About the Lipschitz property of the metric projection in the Hilbert space

We consider the metric projection operator from the real Hilbert space onto a strongly convex set. We prove that the restriction of this operator on the complement of some neighborhood of the strongly convex set is Lipschitz continuous with the Lipschitz constant strictly less than 1. This property characterizes the class of strongly convex sets and (to a certain degree) the Hilbert space. We apply the results obtained to the question concerning the rate of convergence for the gradient projection algorithm with differentiable convex function and strongly convex set.     

Characterizations of strictly convex sets by the uniqueness of support points

Abstract

This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so-called regular sets, for which every extreme point is a unique support point, is introduced. Closed strictly convex sets and their intersections with some other sets are shown to belong to this class. The obtained characterizations are then applied to set-valued maps and to the separation of a convex set and a strictly convex set. Under suitable assumptions, so-called set-valued maps with path property are characterized by strictly convex images of the considered set-valued map.     

无限维空间拟凸映射多目标最优化问题解集的连通性

本文在一个无限格中引入了拟凸、强拟凸和严格拟凸映射。并在约束集为紧凸条件下，证明了相应的多目标规划问题之有效解集和弱有效解集连通性结果。     

Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic Multi-objective Optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method in order to approach this problem. We prove that the Hausdorff–Pompeiu distance between the weakly Pareto sets associated with the Sample Average Approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic Multi-objective Optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the Sample Average Approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic Optimization Problem, i.e. we optimize over the image space of the Stochastic Optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the Sample Average Approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.     

Efficiency in constrained continuous location

We present a geometrical characterization of the efficient, weakly efficient and strictly efficient points for multi-objective location problems in presence of convex constraints and when distances are measured by an arbitrary norm. These results, established for a compact set of demand points, generalize similar characterizations previously obtained for uncontrained problems. They are used to show that, in planar problems, the set of constrained weakly efficient points always coincides with the closest projection of the set of unconstrained weakly efficient points onto the feasible set. This projection property which are known previously only for strictly convex norms, allows to easily construct all the weakly efficient points and provides a useful localization property for efficient and strictly efficient points.     

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

Weakly Convex Sets and Their Properties

In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.     

Asset Pricing under Progressive Taxes and Existence of General Equilibrium

This paper shows that the existence of general equilibrium in a two-period economy with financial markets and progressive anonymous tax system is not at all problematic, provided securities are purely financial. We explore the concepts of weakly and strongly arbitrage-free security price for return and tax system, and prove arbitrage-free asset pricing theorems without short-sale restrictions. A general equilibrium is a set of current and future prices (contingent on uncertain events) and a set of individual plans such that all markets are cleared. The existence of such an equilibrium is proved under the following conditions: continuous, weakly convex, strictly monotone, complete preferences and strictly positive endowments.     

Lower semicontinuous functionals with uniform sublevel sets

In this paper, lower semicontinuous functionals with uniform sublevel sets are investigated, where the sublevel sets are linear shifts of a set in a fixed direction. The extended real-valued functionals are defined on a topological vector space. Conditions are given under which they are proper, finite-valued, continuous, convex, sublinear, strictly quasi-convex, strictly quasi-concave or monotone. We apply the functionals to the separation of not necessarily convex sets.     

广义凸模糊映射

给出广义凸模糊映射、广义弱凸模糊映射等概念和若干特例。其次,构造集合Axf,y、Af,证明当f为下半连续广义弱凸模糊映射时Afx,y为闭弱凸集,进而得到广义凸模糊映射的充分条件。最后,给出广义凸模糊映射的性质,并指出半严格广义凸模糊映射成为严格广义凸模糊映射的条件。     

集值映射最优化问题的严有效解集的连通性及应用

本文对集值映射最优化问题引入严有效解的概念．证明了当目标函数为锥类凸的集值映射时,其目标空间里的严有效点集是连通的;若目标函数为锥凸的集值映射时,其严有效解集也是连通的．作为应用,讨论了超有效解集的连通性．     

On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces

This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problems in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo line search converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.     

