# Juan Enrique Martínez-LegazAutonomous University of Barcelona | UAB · Departamento de Economía y de Historia Económica

Juan Enrique Martínez-Legaz

PhD Mathematics

## About

184

Publications

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2,059

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Citations since 2016

Introduction

Additional affiliations

October 1990 - present

October 1977 - September 1990

## Publications

Publications (184)

We correct an error in Martínez-Legaz et al. (On farthest Bregman Voronoi cells. Optimization. 2022;71:937–947,Theorem 4.1).

We investigate convergence properties of Bregman distances induced by convex representations of maximally monotone operators. We also introduce and study the projection mappings associated with such distances.

We establish a general duality theorem in a generalized conjugacy framework, which generalizes a classical result on the minimization of a convex function over a closed convex cone. Our theorem yields two quasiconvex duality schemes; one of them is of the surrogate duality type and is applicable to problems having an evenly quasiconvex objective fu...

The production correspondence associated with a technology maps every input vector into the set of output vectors that may be obtained by means of those inputs. The cost function induced by a production correspondence assigns to every pair consisting of a vector of input prices and an output vector the smallest possible cost that has to be paid for...

ACT
Let g be a strictly convex function on an evenly convex set X ⊂
Rn with nonempty interior. Assuming that g is differentiable
on int X, we consider the Bregman distance Dg associated with
g. Given a set T ⊂ Rn, whose elements are called sites, and
a particular site s, the farthest g-Bregman Voronoi cell of s,
denoted by F
g
T (s), consists of al...

The main goal in this paper is to devise an approach to explicitly calculate the constant in the Hoffman’s error bound for (not necessarily convex) inequality systems defining convex sets. We give a constructive proof of the Hoffman’s error bound and show that we can use our method to calculate the constant at least in simple cases.

We characterize the closed convex subsets of Rn which have open or closed Gauss ranges. Some special attention is paid to epigraphs of lower semicontinuous convex functions.

We present necessary and sufficient optimality conditions for the minimization of pseudoconvex functions over convex intersections of non necessarily convex sets. To this aim, we use the notion of local normal cone to a closed set at a point, due to Linh and Penot (SIAM J Optim 17:500–510, 2006). The technique we use to obtain the optimality condit...

We observe that a quasiconvex function which is evenly quasiconvex at a point is not necessarily Greenberg–Pierskalla (briefly, G-P) subdifferentiable at that point, but we prove that a quasiconvex function which is upper semicontinuous on the segments of its effective domain is G-P subdifferentiable on the relative interior of this effective domai...

The classic Voronoi cells can be generalized to a higher order version by considering the cells of points for which a given k-element subset of the set of sites consists of the k closest sites. We study the structure of the k-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher order Voronoi cells...

Given an arbitrary set T in the Euclidean space Rn, whose elements are called sites, and a particular site s, the farthest Voronoi cell of s, denoted by FT(s), consists of all points which are farther from s than from any other site. In this paper we study farthest Voronoi cells and diagrams corresponding to arbitrary (possibly infinite) sets. More...

In 1956 Marguerite Frank and Paul Wolfe proved that a quadratic function which is bounded below on a polyhedron P attains its infimum on P. In this work we search for larger classes of sets F with this Frank-and-Wolfe property. We establish the existence of non-polyhedral Frank-and-Wolfe sets, obtain internal characterizations by way of asymptotic...

We introduce and study a notion of co-radiantness for set-valued mappings between nonnegative orthants of Euclidean spaces. We analyze them from an abstract convexity perspective. Our main results consist in representations, in terms of intersections of graphs, of the increasing co-radiant mappings that take closed normal values, by means of elemen...

We introduce a new asymptotic function, which is mainly adapted to quasiconvex functions. We establish several properties and calculus rules for this concept and compare it to previous notions of generalized asymptotic functions. Finally, we apply our new definition to quasiconvex optimization problems: we characterize the boundedness of the functi...

The classic Voronoi cells can be generalized to a higher-order version by considering the cells of points for which a given $k$-element subset of the set of sites consists of the $k$ closest sites. We study the structure of the $k$-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher-order Voronoi...

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...

In 1956 Marguerite Frank and Paul Wolfe proved that a quadratic function which is bounded below on a polyhedron $P$ attains its infimum on $P$. In this work we search for larger classes of sets $F$ with this Frank-and-Wolfe property. We establish the existence of non-polyhedral Frank-and-Wolfe sets, obtain internal characterizations by way of asymp...

We introduce a (possibly infinite) collection of mutually dual nonconvex optimization problems, which share a common optimal value, and give a characterization of their global optimal solutions. As immediate consequences of our general multiduality principle, we obtain Toland–Singer duality theorem as well as an analogous result involving generaliz...

