Book of abstracts. Variational Analysis, Equilibria and Optimization May Università di Pisa Dipartimento di Informatica Sala Gerace

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1 Book of abstracts Variational Analysis, Equilibria and Optimization May 2017 Università di Pisa Dipartimento di Informatica Sala Gerace 29 May 14:15 14:30 Opening 14:30 15:10 Marco Antono López Cerdá (University of Alicante, Spain) Calmness in linear optimization. Some applications 15:10 15:50 Simone Sagratella (Sapienza University of Rome, Italy) A bridge between bilevel programs and Nash games 15:50 16:15 coffee break 16:15 16:55 Oliver Stein (Karlsruhe Institute of Technology, Germany) Feasible roundings for granular optimization 16:55 17:35 Cornel Pintea (Babeş - Bolyai University Cluj, Romania) Generalized monotonicity in the finite dimensional setting. Applications 17:35 18:15 Radu Strugariu ( Gheorghe Asachi Technical University of Iaşi, Romania) Composition set-valued mappings: metric (sub)regularity and fixed points 20:30 Social Dinner at the Restaurant La Clessidra 30 May 9:00 9:40 Juan Enrique Martínez-Legaz (Universität Autonoma de Barcelona, Spain) On Bregman-type distances for convex functions and maximally monotone operators 9:40 10:20 Marius Durea ( Alexandru Ioan Cuza University of Iaşi, Romania) Directional approach to regularity of mappings and applications to optimization 10:20 11:00 Massimiliano Giuli (University of l Aquila, Italy) Ekelands principle for equilibrium problems 11:00 11:25 coffee break 11:25 12:05 Radek Cibulka (University of West Bohemia in Pilsen, Czech Republic) On primal regularity estimates 12:05 12:45 Mihaela Miholca (Technical University Cluj-Napoca, Romania) Ekelands principle for cyclically antimonotone vector bifunctions and weak vector equilibrium problems 12:45 13:25 Riccardo Cambini (University of Pisa, Italy) On the use of the optimal level solution method in generating the efficient frontier of a class of bicriteria programs 13:25 13:30 Concluding remarks

2 Calmness in linear optimization. Some applications Marco A. López Department of Mathematics University of Alicante Abstract. This talk presents an overview of the recent research of our group on calmness in linear optimization. The talk is developed in two parametric settings: the one of canonical perturbations versus the context of full perturbations. While there exists a clear proportionality between the calmness moduli of the feasible set mappings in both contexts, the analysis of the relationship between the calmness moduli of the argmin mappings is much more complicated. We emphasize the fact that we are always providing point-based expressions (only involving the nominal problem s data) for the calmness moduli. Different applications are presented. Specifically, two of them deal with measures of the convergence of certain algorithms: a regularization method for linear programs with complementarity constraints and interior point methods. Finally, we also present an application to the computation of calmness moduli in robust optimization. A bridge between bilevel programs and Nash games Simone Sagratella Department of Computer, Control, and Management Engineering Antonio Ruberti Sapienza University of Rome sagratella@dis.uniroma1.it Abstract. We study connections between optimistic bilevel programs and generalized Nash equilibrium problems. Namely, in a multi-agent framework, we consider both the vertical case, in which there is a leader that can anticipate the strategies of a follower (bilevel problem), and the horizontal case, in which both the agents must decide their strategies simultaneously (Nash game). We propose a new uneven horizontal case in which the agents play a Nash game that incorporates some taste of hierarchy. We define classes of problems for which solutions of the bilevel problem can be computed by finding equilibria of the uneven Nash game. Our study provides the theoretical backbone and the main ideas underlying some useful novel algorithmic developments. Exploiting these results, we tackle some classes of multi-leader common-follower games stemmed from electricity markets.

3 Feasible roundings for granular optimization Oliver Stein Institute of Operations Research Karlsruhe Institute of Technology Abstract. We introduce a new technique to generate good feasible points of mixedinteger nonlinear optimization problems which are granular in a certain sense. Finding a feasible point is known to be NP hard even for mixed-integer linear problems, so that many construction heuristics have been developed. We show, on the other hand, that efficiently solving certain purely continuous optimization problems and rounding their optimal points leads to feasible points of the original mixed-integer problem, as long as the original problem is granular. For the objective function values of the generated feasible points we present computable a-priori and a-posteriori bounds on the deviation from the optimal value, as well as efficiently computable certificates for the granularity of a given problem. Computational examples for several problems from the MIPLIB libraries illustrate that our method is able to outperform standard software. A post processing step to our approach, using integer line search, further improves the results. Generalized monotonicity in the finite dimensional setting. Applications Cornel Pintea Department of Mathematics Babeş - Bolyai University Cluj cpintea@math.ubbcluj.ro Abstract. It is more or less obvious that the inverse images of continuous Minty- Browder monotone operators on Hilbert spaces are convex. Also the local Minty-Browder monotonicity of operators defined on Hilbert spaces is equivalent with its global counterpart. This shows, via some classical characterizations of convexity, that the local convexity of smooth enough real-valued functions is equivalent with its global counterpart. Analytic requirements on a given operators which ensure both the Minty-Browder monotonicity and the local injectivity of the opertor, ensure its global injectivity. We enlarge the class of Minty-Browder monotone operators to the classes of δ- and h- monotone operators and study the connneectedness of the inverse images of such operators in the finite dimensional setting. Although a weak connectedness version of the inverse images is proved for h-monotone operators, it is good enough to ensure the global injectivity when is combined with the local injectivity.

