ {+∞} is a closed, convex function and C ⊂ X is a closed, convex set.

First, we recall that all subsets of X are CEL (see [START_REF] Borwein | Tangential approximations[END_REF]Proposition 2.4]) and every convex function f defined on a finite dimensional space is continuous on int(dom f ) (see, e.g., [5, Corollary 2.109]). We have the following proposition.

Proposition 5.1. The regularity condition

0 ∈ (dom f -C) (22)
implies

N ((x 0 , f (x 0 )); epi f ) ∩ [-N ((x 0 , f (x 0 )); C × R)] = {(0, 0)}. ( 23
)
Proof. Take an arbitrary (x * , β) ∈ N ((x 0 , f (x 0 )); epi f ) ∩ [-N ((x 0 , f (x 0 )); C × R)]. We first observe that for every x 0 ∈ C with (x 0 , f (x 0 )) ∈ epi f, N ((x 0 , f (x 0 )); C × R) = N (x 0 ; C) × N (f (x 0 ); R) = N (x 0 ; C) × {0}.

Thus β = 0. Moreover, we have

x * , x -x 0 + 0.(r -f (x 0 )) ≤ 0, ∀(x, r) ∈ epi f,

and

-x * , x -x 0 ≤ 0, ∀x ∈ C. ( 25
)
On one hand, from (24) one has x * , y -x 0 ≤ 0, ∀y ∈ dom f. Combining this with (25), we obtain x * , y -x ≤ 0, ∀y ∈ dom f, x ∈ C, or

x * , h ≤ 0, ∀h ∈ (dom f -C),

On the other hand, by the assumption 0 ∈ (dom f -C), we can find a neighborhood U of 0 such that x * , h ≤ 0, for all h ∈ U. The later yields x * = 0. Combining this with β = 0, we obtain (x * , β) = (0, 0).

Conclusions

We have obtained exact formulas for the subdifferential and singular subdifferential of the marginal function for parametric convex programs by using intersection formulas and

  use B(x, r) to denote the open ball centered at x with radius r > 0. Let B X (resp. B X * ) denote the closed unit ball of X (resp. X * ) centered at the origin. The closure and the interior of a set A are denoted by clA and intA, respectively. Let ϕ : X → R be an extended real-valued function, where R := [-∞, +∞] = R∪{+∞}∪ {-∞}. One says that ϕ is proper if the domain

Proposition 3 . 1 .

 31 Let C and D be closed and convex subset of X and a > 0. Then the following assertions are equivalent 1. There exists r > 0 satisfying relation (1), 2. There exists s > 0 such that ∂d(x, C ∩ D) ⊂ a∂d(x, C) + a∂d(x, D) ∀x ∈ C ∩ D ∩ B(x 0 , s). Proof. It is given for completeness. The implication 1. ⇒ 2. is obvious. So let us establish the reverse one. Let u ∈ B(x 0 , s 2 )\C ∩ D and ε ∈]0, √ s 4 [. Then, by Lemma 2.1 in [4], there exist

precisely, one has Proposition 3 . 2 .

 32 Let C and D be closed convex subsets of X and x 0 ∈ C ∩ D. Suppose that one of the sets C or D is CEL at x 0 and bd N (x 0 ; C) ∩ [-bd N (x 0 ; D) ] = {0}.

Proposition 4 . 1 .

 41 The marginal function m(•) is convex. Moreover if m is finite at x 0 , m(•) is continuous at x 0 if and only if for every x in a neighborhood of x 0 , m(x) < +∞, i.e., x 0 ∈ int(dom m).

Theorem 5 . 1 .

 51 (see [2, Theorem 4.2]) Let X and Y be Hausdorff locally convex topological vector spaces. Consider the problem (8) with F : X ⇒ Y being a convex multivalued function and f : X × Y → R being a proper convex function. If at least one of the following regularity conditions is satisfied: (a) int(Gr F ) ∩ dom f = ∅, (b) f is continuous at a point (x 0 , y 0 ) ∈ Gr F, then, for any x ∈ dom m, with m(x) = -∞, and for any ȳ ∈ M (x) we have ∂m(x) = (x * ,y * )∈∂f (x,ȳ)

Theorem 5 . 2 .

 52 (see [1, Theorems 3.3 and 3.4]) Consider the problem (8) with F : X ⇒ Y being a convex multifunction whose graph is closed, and f : X × Y → R a proper, closed, convex function. If

2Example 5 . 1 .Remark 5 . 1 .

 5151 Let us consider an example to show that the regularity condition (22) is really stronger than (23). Let X = R 2 and C = {(x, y) ∈ R 2 : y = x, y ≥ 0}. Consider the function f given by f (x, y) = y = 0.For x 0 = y 0 = 0. On one hand, one has N ((x 0 , y 0 , f(x 0 .y 0 )); C × R) = {(0 R 2 , 0)}. So (23) is satisfied. On the other hand dom f = R × {0}. Then (0, 0) / ∈ (dom f -C).In other words, the regularity condition (23) does not imply (22).Let X, Y be finite dimensional spaces. Note that the convex programming problem considered in Section 4 is a particular case of problem[START_REF] Zȃnlinescu | Convex Analysis in General Vector Spaces[END_REF] with X = X × Y and C = Gr F .From Proposition 5.1 we obtain the following statement. Proposition 5.2. Let X, Y be finite dimensional spaces. The regularity conditions (i) and (ii) of Corollary 4.1 are weaker than (20). In the case X, Y are Banach spaces. It is still unclear to us whenever the regularity conditions (i) and (ii) of Corollary 4.1 are weaker than (20), or not? The assertions of Theorem 4.1 and 4.2 are similar to those in [16, Theorems 5.1]. By imposing the strong convexity requirement on (8), we obtain corresponding equalities.

Acknowledgements. The research of Duong Thi Viet An was supported by Thai Nguyen University of Sicences, Vietnam. The research of Abderrahim Jourani was partially supported by the EIPHI Graduate School (contract ANR-17-EURE-0002). The authors would like to thank the two anonymous referees for their very careful readings and valuable suggestions which have helped to greatly improve the presentation.

appropriate assumptions. Comparions with some known results have been given.