Jane J. Ye

Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints

In this paper we study optimization problems with variational inequality constraints in finite dimensional spaces. Kuhn-Tucker type necessary optimality conditions involving coderivatives are given under certain constraint qualifications including one that ensures nonexistence of non-trivial abnormal multipliers. The result is applied to bilevel programming problems to obtain Kuhn-Tucker type necessary optimality conditions. The Kuhn-Tucker type necessary optimality conditions are shown to be satisfied without any constraint qualification by the class of bilevel programming problems where the lower level is a parametric linear quadratic problem.

Optimality conditions for bilevel programming problems

The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minmax problem satisfy the calmness condition automatically. A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementar¬ity lower level problem are given. First order necessary optimality condition are given using nonsmooth analysis. Second order sufficient optimality conditions are also give...

Exact Penalization and Necessary Optimality Conditions for Generalized Bilevel Programming Problems

The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the upper level decision variable, then using certain uniform parametric error bounds as penalty functions gives single level problems equivalent to the GBLP. Several local and global uniform parametric error bounds are presented, and assumptions guaranteeing that they apply are discussed. We then derive Kuhn--Tucker-type necessary optimality conditions by using exact penalty formulations and nonsmooth analysis.

Constraint Qualifications and Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints

A very general optimization problem with a variational inequality constraint, inequality constraints, and an abstract constraint are studied. Fritz John type and Kuhn--Tucker type necessary optimality conditions involving Mordukhovich coderivatives are derived. Several constraint qualifications for the Kuhn--Tucker type necessary optimality conditions involving Mordukhovich coderivatives are introduced and their relationships are studied. Applications to bilevel programming problems are also given.

Optimality Conditions for Optimization Problems with Complementarity Constraints

Optimization problems with complementarity constraints are closely related to optimization problems with variational inequality constraints and bilevel programming problems. In this paper, under mild constraint qualifications, we derive some necessary and sufficient optimality conditions involving the proximal coderivatives. As an illustration of applications, the result is applied to the bilevel programming problems where the lower level is a parametric linear quadratic problem.

On error bounds for lower semicontinuous functions

Abstract.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.

Multiobjective optimization problem with variational inequality constraints

Abstract. We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem.

First-Order and Second-Order Conditions for Error Bounds

For a lower semicontinuous function f on a Banach space X, we study the existence of a positive scalar

Nondifferentiable Multiplier Rules for Optimization and Bilevel Optimization Problems

\mu

Sufficient Conditions for Error Bounds

such that the distance function dS associated with the solution set S of