In this paper we establish the existence of good solutions for a large class of infinite horizon discrete-time optimal control problems. This class contains optimal control problems arising in economic dynamics which describe a model proposed by Robinson, Solow and Srinivasan with nonconcave utility functions representing the preferences of the planner.

This paper is concerned with generalized differentiation of the efficient point multifunctions of parametric vector optimization problems in Banach spaces. We give formulae for computing and/or estimating the Fréchet coderivative (precoderivative) of this multifunction in a broad class of conventional vector optimization problems with the presence of geometric, operator, (finite and infinite) functional constraints. Examples are given to illustrate the obtained results.

Using the concept of Clarke tangent cone, a new notion of proper Clarke epiderivative for a set-valued map is introduced. Its nature and certain properties are investigated. Finally necessary and sufficient optimality conditions for a constrained set-valued optimization problem have been established in terms of proper Clarke epiderivative.

Following the papers [7, 8, 9] and [19], the strong duality between a convex optimization problem with both cone and equality constraints is further studied by means of the concepts of normal cone and of quasi relative interior. In such a way the difficulty that often the ordering cone has empty interior is overcome.

We consider metric regularity of composite multifucntions and establish an inequality on the moduli of metric regularity. Refining the original proofs of the Robinson-Ursescu theorem and Lyusternik-Graves theorem, we give an unified analysis for these two theorems, applicable to the composite of a closed convex multifunction and a continuous function.

Stationarity and regularity concepts for the three typical for variational analysis classes of objects - real-valued functions, collections of sets, and multifunctions - are investigated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. Furthermore, in many cases the corresponding properties appear to be in a sense equivalent. The properties are defined in terms of certain constants which in the case of regularity properties provide also some quantitative characterizations of these properties. The relations between different constants and properties are discussed.

In this paper, $\epsilon$-subgradients for convex set-valued maps are defined. We prove an existence theorem for $\epsilon$-subgradients of convex set-valued maps. Also, we give necessary $\epsilon$- optimality conditions for an $\epsilon$-solution of a convex set-valued optimization problem (CSP). Moreover, using the single-valued function induced from the set-valued map, we obtain theorems describing the $\epsilon$-subgradient sum formula for two convex set-valued maps, and then give necessary and sufficient $\epsilon$-optimality conditions for the problem (CSP).

In this paper, we introduce nondifferentiable multiobjective fractional programming problems with cone constraints over arbitrary closed convex cones, where every component of the objective function contains a term involving the support function of a compact convex set. For this problem, Wolfe and Mond-Weir type duals are proposed. We establish weak and strong duality theorems for a weakly efficient solution under suitable (V, ρ)-invexity assumptions. As special cases of our duality relations, we give some known duality results.

Stability of a general indefinite quadratic program whose constraint set is the intersection of an affine subspace and a closed convex cone is investigated. We present a systematical study of several stability properties of the Karush-Kuhn-Tucker point map, the global solution map, and the optimal value function, assuming that the problem data undergoes small perturbations. Some techniques from our preceding work on stability of indefinite quadratic programs under linear constraints have found further applications and extensions in this paper.

We consider equilibrium problems (EP) with directionally differentiable (not necessarily ${\mathcal C}^1$) bifunctions which are convex with respect to the second variable and we use a gap function approach to solve them. In the first part of the paper we establish a condition under which any stationary point of the gap function solves (EP) and we propose a solution method which uses descent directions of the gap function. In the final section we study the problem when this condition is not satisfied. In this case we use a family of gap functions depending on a parameter $\alpha$ which allows us to overcome the trouble due to the lack of a descent direction.

In this paper, we first present some sufficient conditions for the upper semicontinuity and/or the continuity of the Bregman farthest-point map $Q_C^g$ and the relative farthest-point map $S_C^g$ for a nonempty $D$-maximally approximately compact subset $C$ of a Banach space $X$. We next present certain sufficient conditions as well as equivalent conditions for a Klee set to be singleton in a Banach space $X$. Our results extend and/or improve the corresponding ones of [Bauschke, et al., J. Approx. Theory, 158 (2009), pp. 170-183] to infinite dimensional spaces.