Weighted quasi-equilibrium problems with lower and upper bounds

This paper is devoted to studying the solution existence of weighted quasi-equilibrium problems with lower and upper bounds by using maximal element theorems, a fixed point theorem of set-valued mappings and Fan–KKM theorem, respectively. Some new results are obtained.     

Generalized invexity of nonsmooth functions

In this paper, the concepts of weak invexity and weak quasi invexity are introduced and the relations among several kinds of generalized invexity are studied for nonsmooth functions by means of the properties of limiting subdifferentials. In addition, the relations between generalized invexity of a nonsmooth function and generalized invariant monotonicity of its limiting subdifferential mapping are researched. Our results here are an extension and generalization of those presented by M. Soleimani-damaneh.     

Metric subregularity for proximal generalized equations in Hilbert spaces

In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish the results of Robinson-Ursescu theorem type for prox-regular multifunctions.     

Metric subregularity for composite-convex generalized equations in Banach spaces

In this paper we consider a convex-composite generalized constraint equation in Banach spaces. Using variational analysis technique, in terms of normal cones and coderivatives, we first establish sufficient conditions for such an equation to be metrically subregular. Under the Robinson qualification, we prove that these conditions are also necessary for the metric subregularity. In particular, some existing results on error bound and metric subregularity are extended to the composite-convexity case from the convexity case.     

Computation formulas and multiplier rules for graphical derivatives in separable Banach spaces

In this paper, we present new computation formulas for the contingent epiderivative and hypoderivative of a set-valued map taking values in a Banach space with a shrinking Schauder basis. These formulas are established in terms of the Fourier coefficients, and, in particular, in terms of the derivatives of the component maps associated with the Schauder basis. As an application, we obtain multiplier rules for vector optimization problems in terms of the derivatives of the component maps, extending classical results from smooth multiobjective optimization problems.     

Weak regularity of functions and sets in Asplund spaces

In this paper, we study a new concept of weak regularity of functions and sets in Asplund spaces. We show that this notion includes prox-regular functions, functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension. We establish also that weak regularity is equivalent to Mordukhovich regularity in finite dimension. Finally, we give characterizations of the weak regularity of epi-Lipschitzian sets in terms of their local representations.     

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.     

A proximal bundle method with inexact data for convex nondifferentiable minimization

A proximal bundle method with inexact data is presented for minimizing an unconstrained nonsmooth convex function f$f$. At each iteration, only the approximate evaluations of f$f$ and its ε$\epsilon$-subgradients are required and its search directions are determined via solving quadratic programmings. Compared with the pre-existing results, the polyhedral approximation model that we offer is more precise and a new term is added into the estimation term of the descent from the model. It is shown that every cluster of the sequence of iterates generated by the proposed algorithm is an exact solution of the unconstrained minimization problem.     

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.     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002     

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

We continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina, The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples (this journal)]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel-Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe’s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained.     

Existence results and gap functions for the generalized equilibrium problem with composed functions

In this paper we provide first existence results for solutions of the generalized equilibrium problem with composed functions (GEPC) under generalized convexity assumptions. Then we construct by employing some tools specific to the theory of conjugate duality two gap functions for (GEPC). The importance of these gap functions is to be seen in the fact that they equivalently characterize the solutions of an equilibrium problem. We also prove that for some particular instances of (GEPC) the gap functions we introduce here become among others the celebrated Auslender’s and Giannessi’s gap functions.     

Stability of subdifferentials of nonconvex functions in Banach spaces

We investigate various notions of subdifferentials and superdifferentials of nonconvex functions in Banach spaces. We prove stability results of these subdifferentials and superdifferentials under various kind of convergences. Our proofs rely on a recent variational principle of Deville, Godefroy and Zizler. Connections between our results, the geometry of Banach spaces and existence theorems of viscosity solutions for first and second-order Hamilton-Jacobi equations in infinite-dimensional Banach spaces will be explained.     

Epi-topology on spaces of lower semicontinuous functions into partially ordered spaces and hyperspaces

The concept of lower semicontinuity is extended to functions mapping into partially ordered spaces. A study is made of spaces of such lower semicontinuous functions under the epi-topology. These spaces are subspaces of hyperspaces with the Fell topology. The closure of such a function space in the hyperspace is characterized for certain spaces. A continuous selection theorem is established, showing that most such function spaces are not ech-complete.     

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 second-order optimality conditions in constrained multiobjective optimization

In this paper we study a multiobjective optimization problem with inequality constraints on finite dimensional spaces. A second-order necessary condition for local weak efficiency is proved under strict differentiability assumptions. We also establish a second-order sufficient condition for local firm efficiency of order 2 under ?-stability assumptions. In this way we generalize some corresponding results obtained by P.Q. Khanh and N.D. Tuan, and by the authors.     

Linear and nonlinear error bounds for lower semicontinuous functions

In this paper, we give some characterizations of linear and nonlinear error bounds for lower semicontinuous functions by a new notion, called subslope. And, extend some results of Azé and Corvellec (SIAM J Optim 12:913–927, 2002) and Corvellec and Motreanu (Math Program Ser A 114:291–319, 2008) slightly. Furthermore, we get a sufficient and necessary condition for global linear error bounds.     

Computational complexity of inner and outerj-radii of polytopes in finite-dimensional normed spaces

This paper studies the complexity of computing (or approximating, or bounding) the various inner and outer radii of ann-dimensional convex polytope in the space ⁿ equipped with an _p norm or a polytopal norm. The polytopeP is assumed to be presented as the convex hull of finitely many points with rational coordinates (V-presented) or as the intersection of finitely many closed halfspaces defined by linear inequalities with rational coefficients (-presented). The innerj-radius ofP is the radius of a largestj-ball contained inP; it isP's inradius whenj = n and half ofP's diameter whenj = 1. The outerj-radius measures how wellP can be approximated, in a minimax sense, by an (n — j)-flat; it isP's circumradius whenj = n and half ofP's width whenj = 1. The binary (Turing machine) model of computation is employed. The primary concern is not with finding optimal algorithms, but with establishing polynomial-time computability or NP-hardness. Special attention is paid to the case in whichP is centrally symmetric. When the dimensionn is permitted to vary, the situation is roughly as follows: (a) for general -presented polytopes in _p spaces with 1相似文献   

Exact penalty functions and calmness for mathematical programming under nonlinear perturbations

In the present paper, the effects of nonlinear perturbations of constraint systems are considered over the relationship between calmness and exact penalization, within the context of mathematical programming with equilibrium constraints. Two counterexamples are provided showing that the crucial link between the existence of penalty functions and the property of calmness for perturbed problems is broken in the presence of general perturbations. Then, some properties from variational analysis are singled out, which are able to restore to a certain extent the broken link. Consequently, conditions on the value function associated to perturbed optimization problems are investigated in order to guarantee the occurrence of the above properties.