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].     

Expressions for subdifferential and optimization problems defined by nonsmooth homogeneous functions

In this paper, we prove a theoretical expression for subdifferentials of lower semicontinuous and homogeneous functions. The theoretical expression is a generalization of the Euler formula for differentiable homogeneous functions. As applications of the generalized Euler formula, we consider constrained optimization problems defined by nonsmooth positively homogeneous functions in smooth Banach spaces. Some results concerning Karush–Kuhn–Tucker points and necessary optimality conditions for the optimization problems are obtained.     

Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems

This paper is devoted to the study of nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. We introduce a constraint qualification which is based on the Mordukhovich subdifferential. Then, we derive a Fritz–John type necessary optimality condition. Finally, interrelations between the new and the existing constraint qualifications such as the Mangasarian–Fromovitz, linear independent, and the Slater are investigated.     

Second-order necessary and sufficient conditions in nonsmooth optimization

In this paper we generalize and sharpen R.W. Chaney's results on unconstrained and constrained second-order necessary and sufficient optimality conditions [5–7] for general Lipschitz functions without the semismoothness assumptionCorresponding author.     

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.     

Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory

A nonsmooth version of a three critical point theorem of Ricceri (due to Iannizzotto) is used to obtain three anti-periodic solutions for a second-order impulsive differential inclusions with a perturbed nonlinearity and two parameters.     

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.     

Nonconvex differential calculus for infinite-dimensional multifunctions

The paper is concerned with generalized differentiation of set-valued mappings between Banach spaces. Our basic object is the so-called coderivative of multifunctions that was introduced earlier by the first author and has had a number of useful applications to nonlinear analysis, optimization, and control. This coderivative is a nonconvex-valued mapping which is related to sequential limits of Fréchet-like graphical normals but is not dual to any tangentially generated derivative of multifunctions. Using a variational approach, we develop a full calculus for the coderivative in the framework of Asplund spaces. The latter class is sufficiently broad and convenient for many important applications. Some useful calculus results are also obtained in general Banach spaces.This research was partially supported by the National Science Foundation under grants DMS-9206989 and DMS-9404128, by the USA-Israel grant 94-00237, and by the NATO contract CRG-950360.     

Euler equations involving nonlinearities without growth conditions

A functional is considered, whose Euler equation involves a nonlinearity without growth conditions. It is shown that every minimum point of the functional is a solution of the Euler equation in a suitable weak sense.This work was partially supported by a national research grant financed by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (40%-1992).     

Second-order optimality conditions using approximations for nonsmooth vector optimization problems under inclusion constraints

Second-order necessary conditions and sufficient conditions for optimality in nonsmooth vector optimization problems with inclusion constraints are established. We use approximations as generalized derivatives and avoid even continuity assumptions. Convexity conditions are not imposed explicitly. Not all approximations in use are required to be bounded. The results improve or include several recent existing ones. Examples are provided to show that our theorems are easily applied in situations where several known results do not work.     

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.     

Scalarization and optimality conditions for vector equilibrium problems

In this paper, we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to the vector equilibrium problems without the convexity assumption. Using nonsmooth analysis and the scalarization results, we provided the necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems. By the assumption of convexity, we gave sufficient conditions for those solutions. As applications, we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems.     

A critical point theorem via the Ekeland variational principle

The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais-Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated.     

Optimality conditions for non-finite valued convex composite functions

Burke (1987) has recently developed second-order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued. In this note we present a technique for reducing the infinite valued case to the finite valued one. We then use this technique to extend the results in Burke (1987) to the case in which the convex function may take infinite values. We conclude by comparing these results with those established by Rockafellar (1989) for the piecewise linear-quadratic case.Dedicated to the memory of Robin W. ChaneyResearch supported in part by the National Science Foundation under grants DMS-8602399 and DMS-8803206, and by the Air Force Office of Scientific Research under grant ISSA-860080.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983.     

On the approximate solutions of constrained inverse Lipschitz function problems and inverse function theorems

We present a principle of approximate solutions of constrained inverse Lipschitz function problems. As corollaries and applications of the principle, we obtain a result of convergence of an approximate solutions sequence for the constrained problems, a conclusion relating direct and inverse images of upper and lower limits of a sequence of subsets, and several versions of inverse Lipschitz function theorems. Finally we give local uniqueness criteria for solutions to constrained nonlinear problems in finite dimension spaces.     

Note on a general Palais-Smale condition

In this paper, we establish a qualitative deformation theorem for a locally Lipschitz function satisfying a general Palais-Smale type condition. Then we show that, with some assumptions, this compactness condition implies the coercivity of the function.     

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.     

Controllability and strong controllability of differential inclusions

In this paper, we prove sufficient conditions for controllability and strong controllability in terms of the Mordukhovich subdifferential for two classes of differential inclusions. The first one is the class of sub-Lipschitz multivalued functions introduced by Loewen-Rockafellar (1994) [10]. The second one, introduced recently by Clarke (2005) [18], is the class of multivalued functions which are pseudo-Lipschitz and satisfy the so-called tempered growth condition. To do this, we establish an error bound result in terms of the Mordukhovich subdifferential outside Asplund spaces.     

Morse Theory for Geodesics in Conical Manifolds

The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. In a previous paper [15] we defined these manifolds as submanifolds of with a finite number of conical singularities. To formulate a good Morse theory we use an appropriate definition of geodesic, introduced in the cited work. The main theorem of this paper (see Theorem 3.6, section 3) proofs that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful. In section 5 we compare our theory with the weak slope approach existing in literature. Some examples are also provided.