Lipschitz-like property of an implicit multifunction and its applications

The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the Lipschitz-like property at a given point are established. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results.     

Linearly perturbed polyhedral normal cone mappings and applications

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed polyhedra in reflexive Banach spaces. Our focus point is a positive linear independence condition, which is a relaxed form of the linear independence condition employed recently by Henrion et al. (2010) [1], and Nam (2010) [3]. The formulae obtained allow us to get new results on solution stability of affine variational inequalities under linear perturbations. Thus, our paper develops some aspects of the work of Henrion et al. (2010) [1] Nam (2010) [3] Qui (in press) [12] and Yao and Yen (2009) [6] and [7].     

New results on linearly perturbed polyhedral normal cone mappings

This paper establishes an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the linearly perturbed normal cone mappings in reflexive Banach spaces. In comparison with Nam (2010) [5], Qui (in press) [8], Qui (2011) [7], Trang (2010) [9], the major advantage of our investigation is that here neither the linear independence condition nor the positively linear independence condition are used. Thus, no assumption on the normal vectors of the active constraints at the point in question is needed. Some aspects of the preceding results (Henrion, Mordukhovich and Nam (2010) [3], Nam (2009) [5], Qui (2011) [7], Yao and Yen (2009) [10], Yao and Yen (2009) [11]) are developed.     

Linearly perturbed generalized polyhedral normal cone mappings and applications

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed generalized polyhedra in reflexive Banach spaces. Assume in addition that the generating elements are linearly independent and some qualification condition holds, the Lipschitzian stability of the parameterized variational inequalities over the right-hand side perturbed generalized polyhedra is characterized using the initial data.     

Metric regularity and subdifferential calculus in Banach spaces

In this paper we give verifiable conditions in terms of limiting Fréchet subdifferentials ensuring the metric regularity of a multivalued functionF(x)=–g(x)+D. We apply our results to the study of the limiting Fréchet subdifferential of a composite function defined on a Banach space.     

Exact Formulae for Coderivatives of Normal Cone Mappings to Perturbed Polyhedral Convex Sets

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.     

Approximate subgradients and coderivatives in Rn

We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Fréchet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Fréchet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.Research supported by NSERC and the Shrum Endowment at Simon Fraser University.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

The fuzzy intersection rule in variational analysis with applications

The fuzzy intersection rule for Fréchet normal cones in Asplund spaces was established by Mordukhovich and the author using the extremal principle, which appears more convenient to apply in some applications. In this paper, we present a complete discussion of this rule in various aspects. We show that the fuzzy intersection rule is another characterization of the Asplund property of the space. Various applications are considered as well. In particular, a complete set of fuzzy calculus rules for general lower semicontinuous functions are established.     

The Ekeland variational principle for set-valued maps involving coderivatives

In this paper we use the Fréchet, Clarke, and Mordukhovich coderivatives to obtain variants of the Ekeland variational principle for a set-valued map F and establish optimality conditions for set-valued optimization problems. Our technique is based on scalarization with the help of a marginal function associated with F and estimates of subdifferentials of this function in terms of coderivatives of F.     

Second-order subdifferentials and convexity of real-valued functions

In this paper we show that the positive semi-definiteness (PSD) of the Fréchet and/or Mordukhovich second-order subdifferentials can recognize the convexity of C¹ functions. However, the PSD is insufficient for ensuring the convexity of a locally Lipschitz function in general. A complete characterization of strong convexity via the second-order subdifferentials is also given.     

Metric regularity of a positive order for generalized equations

This paper focuses on the metric regularity of a positive order for generalized equations. More concretely, we establish verifiable sufficient conditions for a generalized equation to achieve the metric regularity of a positive order at its a given solution. The provided conditions are expressed in terms of the Fréchet coderivative/or the Mordukhovich coderivative/or the Clarke one of the corresponding multifunction formulated the generalized equation. In addition, we show that such sufficient conditions turn out to be also necessary for the metric regularity of a positive order of the generalized equation in the case where the multifunction established the generalized equation is closed and convex.     

Subdifferential calculus in Asplund generated spaces

We extend the definition of the limiting Fréchet subdifferential and the limiting Fréchet normal cone from Asplund spaces to Asplund generated spaces. Then we prove a sum rule, a mean value theorem, and other statements for this concept.     

Nonlinear Perturbations of Polyhedral Normal Cone Mappings and Affine Variational Inequalities

This paper establishes an upper estimate for the Fréchet normal cone to the graph of the nonlinearly perturbed polyhedral normal cone mappings in finite dimensional spaces. Under a positive linear independence assumption on the normal vectors of the active constraints at the point in question, the result leads to an upper estimate for values of the Mordukhovich coderivative of such mappings. On the basis, new results on solution stability of parametric affine variational inequalities under nonlinear perturbations are derived.     

Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure

In this paper, we define two new concepts of efficiency for vector optimization with variable ordering structure, namely the sharp and robust efficiencies, and we study their connections with classical concepts of efficiency in vector optimization. Then, we get necessary optimality conditions for them using Fréchet and Mordukhovich calculus coupled with the Gerstewitz’s (Tammer’s) scalarizing functional and openness results for set-valued maps.     

A family of Halley–Chebyshev iterative schemes for non-Fréchet differentiable operators

A modification of some classical third order methods is studied. The main advantage of these methods is that they do not need evaluate any Fréchet derivative. A convergence theorem in Banach spaces, just assuming that the second divided difference is bounded by a nondecreasing function and a punctual condition, is presented. Fréchet differentiability is not required. Finally, a particular example is analyzed where our conditions are fulfilled and the classical ones fail.     

Sharp Minima for Multiobjective Optimization in Banach Spaces

We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.This research was supported by a Central Research Grant of The Hong Kong Polytechnic University (Grant No. G-T 507). Research of the first author was also supported by the National Natural Science Foundation of PR China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, China (Grant No. 2003A002M).     

A nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions

Fuzzy optimization conditions in terms of the Fréchet subdifferential for reflexive spaces were investigated by Borwein, Treiman and Zhu (1998) in [1]. To achieve the nondegenerate form, it is well known that some qualification conditions should be assumed. In this paper, we are going to prove that the nondegenerate fuzzy optimality condition even holds with no qualification conditions in Asplund spaces (in particular, reflexive spaces) for optimization problems with semi-continuous and continuous data. The results are even new in finite-dimensional frameworks.     

Weak sharp efficiency in multiobjective optimization

By using the generalized Fermat rule, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials and the sum rule for Mordukhovich subdifferentials, we establish a necessary optimality condition for the local weak sharp efficient solution of a constrained multiobjective optimization problem. Moreover, by employing the approximate projection theorem, and some appropriate convexity and affineness conditions, we also obtain some sufficient optimality conditions respectively for the local and global weak sharp efficient solutions of such a multiobjective optimization problem.     

Remarks on strict efficiency in scalar and vector optimization

The aim of this paper is to study optimality conditions for strict local minima to constrained mathematical problems governed by scalar and vectorial mappings. Unlike other papers in literature dealing with strict efficiency, we work here with mappings defined on infinite dimensional normed vector spaces. Firstly, we (mainly) consider the case of nonsmooth scalar mappings and we use the Fréchet and Mordukhovich subdifferentials in order to provide optimality conditions. Secondly, we present some methods to reduce the study of strict vectorial minima to the case of strict scalar minima by means of some scalarization techniques. In this vectorial framework we treat separately the case where the ordering cone has non-empty interior and the case where it has empty interior.