Regularization of Monotone Variational Inequalities with Mosco Approximations of the Constraint Sets

In this paper we study the convergence and stability in reflexive, smooth and strictly convex Banach spaces of a regularization method for variational inequalities with data perturbations. We prove that, when applied to perturbed variational inequalities with monotone, demiclosed, convex valued operators satisfying certain conditions of asymptotic growth, the regularization method we consider produces sequences which converge weakly to the minimal-norm solution of the original variational inequality, provided that the perturbed constraint sets converge to the constraint set of the original inequality in the sense of a modified form of Mosco convergence of order ≥1. If the underlying Banach space has the Kadeč–Klee property, then the sequence generated by that regularization method is strongly convergent. Mathematics Subject Classifications (2000) Primary: 47J0G, 47A52; secondary: 47H14, 47J20.     

Über die Approximation von lokal konvexen Mengen

While convex sets in Euclidean space can easily be approximated by convex sets with C -boundary, the C -approximation of convex sets in Riemannian manifolds is a non-trivial problem. Here we prove that C-approximation is possible for a compact, locally convex set C in a Riemannian manifold if (i) C has strictly convex boundary or if (ii) the sectional curvature is positive or negative on C.The proofs are based on a detailed analysis of the distance function from C, on results from [1] and on the Greene-Wu approximation process for convex functions ([5], [6]). Finally, using similar methods, a partial tubular neighborhood with geodesic fibres is constructed for the boundary of a locally convex set. This construction is essential for some results in [2].     

A theorem about affine maps in Banach spaces

It is shown that a weakly compact convex set in a strictly convex space cannot be decomposed into two non-empty, disjoint sets which are similar to each other in the sense that one is the image of the other under a non-expansive affine map.     

Strong Restricted-Orientation Convexity

Strong restricted-orientation convexity is a generalization of standard convexity. We explore the properties of strongly convex sets in multidimensional Euclidean space and identify major properties of standard convex sets that also hold for strong convexity. We characterize strongly convex flats and halfspaces, and establish the strong convexity of the affine hull of a strongly convex set. We then show that, for every point in the boundary of a strongly convex set, there is a supporting strongly convex hyperplane through it. Finally, we show that a closed set with nonempty interior is strongly convex if and only if it is the intersection of strongly convex halfspaces; we state a condition under which this result extends to sets with empty interior.     

Relations between the convexity of a set and the differentiability of its support function

It is known that, in finite dimensions, the support function of a compact convex set with nonempty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper, we realize a thorough study of the relations between the differentiability of the support function on the interior of its domain and the convexity of the set, mainly for unbounded sets. Then, we revisit some results related to the differentiability of the cost function associated to a production function.     

几种凸模糊集间的转换关系

在引用扎德所定义的凸模糊集、强凸模糊集、严格凸模糊集等概念的基础上，探讨了这三种凸模糊集间的转换条件，得到凸模糊集与强凸模糊集、强凸模糊集与严格凸模糊集间的等价条件。     

Rigid pentagons in hypercubes

Two sets of vertices of a hypercubes in ⁿ and ^m are said to be equivalent if there exists a distance preserving linear transformation of one hypercube into the other taking one set to the other. A set of vertices of a hypercube is said to be weakly rigid if up to equivalence it is a unique realization of its distance pattern and it is called rigid if the same holds for any multiple of its distance pattern. A method of describing all rigid and weakly rigid sets of vertices of hypercube of a given size is developed. It is also shown that distance pattern of any rigid set is on the face of convex cone of all distance patterns of sets of vertices in hypercubes.Rigid pentagons (i.e. rigid sets of size 5 in hypercubes) are described. It is shown that there are exactly seven distinct types of rigid pentagons and one type of rigid quadrangle. It is also shown that there is a unique weakly rigid pentagon which is not rigid. An application to the study of all rigid pentagons and quadrangles inL ¹ having integral distance pattern is also given.This work was done during a visit of both the authors to Mehta Research Institute, Allahabad, India.