The classical Frank and Wolfe theorem states that a quadratic function which is bounded below on a convex polyhedron P attains its inﬁmum on P. We investigate whether more general classes of convex sets F can be identiﬁed which have this Frank-and-Wolfe property. We show that the intrinsic characterizations of Frank-and-Wolfe sets hinge on asymptot...

We introduce and study the class of weakly Motzkin predecomposable sets, which are those sets in ℝn that can be expressed as the Minkowski sum of a bounded convex set and a convex cone, none of them being necessarily closed. This class contains that of Motzkin predecomposable sets, for which the bounded components are compact, which in turn contain...

This Special Issue of the Journal ‘Optimization’ is dedicated to the ‘International Seminar on Optimization and Related Areas’ (ISORA), which was held at the Instituto de Matemática y Ciencias Afines (IMCA) in Lima (Peru) on October 5–9, 2015.
This series of seminars begun in 1993, as a tribute to the memory of Eugen Blum, a Swiss mathematician wh...

The paper provides a new subdifferential characterization for Motzkin decomposable (convex) functions. This characterization leads to diverse stability properties for such a decomposability for operations like addition and composition.

We analyze a sequential decision making process, in which at each step the decision is made in two stages. In the first stage a partially optimal action is chosen, which allows the decision maker to learn how to improve it under the new environment. We show how inertia (cost of changing) may lead the process to converge to a routine where no furthe...

Given an arbitrary set T in the Euclidean space whose elements are called sites, and a particular site s, the Voronoi cell of s, denoted by VT(s), consists of all points closer to s than to any other site. The Voronoi mapping of s, denoted by ψs, associates to each set T ∋ s the Voronoi cell VT(s) of s w.r.t. T. These Voronoi cells are solution set...

In this paper we provide several characterizations of Minkowski sets, i.e. closed, possibly unbounded, convex sets which are representable as the convex hulls of their sets of extreme points. The equality between the relative boundary of a closed convex set containing no lines and its Pareto-like associated set ensures the Minkowski property of the...

We present a generalization of the strong Fitzpatrick inequality in the context of reflexive Banach spaces, involving a twisted bigger conjugate function. We also introduce a related family of gap functions for maximal monotone inclusion problems.

We consider the problem of locating a facility amongst a given collection of attraction and repulsion points. The goal is to find a location x in the Euclidean space ℝⁿ for a facility, such that the difference between the weighted sum of distances from x to the attraction points and the weighted sum of distances to the repulsion points is minimized...

It is known that complementarity functions play an important role in dealing with complementarity problems. The most well known complementarity problem is the symmetric cone complementarity problem (SCCP) which includes nonlinear complementarity problem (NCP), semidefinite complementarity problem (SDCP), and second-order cone complementarity proble...

We introduce two Moreau conjugacies for extended real-valued functions
h on a separated locally convex space. In the first scheme, the biconjugate of h coincides
with its closed convex hull, whereas, for the second scheme, the biconjugate
of h is the evenly convex hull of h. In both cases, the biconjugate coincides with the
supremum of the minorant...

We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper fun...

We present necessary and sufficient conditions for a monotone bifunction to be maximally monotone, based on a recent characterization of maximally monotone operators. These conditions state the existence of solutions to equilibrium problems obtained by perturbing the defining bifunction in a suitable way.

We present two generalized conjugation schemes for lower semi-continuous functions defined on a real Banach space whose norm is Fréchet differentiable off the origin, and sketch their applications to optimization duality theory. Both approaches are based upon a new characterization of lower semi-continuous functions as pointwise suprema of a specia...

We introduce a subfamily of additive enlargements of a maximally monotone
operator. Our definition is inspired by the early work of Simon Fitzpatrick.
These enlargements constitute a subfamily of the family of enlargements
introduced by Svaiter. When the operator under consideration is the
subdifferential of a convex lower semicontinuous proper fun...

In the present note we point out and rectify an incorrect result in Corollary 2.5 of Martinez-Legaz, Some generalizations of Rockafellar's surjectivity theorem (Pac. J. Optim., 4 (2008), no. 3, 527-535). The revision of that result provides further insight on the extent to which the surjectivity theorem can be generalized and on the role played by...

In this work, we achieve a complete characterization of the existence of a saddle value, for bifunctions which are convex, proper, and lower semi continuous in their first argument, by considering new suitably defined notions of special directions of recession. As special cases, we obtain some recent results of Lagrangian duality theory on zero dua...

