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.     

Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities

In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12].     

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.     

Pointwise well-posedness in set optimization with cone proper sets

This paper deals with the well-posedness property in the setting of set optimization problems. By using a notion of well-posed set optimization problem due to Zhang et al. (2009) [18] and a scalarization process, we characterize this property through the well-posedness, in the Tykhonov sense, of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well-posed. Our approach is based just on a weak boundedness assumption, called cone properness, that is unavoidable to obtain a meaningful set optimization problem.     

The principle of symmetric criticality for non-differentiable mappings

Let X be a Banach space on which a symmetry group G linearly acts and let J be a G-invariant functional defined on X. In 1979, R. Palais (Comm. Math. Phys. 69 (1979) 19) gave some sufficient conditions to guarantee the so-called “Principle of Symmetric Criticality”: every critical point of J restricted on the subspace of G-symmetric points becomes also a critical point of J on the whole space X. This principle is generalized to the case where J is not differentiable within the setting which does not require the full variational structure under the hypothesis that the action of G is isometry or G is compact.     

Convergence of set-valued mappings: Equi-outer semicontinuity

The concept of equi-outer semicontinuity allows us to relate the pointwise and the graphical convergence of set-valued-mappings. One of the main results is a compactness criterion that extends the classical Arzelà-Ascolì theorem for continuous functions to this new setting; it also leads to the exploration of the notion of continuous convergence. Equi-lower semicontinuity of functions is related to the outer semicontinuity of epigraphical mappings. Finally, some examples involving set-valued mappings are re-examined in terms of the concepts introduced here.Research supported in part by a grant of the National Science Foundation.     

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.     

Point-based sufficient conditions for metric regularity of implicit multifunctions

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.     

General existence theorems, alternative theorems and applications to minimax problems

We establish general theorems on maximal elements, coincidence points and nonempty intersections for set-valued mappings on GFC-spaces and show their equivalence. Applying them we derive equivalent forms of alternative theorems. As applications, we develop in detail general types of minimax theorems. The results obtained improve or include as special cases several recent ones in the literature.     

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.     

On mappings covering at a nonlinear rate and their perturbation stability

The present paper contains a study of covering (alias, openness) properties at a nonlinear rate for set-valued mappings between metric spaces. Such study is focussed on the stability of these properties in the presence of perturbations. A crucial result valid for linear openness, known as Milyutin’s theorem, is extended to set-valued mappings covering at a nonlinear rate under possibly non-Lipschitz perturbations. Consequently, a Lyusternik type theorem is derived from such extension and a general penalization principle for constrained optimization problems, which exploits nonlinear covering properties, is presented.     

Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities

Multiple critical points theorems for non-differentiable functionals are established. Applications both to elliptic variational-hemivariational inequalities and eigenvalue problems with discontinuous nonlinearities are then presented.     

Error bounds for vector-valued functions: Necessary and sufficient conditions

In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new derivative-like objects (slopes and subdifferentials) are introduced and a general classification scheme of error bound criteria is presented.     

An eigenvalue problem for a hemivariational inequality involving a nonlinear compact operator

The present paper deals with an eigenvalue problem arising in hemivariational inequalities involving a nonlinear compact operator. This problem has been studied concerning the existence of its solution by applying a critical point approach suitable for nonconvex, nonsmooth energy functions. The method is based on Ekeland's variational principle and on the results of Chang and Szulkin.     

Necessary optimality conditions for bilevel minimization problems

In this paper we study bilevel minimization problems. Using the implicit function theorem, variational analysis and exact penalty results we establish necessary optimality conditions for these problems.     

Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S-iteration process recently introduced by Sahu in [14] converges strongly to a unique fixed point of a mapping T, where T is κ-strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ_i-strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980) [1] and Yamada (2001) [15] to a general class of λ-strictly pseudocontractive mappings in uniformly smooth Banach spaces.     

Openness properties for parametric set-valued mappings and implicit multifunctions

The aim of this paper is to obtain some openness results in terms of normal coderivative for parametric set-valued mappings acting between infinite dimensional spaces. Then, implicit multifunction results are obtained by simply specializing the openness results. Moreover, we study a kind of metric regularity of the implicit multifunction. The results of the paper generalize several recent results in literature.     

The hypermetric cone is polyhedral

The hypermetric coneH _n is the cone in the spaceR ^n(n–1)/2 of all vectorsd=(d _ij)_1i<jn satisfying the hypermetric inequalities: –_1ijn z _j z _j d _ij 0 for all integer vectorsz inZ ⁿ with –_1in z _i =1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneH _n is polyhedral.