Lipschitzian Selection Research Articles

Subgradients of the Value Function in a Parametric Convex Optimal Control Problem

Motivated by our recent works on the optimal value function in parametric optimal control problems under linear state equations, in this paper we study of the first-order behavior of the value function of a parametric convex optimal control problem with a convex cost function and linear state equations. By establishing an abstract result on the subdifferential of the value function to a parametric convex mathematical programming problem, we derive a formula for computing the subdifferential and the singular subdifferential of the value function to a parametric convex optimal control problem. By virtue of the convexity, several assumptions used in the above papers, like the existence of a local upper Lipschitzian selection of the solution map, as well as the V-inner semicontinuity of the solution map, are no longer needed.

Differential stability of convex optimization problems under inclusion constraints

Motivated by the recent work of Mordukhovich et al. [Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B. 2009;116:369–396] on the optimal value function in parametric programming under inclusion constraints, this paper presents some new results on differential stability of convex optimization problems under inclusion constraints and functional constraints in Hausdorff locally convex topological vector spaces. By using the Moreau–Rockafellar theorem and appropriate regularity conditions, we obtain formulas for computing the subdifferential and the singular subdifferential of the optimal value function. By virtue of the convexity, several assumptions used in the above paper by Mordukhovich et al., like the nonemptyness of the Frechet upper subdiffential of the objective function, the existence of a local upper Lipschitzian selection of the solution map, as well as the -inner semicontinuity and the -inner semicompactness of the solution map, are no longer needed....

Differential stability of convex optimization problems under inclusion constraints

Motivated by the recent work of Mordukhovich et al. [Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B. 2009;116:369–396] on the optimal value function in parametric programming under inclusion constraints, this paper presents some new results on differential stability of convex optimization problems under inclusion constraints and functional constraints in Hausdorff locally convex topological vector spaces. By using the Moreau–Rockafellar theorem and appropriate regularity conditions, we obtain formulas for computing the subdifferential and the singular subdifferential of the optimal value function. By virtue of the convexity, several assumptions used in the above paper by Mordukhovich et al., like the nonemptyness of the Fréchet upper subdiffential of the objective function, the existence of a local upper Lipschitzian selection of the solution map, as well as the -inner semicontinuity and the -inner semicompactness of the solution map, are no longer needed. Relationships between our results and the corresponding ones in Aubin’s book [Optima and equilibria. An introduction to nonlinear analysis. 2nd ed. New York (NY): Springer; 1998] are discussed.

Continuous Interpolation of Solutions of Lipschitz Inclusions

We show that given any finite set of trajectories of a Lipschitz differential inclusion there exists a continuous selection from the set of its solutions that interpolates the given trajectories. In addition, we present a result on lipschitzian selections.

Continuous and Lipschitzian selections from a measurable set-valued map

In this paper we construct for a given measurable set-valued map K: (0, T) -+ R” with convex and compact values a lower semicontinuous set-valued map H( .) with convex compact values such that H(t) c K(t) for almost all t E (0, T). Moreover, H( ) contains every local continuous selection from K( ), i.e., every continuous function r: (a, b) H R”, (a, 6) c (0, T) such that r(t) E K(t) for almost all t E (a, b). Then we construct a set-valued map f( .) with convex compact values such that L(t) c K(t) for almost all t E (0, T), and L( .) contains every Lipschitzian selection from K( .) defined on the previously given open set PC (0, T) with the Lipschitz constant less or equal to k 3 0. If L( .) is not identically equal to the empty set then it is continuous on P. In both cases we define the maps H( .) and L( .) using the support function. One motivation for construction of these regularizations of the setvalued map K( .) comes from viability theory (see [ 1, 21). If a set-valued map F: Graph(K) c-f R” is given we may regard K( .) as a viability map and we consider the following viability problem

Uniform approximation by a non-convex cone of continuous functions

Let Π be a collection of subsets of a compact set S in a normed linear space and K be all continuous functions ƒ on S whose level sets, {s: ƒ(s) ⩽ α}, are in Π for all α. Then K is a cone which is not necessarily convex. The problem under consideration is to find a best uniform approximation to a continuous function on S from K. In this article, under certain conditions on Π, extremal best approximations are identified, a best approximation and its uniqueness are characterized, and Lipschitzian selections are determined. The results are illustrated by approximation problems. Analysis is also presented for the special case when K is a convex cone. Applications are given to normed vector lattices and the isotone approximation problem on order-intervals.

Duality and Lipschitzian selections in best approximation from nonconvex cones

Duality relationships in finding a best approximation from a nonconvex cone in a normed linear space in general and in the space of bounded functions in particular, are investigated. The cone and the dual problems are defined in terms of positively homogeneous super-additive functional on the space. Conditions are developed on the cone so that the duality gap between a pair of primal and dual problems does not exist. In addition, Lipschitz continuous selections of the metric projection are identified. The results are specialized to a convex cone. Applications are indicated to approximation problems.

Lipschitzian selections in best approximation by continuous functions

The problem under consideration is to find a best uniform approximation to a function ƒ from a set K in the space of continuous functions. Conditions are derived on K such that the selection operator mapping ƒ to one of its best approximations is Lipschitzian. Their application is illustrated by approximation problems.

Lipschitzian selections in approximation from nonconvex sets of bounded functions

Consider the problem of finding a best uniform approximation to a function ƒ from a nonconvex set K in the space of bounded functions. Conditions are developed on K so that the operator mapping ƒ to one of its best approximations ƒ′ is Lipschitzian with some constant C and is optimal Lipschitzian, i.e., has the smallest C among all such operators.

Uniform approximation by quasi-convex and convex functions

Given a bounded function f defined on a convex subset of R n , the two problems considered are to find a quasi-convex (convex) function which is a best approximation to f under the uniform norm. It is shown that if f is the greatest quasi-convex (convex) minorant of f, then f′ = f + c , for some c ≧ 0, is the maximal best quasi-convex (convex) approximation to f. Furthermore, the nonlinear operator T defined by T( f) = f′ is a Lipschitzian selection operator with some least constant C( T), where C( T) ≤ C( T′) for all Lipschitzian operators T′ which map f to one of its best approximations. Thus T is optimal in this sense.

Optimal Lipschitzian selection operator in quasi-convex optimization

Given a bounded real function f defined on a closed bounded real interval I, the problem is to find a quasi-convex function f′ which minimizes the supremum of ¦ f(s) − f′(s)¦ for all s in I, over the class of all quasi-convex functions f′ on I. Such a nearest function f′ is not unique in general. For each f, is it possible to select a nearest f′ so that the selection operator f → f′ satisfies a Lipschitz condition with some constant C? If so, does there exist an operator for which C is minimum? It is shown that there exists a maximal nearest function f so that the operator f → f is such an optimal Lipschitzian selection operator with C = 2.

Lipschitz condition in minimum norm problems on bounded functions

Consider the Banach space of bounded functions with uniform norm. Given an element ƒ and a closed convex set in this space, the minimum norm problem is to find an element in the convex set nearest to ƒ. Such a nearest point is not unique in general. For each ƒ in the space, is it possible to select a nearest element ƒ′ so that the selection operator ƒ → ƒ′ satisfies a Lipschitz condition with some constant C? If so, does there exist an operator for which C is minimum? This article determines the required Lipschitzian selection operators with smallest possible constants for the minimum norm problem in three cases of special interest.

Applications of the Hahn-Banach Theorem in Approximation Theory

Applications of the Hahn-Banach Theorem in Approximation Theory