On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

A generalized mixed vector variational-like inequality problem

In this paper, we introduce relaxed η-α-P-monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of η-complete semicontinuity and η-strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying the Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.     

The Dirichlet problem for Baire-one functions

Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.     

Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Minimum L1-Distance Projection onto the Boundary of a Convex Set: Simple Characterization

We show that the minimum distance projection in the L ₁-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec (Ref. 1).     

Generalized Vector Variational Inequalities

In this paper, we introduce a generalized vector variational inequality problem (GVVIP) which extends and unifies vector variational inequalities as well as classical variational inequalities in the literature. The concepts of generalized C-pseudomonotone and generalized hemicontinuous operators are introduced. Some existence results for GVVIP are obtained with the assumptions of generalized C-pseudomonotonicity and generalized hemicontinuity. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained.     

Duality for <Emphasis Type="Italic">ε</Emphasis>-Variational Inequality

In this paper, following the method in the proof of the composition duality principle due to Robinson and using some basic properties of the ε-subdifferential and the conjugate function of a convex function, we establish duality results for an ε-variational inequality problem. Then, we give Fenchel duality results for the ε-optimal solution of an unconstrained convex optimization problem. Moreover, we present an example to illustrate our Fenchel duality results for the ε-optimal solutions. The authors thank the referees for valuable suggestions and comments. This work was supported by Grant No. R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2^Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ^* ε K such that f g(χ^*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.     

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

Equilibrium Problems with Applications to Eigenvalue Problems

In this paper, we consider equilibrium problems and introduce the concept of (S)₊ condition for bifunctions. Existence results for equilibrium problems with the (S)₊ condition are derived. As special cases, we obtain several existence results for the generalized nonlinear variational inequality studied by Ding and Tarafdar (Ref. 1) and the generalized variational inequality studied by Cubiotti and Yao (Ref. 2). Finally, applications to a class of eigenvalue problems are given.     

ProperD-solutions of multiobjective programming problems with set functions

A special class of solutions for multiobjective programming problems with set functions is considered. A subset of nondominated solutions, called properD-solution set, with respect to a given domination structure is characterized under two situations, with and without inequality constraints.The authors greatly appreciate valuable comments received from the referees.     

An Analog of the Krein--Mil'man Theorem for Strongly Convex Hulls in Hilbert Space

We prove the following theorem: in Hilbert space a closed bounded set is contained in the strongly convex R-hull of its R-strong extreme points. R-strong extreme points are a subset of the set of extreme points (it may happen that these two sets do not coincide); the strongly convex R-hull of a set contains the closure of the convex hull of the set.     

Optimality Conditions for Metrically Consistent Approximate Solutions in Vector Optimization

In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ε-efficient concept. Several properties of the ε-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining the same precision in the vector problem as in the scalarization. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), Project MTM2006-02629 and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for helpful comments and suggestions.     

Typical convex program is very well posed

In this paper we consider the collection of convex programming problems with inequality and equality constraints, in which every problem of the collection is obtained by linear perturbations of the cost function and right-hand side perturbation of the constraints, while the ``core' cost function and the left-hand side constraint functions are kept fixed. The main result shows that the set of the problems which are not well-posed is σ-porous in a certain strong sense. Our results concern both the infinite and finite dimensional case. In the last case the conclusions are significantly sharper. Research of A. Ioffe was supported in part by the US-Israel Binational Fund under the grant 2000157. research of R. E. Lucchetti was partially supported by Ministero dell'Istruzione, dell'Università e della Ricerca (COFIN 2001).     

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.      

Strict feasibility of pseudomonotone set-valued variational inequalities

In this article, we use degree theory developed in Kien et al. [B.T. Kien, M.-M. Wong, N.C. Wong, and J.C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res. 193 (2009), pp. 12–22.] to prove a result on the existence of solutions to set-valued variational inequality under a weak coercivity condition, provided that the set-valued mapping is upper semicontinuous with nonempty compact convex values. If the set-valued mapping is pseudomonotone in the sense of Karamardian and upper semicontinuous with nonempty compact convex values, it is shown that the set-valued variational inequality is strictly feasible if and only if its solution set is nonempty and bounded.     

Geometric Tomography of Convex Cones

The parallel X-ray of a convex set K⊂ℝ ⁿ in a direction u is the function that associates to each line l, parallel to u, the length of K∩l. The problem of finding a set of directions such that the corresponding X-rays distinguish any two convex bodies has been widely studied in geometric tomography. In this paper we are interested in the restriction of this problem to convex cones, and we are motivated by some applications of this case to the covariogram problem. We prove that the determination of a cone by parallel X-rays is equivalent to the determination of its sections from a different type of tomographic data (namely, point X-rays of a suitable order). We prove some new results for the corresponding problem which imply, for instance, that convex polyhedral cones in ℝ³ are determined by parallel X-rays in certain sets of two or three directions. The obtained results are optimal.     

<Emphasis Type="Italic">ε</Emphasis>-Optimality and <Emphasis Type="Italic">ε</Emphasis>-Lagrangian Duality for a Nonconvex Programming Problem with an Infinite Number of Constraints

In this paper, ε-optimality conditions are given for a nonconvex programming problem which has an infinite number of constraints. The objective function and the constraint functions are supposed to be locally Lipschitz on a Banach space. In a first part, we introduce the concept of regular ε-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to ε and are obtained by weakening the classical complementarity conditions. Furthermore, they are satisfied without assuming any constraint qualification. Then, we prove that these conditions are also sufficient for ε-optimality when the constraints are convex and the objective function is ε-semiconvex. In a second part, we define quasisaddlepoints associated with an ε-Lagrangian functional and we investigate their relationships with the generalized KKT conditions. In particular, we formulate a Wolfe-type dual problem which allows us to present ε-duality theorems and relationships between the KKT conditions and regular ε-solutions for the dual. Finally, we apply these results to two important infinite programming problems: the cone-constrained convex problem and the semidefinite programming problem.     

Lipschitz continuity ofV-subdifferentials of convex operators

It is proved that theV-subdifferential of a convex operator is locally Lipschitzian on the set of points at which it is continuous and subdifferentiable.Proposition 2.2 was originally stated for Holder continuity ofV-subdifferentials. The author would like to thank J. P. Penot for a very helpful suggestion which led to the present form of this proposition.