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By a scalarization method and properties of semi-algebraic sets, it is proved that both the Pareto solution set and the weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. Consequences of the results for vector optimization problems are discussed in details. The results of this paper solve in the affirmative some open questions for the case of general problems without requiring monotonicity of the operators involved. [ABSTRACT FROM AUTHOR]

The problem of finding minima of weakly sequentially lower semicontinuous functions on reflexive Banach spaces is studied by means of convex and nonconvex subdifferentials. Finding a descent direction for a non-stationary point is a question of importance for many optimization algorithms. The existence or non-existence of such a direction is clarified through several theorems and a series of selective examples. For the general problem, a notion called radius of descent is proposed and shown to be useful for the analysis related to descent directions. [ABSTRACT FROM AUTHOR]

In this paper, without using any regularity assumptions, we derive a new exact formula for computing the Fréchet coderivative and an exact formula for the Mordukhovich coderivative of normal cone mappings to perturbed polyhedral convex sets. Our development establishes generalizations and complements of the existing results on the topic. An example to illustrate formulae is given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the problems under consideration reduce to linear semi-infinite optimization. Examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper deals with the generalized Clarke epiderivative of the extremum (or efficient point) multifunction in parametric vector optimization problems. The formulas for computing and/or estimating the generalized Clarke epiderivative of this extremum multifunction are given in terms of the Clarke tangent cone to the graph of a multifunction or the constraint mapping and/or the Fréchet derivative of the objective function. An application to semi-infinite programming is given. [ABSTRACT FROM AUTHOR]

Abstract: We obtain some point-based sufficient conditions for the metric regularity in Robinson’s sense of implicit multifunctions in a finite-dimensional setting. The new implicit function theorem (which is very different from the preceding results of Ledyaev and Zhu [Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238], Ngai and Théra [H.V. Ngai, M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12 (2004) 195–223], Lee, Tam and Yen [G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22]) can be used for analyzing parametric constraint systems as well as parametric variational systems. Our main tools are the concept of normal coderivative due to Mordukhovich and the corresponding theory of generalized differentiation. [Copyright &y& Elsevier]

In this paper, we investigate the strong convergence of an inexact proximalpoint algorithm. It is known that the proximal-point algorithm converges weakly to a solution of a maximal monotone operator, but fails to converge strongly. Solodov and Svaiter (Math. Program. 87:189–202, 2000) introduced a new proximal-type algorithm to generate a strongly convergent sequence and established a convergence result in Hilbert space. Subsequently, Kamimura and Takahashi (SIAM J. Optim. 13:938–945, 2003) extended the Solodov and Svaiter result to the setting of uniformly convex and uniformly smooth Banach space. On the other hand, Rockafellar (SIAM J. Control Optim. 14:877–898, 1976) gave an inexact proximal-point algorithm which is more practical than the exact one. Our purpose is to extend the Kamimura and Takahashi result to a new inexact proximal-type algorithm. Moreover, this result is applied to the problem of finding the minimizer of a convex function on a uniformly convex and uniformly smooth Banach space. [ABSTRACT FROM AUTHOR]

In the present paper, we study the finite termination of sequences generated by inexact proximal point algorithms for finding zeroes of a maximal monotone (set-valued) operator $$T$$ on a Hilbert space. Under some mild conditions, we get that a sequence generated by inexact proximal point algorithm stops after a finite number of iterations. Our results extend the corresponding results in Rockafellar (SIAM J Control Optim 14:877-898, ). In particular, for optimization problems, our results improve corresponding results in Ferris (Math Progr 50:359-366, ). As applications, we obtain finite termination of projected gradient method. [ABSTRACT FROM AUTHOR]

In this paper, we study second-order necessary optimality conditions for a discrete optimal control problem with a nonconvex cost function and control constraints. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we derive second-order optimality conditions for a discrete optimal control problem. [ABSTRACT FROM AUTHOR]

This paper presents an exact formula for computing the normal cones of the constraint set mapping including the Clarke normal cone and the Mordukhovich normal cone in infinite programming under the extended Mangasarian-Fromovitz constraint qualification condition. Then, we derive an upper estimate as well as an exact formula for the limiting subdifferential of the marginal/optimal value function in a general Banach space setting. [ABSTRACT FROM AUTHOR]

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions. [ABSTRACT FROM AUTHOR]

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study a class of generalized mixed vector equilibrium problems for multifunctions, which includes as special cases a number of generalized vector variational inequalities, generalized vector variational-like inequalities and generalized vector equilibrium problems. We first establish a new vector version of Minty's lemma, and then prove an existence theorem of solutions for the generalized mixed vector equilibrium problems by using this result. Furthermore, this existence theorem is applied to obtain a new existence result of solutions for a class of generalized vector variational-like inequalities. [ABSTRACT FROM AUTHOR]

In this paper, a gap function for a system of vector equilibrium problems is introduced and studied. Some necessary and sufficient conditions for the system of vector equilibrium problems are established. Characterizations of the solutions set for the system of vector equilibrium problems are also derived. Furthermore, some existence results of solutions for the system of vector equilibrium problems are proved. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the existence of solutions for a class of noncoercive variational inequalities involving a p-Laplacian type operator. Our approach is based essentially on equilibrium problems and arguments from recession analysis. Our results are of two types: the first is obtained in a monotone framework; the second is obtained without a monotonicity assumption. [ABSTRACT FROM AUTHOR]

In this paper, we introduce mathematical programs with vector optimization constraints. For these problems, we establish two models in both the weak Pareto solution and Pareto solution setting. Some new existence results are obtained under rather weak conditions. We establish also equivalences between mathematical programs with vector optimization constraints and mathematical programs with vector variational inequality constraints. [ABSTRACT FROM AUTHOR]

We consider a general class of equilibrium problems which involve a single-valued mapping and a nonsmooth bifunction. Such mixed equilibrium problems are solved with a combined relaxation method using an auxiliary iteration of a splitting-type method for constructing a separating hyperplane. We prove the convergence of the method under the assumption that the dual of the mixed equilibrium problem is solvable. Convergence rates are also derived. [ABSTRACT FROM AUTHOR]

The auxiliary principle technique is extended to study the generalized strongly nonlinear mixed variational-like inequality problem for set-valued mappings without compact values. We establish first the existence of a solution of the related auxiliary problem. Then, the iterative algorithm for solving that problem is given by using this existence result. Moreover, the existence of a solution of the original problem and the convergence of iterative sequences generated by the algorithm are both derived. [ABSTRACT FROM AUTHOR]

This paper studies the solution stability of a parametric optimal control problem with linear state equation, control constraints, and convex cost functions. By reducing the problem to a parametric programming problem and a parametric variational inequality, we obtain the lower semicontinuity of the solution map to a parametric optimal control problem. [ABSTRACT FROM AUTHOR]