This article surveys the main contributions of K.-H. Elster to the theory of generalized conjugate functions and its applications to duality in nonconvex optimization.

We introduce and study the family of sets in a finite dimensional Euclidean space which can be written as the Minkowski sum of a compact and convex set and a convex cone (not necessarily closed). We establish several properties of the class of such sets, called Motzkin predecomposable, some of which hold also for the class of Motzkin decomposable s...

We obtain a simple integration formula for the Fenchel subdifferentials on Euclidean spaces and analyze some of its consequences. For functions defined on locally convex spaces, we present a similar result in terms of 𝜖-subdifferentials.

In this article, we extend the definition of γ-active constraints for linear semi-infinite programming to a definition applicable to convex semi-infinite programming, by two approaches. The first approach entails the use of the subdifferentials of the convex constraints at a point, while the second approach is based on the linearization of the conv...

We present necessary and sufficient optimality conditions for the minimization of pseudoconvex functions over convex sets defined by non necessarily convex functions, in terms of tangential subdifferentials. Our main result unifies a recent KKT type theorem obtained by Lasserre for differentiable functions with a nonsmooth version due to Dutta and...

We show that suitable restatements of the classical Weierstrass extreme value theorem give necessary and sufficient conditions for the existence of a global minimum and of both a global minimum and a global maximum.

Fitzpatrick proved that maximal monotone operators in topological vector spaces are representable by lower semi-continuous convex functions. A monotone operator is representable if it can be represented by a lower-semicontinuous convex function. The smallest representable extension of a monotone operator is its representable closure. The intersecti...

We give a necessary and sufficient condition for a difference of convex (DC, for short) functions, defined on a normed space, to be Lipschitz continuous. Our criterion relies on the intersection of the ε-subdifferentials of the involved functions.

We give sufficient conditions for the infimum of a quasiconvex vector function f over an intersection to agree with the supremum of the infima of f over the R
i
’s.

We analyze the least increment function, a convex function of n variables associated to an n-person cooperative game. Another convex representation of cooperative games, the indirect function, has previously been studied. At every point the least increment function is greater than or equal to the indirect function, and both functions coincide in th...

We give a necessary and sufficient condition for a difference of convex (DC,
for short) functions, defined on a locally convex space, to be Lipschitz
continuous. Our criterion relies on the intersections of the
"epsilon-subdifferentials of the involved functions.

We introduce two new families of properties on convex sets of R n, in order to establish new theorems regarding open and closed separation of a convex set from any outside point by linear operators from R n to R m, in the sense of the lexicographical order of R m, for each m ∈ {1,..., n}. We thus obtain lexicographical extensions of well known sepa...

A characterization of d.c. functions f:Ω→R in terms of the quasidifferentials of f is obtained, where Ω is an open convex set in a real Banach space. Recall that f is called d.c. (difference of convex) if it can be represented as a difference of two finite convex functions. The relation of the obtained results with known characterizations is discus...

In this paper we discuss symmetrically self-dual spaces, which are simply
real vector spaces with a symmetric bilinear form. Certain subsets of the space
will be called q-positive, where q is the quadratic form induced by the
original bilinear form. The notion of q-positivity generalizes the classical
notion of the monotonicity of a subset of a pro...

A weighted average worth per capita formula is presented for any semivalue of a TU game. Further, this formula is used to derive a characterisation of the class of games with the property that a given semivalue belongs to the power core of the game, by means of a linear system of inequalities. It is shown that for the Shapley value, the only effici...

A subset of a locally convex space is called e-convex if it is the intersection of a family of open halfspaces. An extended real-valued function on such a space is called e-convex if its epigraph is e-convex. In this paper we introduce a suitable support function for e-convex sets as well as a conjugation scheme for e-convex functions.

We give sufficient conditions for the infimum of a quasiconvex function f over the intersection i∈I Ri to agree with the supremum of the infima of f over the Ri's. We apply these results to the distance function in a normed space.

A generalization of Rockafellar’s surjectivity theorem was provided in [J.-E. Martínez-Legaz, Pac. J. Optim. 4, No. 3, 527–535 (2008; Zbl 1198.47071)], replacing the duality mapping by any maximal monotone operator having a finite-valued Fitzpatrick function. The present paper extends this result to the nonreflexive setting for maximal monotone ope...

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous
result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T,S which share the same convex-like domain D coincide whenever T(x)ÇS(x) ¹ ÆT(x)\cap S(x)\not=\emptyset for ev...

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. The main result in this paper establishes that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded....

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These ch...

Many papers on both scalar and multiobjective optimization prob-lems use the assumption that the objective and constraint functions are invex with respect to the same function η. In this note we characterize the finite families of functions for which this condition holds.

