Constraint qualifications for nonsmooth programming

ABSTRACT

By using an implicit function theorem and a result of error bound, we provide new constraint qualifications ensuring the Karush–Kuhn–Tuker necessary optimality conditions for both smooth and nonsmooth optimization problems in normed spaces or Banach spaces.     

Well-Posedness Under Relaxed Semicontinuity for Bilevel Equilibrium and Optimization Problems with Equilibrium Constraints

Bilevel equilibrium and optimization problems with equilibrium constraints are considered. We propose a relaxed level closedness and use it together with pseudocontinuity assumptions to establish sufficient conditions for well-posedness and unique well-posedness. These conditions are new even for problems in one-dimensional spaces, but we try to prove them in general settings. For problems in topological spaces, we use convergence analysis while for problems in metric cases we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescu’s measures of noncompactness of approximate solution sets. Besides some new results, we also improve or generalize several recent ones in the literature. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped.     

Duality and saddle-points for convex-like vector optimization problems on real linear spaces

Usually, finite dimensional linear spaces, locally convex topological linear spaces or normed spaces are the framework for vector and multiojective optimization problems. Likewise, several generalizations of convexity are used in order to obtain new results. In this paper we show several Lagrangian type duality theorems and saddle-points theorems. From these, we obtain some characterizations of several efficient solutions of vector optimization problems (VOP), such as weak and proper efficient solutions in Benson’s sense. These theorems are generalizations of preceding results in two ways. Firstly, because we consider real linear spaces without any particular topology, and secondly because we work with a recently appeared convexlike type of convexity. This new type, designated GVCL in this paper, is based on a new algebraic closure which we named vector closure. This research for the second author was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.     

The Fermat rule for multifunctions on Banach spaces

Using variational analysis, we study vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In particular, in terms of the coderivatives, we present Fermat’s rules as necessary conditions for an optimal solution of the above problems. As applications, we also provide some necessary conditions (in terms of Clarke’s normal cones or the limiting normal cones) for Pareto efficient points.This research was supported by a postdoctoral fellowship scheme (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong. Research of the first author was also supported by the National Natural Science Foundation of P. R. China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, P. R. China (Grant No. 2003A002M).     

Asymptotic Conditions for Degree Diminution Along Prescribed Lines

One of the problems in bivariate polynomial interpolation is the choice of a space of polynomials suitable for interpolating a given set of data. Depending on the number of data, a usual space is that of polynomials in 2 variables of total degree not greater than k. However, these spaces are not enough to cover many interpolation problems. Here, we are interested in spaces of polynomials of total degree not greater than k whose degree diminishes along some prescribed directions. These spaces arise naturally in some interpolation problems and we describe them in terms of polynomials satisfying some asymptotic interpolation conditions. This provides a general frame to the interpolation problems studied in some of our recent papers.     

Error analysis for finite element methods for steady natural convection problems

We analyze the error in finite element methods in approximating, so-called, free or natural convection problems. We also include the effects of conducting solid walls in our analysis. Under a uniqueness condition on the Rayleigh and Prandtl numbers (which we derive), we give direct, quasioptimal error estimates for “div-stable” finite element spaces for the fluid variables and general conforming finite element spaces for the temperature. At larger Rayleigh numbers, we give analogous, asymptotic error estimates, basing this analysis upon local uniqueness properties of the true solution (u p T), which we also establish.     

Energy stability for a class of two-dimensional interface linear parabolic problems

We consider parabolic equations in two-dimensions with interfaces corresponding to concentrated heat capacity and singular own source. We give an analysis for energy stability of the solutions based on special Sobolev spaces (the energies also are given by the norms of these spaces) that are intrinsic to such problems. In order to define these spaces we study nonstandard spectral problems in which the eigenvalue appears in the interfaces (conjugation conditions) or at the boundary of the spatial domain. The introducing of appropriate spectral problems enable us to precise the values of the parameters which control the energy decay. In fact, in order for numerical calculation to be carried out effectively for large time, we need to know quantitatively this decay property.     

Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications

We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.

