Well-Posedness by Perturbations of Variational Problems

In this paper, we consider the extension of the notion of well-posedness by perturbations, introduced by Zolezzi for optimization problems, to other related variational problems like inclusion problems and fixed-point problems. Then, we study the conditions under which there is equivalence of the well-posedness in the above sense between different problems. Relations with the so-called diagonal well-posedness are also given. Finally, an application to staircase iteration methods is presented.     

The Equivalence of Several Basic Theorems for Subdifferentials

Several different basic tools are used for studying subdifferentials. They are a nonlocal fuzzy sum rule in (Borwein et al., 1996; Zhu, 1996), a multidirectional mean value theorem in (Clarke and Ledyaev, 1994; Clarke et al., 1998), local fuzzy sum rules in (Ioffe, 1984, 1990) and an extremal principle in (Kruger and Mordukhovich, 1980; Mordukhovich, 1976, 1980, 1994). We show that all these basic results are equivalent and discuss some interesting consequences of this equivalence.     

Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.     

Extremal Solutions for Quasilinear Elliptic Inclusions in All of ℝN with State-Dependent Subdifferentials

This paper deals with quasilinear elliptic differential inclusions defined in all of ^N and governed in general by a nonpotential quasilinear elliptic operator of the Leray–Lions type and a multivalued term in form of a (nonmonotone) state-dependent subdifferential. We prove the existence of entire extremal solutions within a sector of an ordered pair of appropriately defined upper and lower solutions without imposing any condition at infinity. Therefore, standard variational methods cannot be applied here. Furthermore, due to the unboundedness of the domain and due to lack of monotonicity of the operators involved, no comparison results are available such that the problem under consideration becomes even more difficult.     

On optimality conditions for vector variational inequalities

Optimality conditions for weak efficient, global efficient and efficient solutions of vector variational inequalities with constraints defined by equality, cone and set constraints are derived. Under various constraint qualifications, necessary optimality conditions for weak efficient, global efficient and efficient solutions in terms of the Clarke and Michel–Penot subdifferentials are established. With assumptions on quasiconvexity of constraint functions sufficient optimality conditions are also given.     

Parametric Method for Global Optimization

This paper considers constrained and unconstrained parametric global optimization problems in a real Hilbert space. We assume that the gradient of the cost functional is Lipschitz continuous but not smooth. A suitable choice of parameters implies the linear or superlinear (supergeometric) convergence of the iterative method. From the numerical experiments, we conclude that our algorithm is faster than other existing algorithms for continuous but nonsmooth problems, when applied to unconstrained global optimization problems. However, because we solve 2n + 1 subproblems for a large number n of independent variables, our algorithm is somewhat slower than other algorithms, when applied to constrained global optimization.This work was partially supported by the NATO Outreach Fellowship - Mathematics 219.33.We thank Professor Hans D. Mittelmann, Arizona State University, for cooperation and support.     

Mean-Value Theorem with Small Subdifferentials

We prove a mean-value theorem for lower semicontinuous functions on a large class of Banach spaces which contains the class of Asplund spaces, in particular reflexive Banach spaces and Banach spaces with a separable dual. It involves the lower subdifferential (or contingent subdifferential) and the Fréchet subdifferentials, which are among the smallest subdifferentials known to date. It follows that the estimates which it provides require weak assumptions and are accurate. When the function is locally Lipschitzian, we get a simple statement which refines the Lebourg mean-value theorem.     

On Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Equilibrium Constraints

In this article, necessary conditions of Fritz John type for weak efficient solutions of a nonsmooth vector equilibrium problem involving equilibrium constraints (VEPEC) in terms of the Clarke subdifferentials are established. Under constraint qualifications which are suitable for (VEPEC), necessary conditions of Kuhn-Tucker type for efficiency are derived. Under assumptions on generalized convexity of data, sufficient conditions for efficiency are developed. Some applications to vector variational inequalities and vector optimization problems with equilibrium constraints are also given.     

Periodic solutions to nonlinear one dimensional wave equation with -dependent coefficients

This paper deals with -periodicity and regularity of solutions to the one dimensional nonlinear wave equation with -dependent coefficients

     

On Conjugations for Functions with Values in Extensions of Ordered Groups

We continue our study of generalized conjugations for functions with values in the canonical enlargement of a complete ordered group, started in [10], which encompass various kinds of known conjugations and polarities. We obtain extensions, to this framework, of some results on d.c. duality theory and subdifferentials, and we give some applications to conjugations and subdifferentials for functions with values in .