There are infinitely many ways of representing a d.c. function as a difference of convex functions. In this paper we analyze how the computational efficiency of a d.c. optimization algorithm depends on the representation we choose for the objective function, and we address the problem of characterizing and obtaining a computationally optimal repres...

In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity
result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a
generalization of Fenchel’s duality theorem in the framework of abstract convexity and give some...

In his recent book [“From Hahn–Banach to monotonicity” (Lecture Notes in Mathematics 1693; Berlin: Springer) ( 2 2008; Zbl 1131.47050)], St. Simons has introduced the notion of SSD space to provide an abstract algebraic framework for the study of monotonicity. Graphs of (maximal) monotone operators appear to be (maximally) q-positive sets in suitab...

We prove some generalizations of Rockafellar's surjectivity theorem and related results, which consist in replacing the duality mapping by another maximal monotone operator satisfying suitable conditions.

We present a general tabu search iterative algorithm to solve abstract problems on metric spaces. At each iteration, if the current solution turns out to be unacceptable then a neighborhood of unacceptable solutions is determined and excluded for further exploration, in such a way that, under mild assumptions, an acceptable solution is asymptotical...

We compare the Fenchel-Moreau second conjugates associated to an arbitrary coupling function p X x W (R) over bar = [-infinity +infinity] between two sets X and W with the second level set conjugates associated to the same coupling. For a coupling phi: R-n x R-n -> R = (-infinity +infinity) that is additively homogeneous in one (or both) of the var...

We prove that if (X,d) is a metric space, C is a closed subset of X and x∈X, then the distance of x to R∩S agrees with the maximum of the distances of x to R and S, for every closed subsets R,S⊂C such that C=R∪S, if and only if C is x-boundedly connected.

Given a convex function f defined for positive real variables, the so-called Csiszár f-divergence is a function If defined for two n-dimensional probability vectors p=(p1,…,pn) and q=(q1,…,qn) as If(p,q):=∑i=1nqif(piqi). For this generalized measure of entropy to have distance-like properties, especially symmetry, it is necessary for f to satisfy t...

It is well known that the Fitzpatrick function of a maximal monotone operator is minimal in the class of convex functions bounded below by the duality product. Our main result establishes that, in the setting of reflexive Banach spaces, the converse also holds; that is, every such minimal function is the Fitzpatrick function of some maximal monoton...

This paper determines the precise connection between the curvature properties of an objective function and the ray-curvature properties of its dual. When the objective function is interpreted as a Bernoulli or cardinal utility function, our results characterize the relationship between an agent’s attitude towards income risks and her attitude towar...

We characterize the l.s.c. convex functionsf:(0,+∞)→ℝ ¯ that satisfy f(x)=xf(1/x) in terms of their Fenchel conjugates. We also introduce a suitable duality theory for such functions and characterize the associated support sets.

We consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli. We establish a new dual
formulation for this equilibrium problem using the classical Fenchel conjugation, thus generalizing the classical convex duality
theory for optimization problems.

We study an approximation method for sets and functions which erases comers but keeps smooth parts. Basic properties of such a method are pointed out in a general and simple way. Several convergence results are provided, essentially in the framework of variational analysis.

We give simpler proofs of some known conjugation formulas and subdifferential formulas of convex analysis and we give some new interconnections between them, showing how each of them follows from the others.

Several characterizations of convexity for totally balanced games are presented. As a preliminary result, it is first shown
that the core of any subgame of a nonnegative totally balanced game can be easily obtained from the maximum average value
(MAV) function of the game. This result is then used to get a characterization of convex games in terms...

This article deals with systems of infinitely many inequalities involving functions that are positively homogeneous over a nonempty convex cone of the Euclidean space. Generalized convex conjugation theory is applied to derive a Farkas-type and a Gale-type theorem for this kind of systems. These results are particularized for linear and min-type in...

We study increasing quasiconcave functions which are co-radiant. Such functions have frequently been employed in microeconomic analysis. The study is carried out in the contemporary framework of abstract convexity and abstract concavity. Various properties of these functions are derived. In particular we identify a small natural infimal generator o...

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the o...

This article presents an approach to generalized convex duality theory based on Fenchel-Moreau conjugations; in particular,
it discusses quasiconvex conjugation and duality in detail. It also describes the related topic of microeconomics duality
and analyzes the monotonicity of demand functions.

We give an elementary proof of a theorem of Brickman, which establishes the convexity of the image of the unit sphere of a space with dimension at least three under a vector mapping into R2 whose component functions are quadratic forms. We also discuss some consequences of this theorem regarding the range of that mapping.