     

Vectorial Ekeland variational principle for systems of equilibrium problems and its applications

For a family of vector-valued bifunctions,we introduce the notion of sequentially lower monotonity,which is strictly weaker than the lower semi-continuity of the second variables of the bifunctions.Then,we give a general version of vectorial Ekeland variational principle(briefly,denoted by EVP) for a system of equilibrium problems,where the sequentially lower monotone objective bifunction family is defined on products of sequentially lower complete spaces(concerning sequentially lower complete spaces,see Zhu et al(2013)),and taking values in a quasi-ordered locally convex space.Besides,the perturbation consists of a subset of the ordering cone and a family {p_i}_(i∈I) of negative functions satisfying for each i∈I,p_i(x_i,y_i) = 0 if and only if x_i=y_i.From the general version,we can deduce several particular equilibrium versions of EVP,which can be applied to show the existence of solutions for countable systems of equilibrium problems.In particular,we obtain a general existence result of solutions for countable systems of equilibrium problems in the setting of sequentially lower complete spaces.By weakening the compactness of domains and the lower semi-continuity of objective bifunctions,we extend and improve some known existence results of solutions for countable system of equilibrium problems in the setting of complete metric spaces(or Fréchet spaces).When the domains are non-compact,by using the theory of angelic spaces(see Floret(1980)),we generalize some existence results on solutions for countable systems of equilibrium problems by extending the framework from reflexive Banach spaces to the strong duals of weakly compactly generated spaces.     

The Lagrange multiplier rule for super efficiency in vector optimization

In this paper, we study constrained multiobjective optimization problems with objectives being closed-graph multifunctions in Banach spaces. In terms of the coderivatives and Clarke's normal cones, we establish Lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems.     

Existence of solutions for generalized implicit vector variational-like inequalities

In this paper, we introduce a new class of generalized implicit vector variational-like inequalities in Hausdorff topological vector spaces and Banach spaces which contain implicit vector equilibrium problems, implicit vector variational inequalities and implicit vector complementarity problems as special cases. We derive some new results by using the KKM–Fan theorem, under compact and noncompact assumptions on underlying convex sets.     

Banach空间一阶非线性微分方程终值问题解的存在性

在Fréchet空间中利用推广的Tychonov不动点定理研究了Banach空间中一阶非线性微分方程终值问题解的存在性.     

Error bounds for systems of lower semicontinuous functions in Asplund spaces

In this paper, using the Fréchet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.     

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.     

Variational Inequality and Complementarity Problem for Set－Valued Mapping with Application

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Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces

In hyperconvex metric spaces we consider best approximation, invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive.     

Nonlocal boundary problems for a third-order one-dimensional nonlinear pseudoparabolic equation

We consider initial boundary value problems for a third-order nonlinear pseudoparabolic equation with one space dimension. The boundary condition is given by an integral; the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet or Neumann counterparts. By means of appropriate elliptic estimates we are able to seek solutions not only in the weighted spaces but also in the usual Sobolev spaces. The procedure is carried out in a unified way. Our results characterize a regularity of the pseudoparabolic operator that is different from that of the parabolic operator.     

General Ekeland's Variational Principle for Set-Valued Mappings

In this paper, we introduce the concept of approximate solutions for set-valued optimization problems. A sufficient condition for the existence of approximate solutions is obtained. A general Ekeland's variational principle for set-valued mappings in complete ordered metric spaces and complete metric spaces are derived. These results are generalizations of results for vector-valued functions in Refs. 1–4.     

On the Weak Semi-continuity of Vector Functions and Minimum Problems

Lower semi-continuity from above or upper semi-continuity from below has been used by many authors in recent papers. In this paper, we first study the weak semi-continuity for vector functions having particular form as that of Browder in ordered normed vector spaces; we obtain several new results on the lower semi-continuity from above or upper semi-continuity from below for these vector functions. Our results generalize some well-known results of Browder in scalar case. Secondly, we study the minimum or maximum problems for vector functions satisfying lower semi-continuous from above or upper semi-continuous from below conditions; several new results on the existence of minimal points or maximal points are obtained. We also use these results to study vector equilibrium problems and von Neumann’s minimax principle in ordered normed vector spaces.     

Exceptional Families of Elements for Variational Inequalities in Banach Spaces

In keeping with very recent efforts to establish a useful concept of an exceptional family of elements for variational inequality problems rather than complementarity problems as in the past, we propose such a concept. It generalizes previous ones to multivalued variational inequalities in general normed spaces and allows us to obtain several existence results for variational inequalities corresponding to earlier ones for complementarity problems. Compared with the existing literature, we consider problems in more general spaces and under considerably weaker assumptions on the defining map.