Jiye Han

Exceptional families and existence theorems for variational inequality problems

This paper introduces the concept of exceptional family for nonlinear variational inequality problems. Among other things, we show that the nonexistence of an exceptional family is a sufficient condition for the existence of a solution to variational inequalities. This sufficient condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. From the results in this paper, we believe that the concept of exceptional families of variational inequalities provides a new powerful tool for the study of the existence theory for variational inequalities.

Exceptional Family of Elements for a Variational Inequality Problem and its Applications

This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P* -mappings is also obtained.

Exact Penalty Functions for Convex Bilevel Programming Problems

In this paper, we propose a new constraint qualification for convex bilevel programming problems. Under this constraint qualification, a locally and globally exact penalty function of order 1 for a single-level reformulation of convex bilevel programming problems is given without requiring the linear independence condition and the strict complementarity condition to hold in the lower-level problem. Based on these results, locally and globally exact penalty functions for two other single-level reformulations of convex bilevel programming problems can be obtained. Furthermore, sufficient conditions for partial calmness to hold in some single-level reformulations of convex bilevel programming problems can be given.

An entropic regularization approach for mathematical programs with equilibrium constraints

A new smoothing approach based on entropic regularization is proposed for solving a mathematical program with equilibrium constraints (MPEC). With some known smoothing properties of the entropy function and keeping real practice in mind, we reformulate an MPEC problem as a smooth nonlinear programming problem. In this way, a difficult MPEC problem becomes solvable by using available nonlinear optimization software. To support our claims, we use an online solver and test the performance of the proposed approach on a set of well-known test problems.

A Nonmonotone Trust Region Method for Nonlinear Programming with Simple Bound Constraints

Abstract. In this paper we propose a nonmonotone trust region algorithm for optimization with simple bound constraints. Under mild conditions, we prove the global convergence of the algorithm. For the monotone case it is also proved that the correct active set can be identified in a finite number of iterations if the strict complementarity slackness condition holds, and so the proposed algorithm reduces finally to an unconstrained minimization method in a finite number of iterations, allowing a fast asymptotic rate of convergence. Numerical experiments show that the method is efficient.

Two interior-point methods for nonlinear P * t-complementarity problems

Two interior-point algorithms using a wide neighborhood of the central path are proposed to solve nonlinear P*-complementarity problems. The proof of the polynomial complexity of the first method requires the problem to satisfy a scaled Lipschitz condition. When specialized to monotone complementarity problems, the results of the first method are similar to those in Ref. 1. The second method is quite different from the first in that the global convergence proof does not require the scaled Lipschitz assumption. However, at each step of this algorithm, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied.

On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that under some milder than usual assumptions the proposed algorithm finds an exact solution of VLCP in a finite number of iterations. Some computational results are included to illustrate the potential of this approach.

Newton and quasi-Newton methods for normal maps with polyhedral sets

AbstractWe present a generalized Newton method and a quasi-Newton method forsolving

A class of nonmonotone trust region algorithms for unconstrained optimization problems

H(x): = F(\prod {_c } (x)) + x - \prod {_c } (x) = 0