4 Composition set-valued mappings: metric (sub)regularity and fixed points Radu Strugariu Department of Mathematics Gheorghe Asachi Technical University of Iaşi Abstract. We underline the importance of the parametric regularity and subregularity properties of set-valued mappings, defined with respect to fixed sets. We show that the subregularity appears naturally for some very simple mappings which play an important role in the theory of metric regularity. The main result, which goes back to the Milyutin principle, concernes the preservation of metric subregularity at generalized compositions. As applications, we present, on purely metric setting, several fixed point assertions for set-valued mappings in local and global frameworks. On Bregman-type distances for convex functions and maximally monotone operators Juan Enrique Martínez-Legaz Departament d Economia i d Historia Economica Universitat Autonoma de Barcelona JuanEnrique.Martinez.Legaz@uab.cat Abstract. In this joint paper with Regina S. Burachik, given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.

5 Directional approach to regularity of mappings and applications to optimization Marius Durea Faculty of Mathematics Alexandru Ioan Cuza University of Iaşi Abstract. The classical notions of regularity for set-valued maps do not necessarily make a distinction between minima and maxima. This aspect motivates us to introduce and study, starting from a special form of a minimal time function, three directional regularity concepts for set-valued mappings acting between normed vector spaces. These concepts are directional metric regularity, directional linear openness, and directional Aubin continuity, and take into account sets of directions in both input and output spaces. In order to link these concepts to vector optimization problems we prove a generalized form of the Ekeland Variational Principle and then we derive necessary conditions and sufficient conditions for directional regularities in terms of Fréchet generalized differentiation. Finally, on the basis of the concepts and facts developed before, we deal with necessary and sufficient optimality conditions for minimality in vector optimization problems governed by set-valued mappings. Ekeland s principle for equilibrium problems Massimiliano Giuli Department of Information Engineering, Computer Science and Mathematics University of L Aquila massimiliano.giuli@univaq.it Abstract. The triangle inequality property is a basic assumption for the Ekeland s variational principle for equilibrium problems to hold. This property has been used to study the existence of solutions of equilibrium problems and arbitrary systems of equilibrium problems which do not involve any convexity concept, neither for the domain nor for the bifunction. The main aim of this talk is to extend the the Ekeland s variational principle to a larger class of bifunctions. We replace the triangle inequality property by a condition, that is shown to be equivalent to a certain monotonicity of the bifunction. Moreover we prove the existence of solutions in compact and noncompact settings. The proofs of existence don t rely on the Ekeland s variational principle but they are performed using only elementary results. This fact allows us to remove the metric structure on the topological space and additional technical conditions.

6 On primal regularity estimates Radek Cibulka Department of Mathematics, Faculty of Applied Sciences University of West Bohemia in Pilsen Abstract. We survey some regularity statements in variational analysis. We focus on theorems guaranteeing the openness at a linear rate of a mapping around the reference point, which is known to be equivalent to metric regularity. In particular we are going to discuss techniques used in the proofs of such statements. We start with regularity criterion by A.D. Ioffe, which can be traced back to the work of M. Fabian and D. Preiss, that substitutes complicated iterative procedures. When time permits we intend to present some applications of theoretical results. The lecture is based on papers co-authored by M. Fabian and A.D. Ioffe. Ekeland s principle for cyclically antimonotone vector bifunctions and weak vector equilibrium problems Mihaela Miholca Department of Mathematics Technical University Cluj-Napoca mihaela.miholca@yahoo.com Abstract. In this paper, we extend the notion of cyclic antimonotonicity (known for scalar bifunctions) to the vector case, in order to obtain some results on Ekeland s principle for vector equilibrium problems. We characterize the cyclic antimonotonicity in terms of a suitable approximation from below of the vector bifunction, which alows us to avoid the demanding triangle inequality property, usually required in the literature, when dealing with Ekeland s principle for bifunctions. Furthermore, a result for weak vector equilibria in the absence of convexity assumptions is given, without passing through the existence of approximate solutions.

7 On the use of the optimal level solution method in generating the efficient frontier of a class of bicriteria programs Riccardo Cambini Department of Economics and Management University of Pisa riccardo.cambini@unipi.it Abstract. First a very brief survey on the optimal level solutions method is provided and the related algorithm scheme is described in details. Then a particular class of bicriteria maximization problems over a compact polyhedron is considered and studied. Specifically speaking, the first component of the objective function is the ratio of powers of affine functions while the second component is linear. Several theoretical properties are provided and stated, such as the pseudoconcavity of the first fractional objective function, the connectedness and compactness of the efficient frontier and of the set of efficient points. Finally, an algorithm for generating the whole efficient frontier is proposed and a numerical example to clarify the use of the algorithm is presented. The talk is based on joint research with L. Carosi.

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