Constantin Zălinescu

Subdifferential Calculus Rules in Convex Analysis: A Unifying Approach Via Pointwise Supremum Functions

We provide a rule to calculate the subdifferential set of the pointwise supremum of an arbitrary family of convex functions defined on a real locally convex topological vector space. Our formula is given exclusively in terms of the data functions and does not require any assumption either on the index set on which the supremum is taken or on the involved functions. Some other calculus rules, namely chain rule formulas of standard type, are obtained from our main result via new and direct proofs.

Set-valued Optimization

Introduction.- Order Relations and Ordering Cones.- Continuity and Differentiability.- Tangent Cones and Tangent Sets.- Nonconvex Separation Theorems.- Hahn-Banach Type Theorems.- Hahn-Banach Type Theorems.- Conjugates and Subdifferentials.- Duality.- Existence Results for Minimal Points.- Ekeland Variational Principle.- Derivatives and Epiderivatives of Set-valued Maps.- Optimality Conditions in Set-valued Optimization.- Sensitivity Analysis in Set-valued Optimization and Vector Variational Inequalities.- Numerical Methods for Solving Set-valued Optimization Problems.- Applications.

Lipschitz properties of the scalarization function and applications

The scalarization functions were used in vector optimization for a long period. Similar functions were introduced and used in economics under the name of shortage function or in mathematical finance under the name of (convex or coherent) measures of risk. The main aim of this article is to study Lipschitz continuity properties of such functions and to give some applications for deriving necessary optimality conditions for vector optimization problems using the Mordukhovich subdifferential.

Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems

In this paper, we study conditioning problems for convex and nonconvex functions defined on normed linear spaces. We extend the notion of upper Lipschitzness for multivalued functions introduced by Robinson, and show that this concept ensures local conditioning in the nonconvex case via an abstract subdifferential; in the convex case, we obtain complete characterizations of global conditioning in terms of an extension of the upper-Lipschitz property.

Stability for a Class of Nonlinear Optimization Problems and Applications

The aim of this chapter is to give a unified approach to some problems in nonlinear optimization using asymptotic cones, recession functions and asymptotically compact sets. Thus we establish a stability result for a class of nonconvex programming problems which turns out to be equivalent to Dedieu’s criterion for the closedeness of the image of a closed set by a multifunction. Also we obtain a formula for the recession function of the marginal function for the first time. This formula seems to be important and new also in the finite dimensional case. The convex version of the stability result is used to reobtain formulae for the conjugates, ϵ-subdifferentials and recession functions of some convex functions, results which are comparable with those of McLinden. It is also shown that in some cases one can perturbe the objective function of a family of convex problems such that the resulting problems have optimal solutions; the behaviour of the values of these perturbed problems and their solutions is also investigated. Another result establishes the relationship between conically compact sets introduced by Isac and Thera and asymptotic cones.

comparison of existence results for efficient points

Existence results of maximal points with respect to general binary relations were stated by Hazen and Morin (Ref. 1) and by Gajek and Zagrodny (Ref. 2). In this paper, we point out that the natural framework for this problem is that of transitive and reflexive relations (preorders). The aim of this paper is to discuss existence results for maximal points with respect to general transitive relations in such a way that, when considering them for preorders defined by convex cones, we are able to recover most known existence results for efficient points; the quasi-totality of them, with their (short) proofs, is presented, too.

Scalar convergence of convex sets

Abstract We study the scalar convergence of sequences of convex sets defined by lim(sup ϑ ( A n )) = ϑ ( A ) for all ϑ in dual space. New properties are given. Relationship between scalar convergence and other known convergences is examined. Two natural distinct uniformities on nonvoid closed convex sets define the scalar convergence. The associated topology is the weakest such that A → d ( A , H ) is continuous for each hyperplane H .

Vector Variational Principles for Set-Valued Functions

Deriving existence results and necessary conditions for approximate solutions of nonlinear optimization problems under week assumptions is an interesting and modern field in optimization theory. It is of interest to show corresponding results for optimization problems without any convexity and compactness assumptions. Ekeland’s variational principle is a very deep assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighborhood of an approximate solution of the original problem. The importance of Ekeland’s variational principle in nonlinear analysis is well known. Especially, this assertion is very useful for deriving necessary conditions under certain differentiability assumptions. In optimal control Ekeland’s principle can be used in order to prove an e-maximum principle in the sense of Pontryagin and in approximation theory for deriving e-Kolmogorov conditions.

Bounded (Hausdorff) Convergence: Basic Facts and Applications

We present a survey of some uses of a remarkable convergence on families of sets or functions. We evoke some of its applications and stress some calculus rules. The main novelty lies in the use of a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. This notion yields criteria for the study of boundedness properties.

A New Proof of the Maximal Monotonicity of the Sum Using the Fitzpatrick Function

A classical result of Rockafellar [6] states that the sum of two maximal monotone multifunctions on a reflexive Banach space is maximal monotone when the interior of the domain of one of them intersects the domain of the other. There are several proofs of this important result. The original proof of Rockafellar [6] uses some results of Browder [1]; put together, the proof is quite involved. The proof furnished in the recent book by Simons [7] uses minimax theorems. We give another (short) proof of this result in reflexive spaces using the Fitzpatrick function associated to a monotone multifunction and a result on the conjugate of the sum of two convex functions. We recall first some notation and results related to convex analysis. For this propose, consider a separated locally convex space E and E∗ its topological dual; we get so the dual system (E, E∗, 〈·, ·〉), where 〈x, x∗〉 := x∗(x) for x ∈ E and x∗ ∈ E∗. We endow E∗ with the weak-star topology w∗ := σ(E∗, E), and so the topological dual of E∗ is identified with E. As usual, having a subset A of E, we use the notation intA, cl A or A, co A, coA and aff A for the interior, closure, convex hull, closed convex hull, and the affine hull of A, respectively; moreover, A and A denote the core (algebraic interior) and the intrinsic core of A, while A is A when aff A is closed and A is the empty set otherwise. The domain, the epigraph and the conjugate of f : E → R are introduced by