Optimality conditions for mixed discrete bilevel optimization problems

In this article, we consider bilevel optimization problems with discrete lower level and continuous upper level problems. Taking into account both approaches (optimistic and pessimistic) which have been developed in the literature to deal with this type of problem, we derive some conditions for the existence of solutions. In the case where the lower level is a parametric linear problem, the bilevel problem is transformed into a continuous one. After that, we are able to discuss local optimality conditions using tools of variational analysis for each of the different approaches. Finally, we consider a simple application of our results namely the bilevel programming problem with the minimum spanning tree problem in the lower level.     

Necessary optimality conditions for bilevel minimization problems

In this paper we study bilevel minimization problems. Using the implicit function theorem, variational analysis and exact penalty results we establish necessary optimality conditions for these problems.     

Test problem generation for quadratic-linear pessimistic bilevel optimization

A generation method of quadratic-linear bilevel optimization test problems in a pessimistic formulation is proposed and justified. Propositions about the exact form and the number of local and global pessimistic solutions in generated problems are proved.     

Optimality conditions for vector equilibrium problems

In this paper, we present the necessary and sufficient conditions for weakly efficient solution, Henig efficient solution, globally efficient solution, and superefficient solution to the vector equilibrium problems with constraints. As applications, we give the necessary and sufficient conditions for corresponding solution to the vector variational inequalities and vector optimization problems.     

Optimality conditions for nonlinear infinite programming problems

An infinite programming problem consists in minimizing a functional defined on a real Banach space under an infinite number of constraints. The main purpose of this article is to provide sufficient conditions of optimality under generalized convexity assumptions. Such conditions are necessarily satisfied when the problem possesses the property that every stationary point is a global minimizer.     

Necessary optimality conditions for bilevel set optimization problems

Bilevel programming problems are hierarchical optimization problems where in the upper level problem a function is minimized subject to the graph of the solution set mapping of the lower level problem. In this paper necessary optimality conditions for such problems are derived using the notion of a convexificator by Luc and Jeyakumar. Convexificators are subsets of many other generalized derivatives. Hence, our optimality conditions are stronger than those using e.g., the generalized derivative due to Clarke or Michel-Penot. Using a certain regularity condition Karush-Kuhn-Tucker conditions are obtained.      

Optimality conditions for reflecting boundary control problems

We consider a control problem with reflecting boundary and obtain necessary optimality conditions in the form of the maximum Pontryagin principle. To derive these results we transform the constrained problem in an unconstrained one or we use penalization techniques of Morreau-Yosida type to approach the original problem by a sequence of optimal control problems with Lipschitz dynamics. Then nonsmooth analysis theory is used to study the convergence of the penalization in order to obtain optimality conditions.     

First-order necessary optimality conditions for general bilevel programming problems

We formulate in this paper several versions of the necessary conditions for general bilevel programming problems. The technique used is related to standard methods of nonsmooth analysis. We treat separately the following cases: Lipschitz case, differentiable case, and convex case. Many typical examples are given to show the efficiency of theoretical results. In the last section, we formulate the general multilevel programming problem and give necessary conditions of optimality in the general case. We illustrate then the application of these conditions by an example.Lecturer, Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, Montreal, Canada.The author is indebted to Professor M. Florian for support and encouragement in the writing of this paper.     

Second order optimality conditions for bilevel set optimization problems

In this paper, we introduce a new notion of augmenting function known as indicator augmenting function to establish a minmax type duality relation, existence of a path of solution converging to optimal value and a zero duality gap relation for a nonconvex primal problem and the corresponding Lagrangian dual problem. We also obtain necessary and sufficient conditions for an exact penalty representation in the framework of indicator augmented Lagrangian.     

Optimality conditions for state constrained nonlinear control problems

The problem of optimal control of nonlinear control and state constrained control problems, where the state constraint may involve differential operators and the cost functionals may be nonsmooth, is studied. For this class of problems, necessary optimality conditions using techniques from infinite dimensional optimization theory adapted to the framework of control problems are derived. It is shown that the underlying structure admits a considerable relaxation of the classical constraint qualifications. The theory then is applied to examples of various nonlinear elliptic equations and state constraints.     

Optimality conditions for discrete calculus of variations problems

Nonlinear discrete calculus of variations problems with variable endpoints and with equality type constraints on trajectories are considered. We derive new nontrivial first- and second-order necessary optimality conditions.     

Optimality conditions for directionally differentiable multi-objective programming problems

This paper is concerned with the optimality for multi-objective programming problems with nonsmooth and nonconvex (but directionally differentiable) objective and constraint functions. The main results are Kuhn-Tucker type necessary conditions for properly efficient solutions and weakly efficient solutions. Our proper efficiency is a natural extension of the Kuhn-Tucker one to the nonsmooth case. Some sufficient conditions for an efficient solution to be proper are also given. As an application, we derive optimality conditions for multi-objective programming problems including extremal-value functions.This work was done while the author was visiting George Washington University, Washington, DC.     

Optimality conditions in optimization problems with convex feasible set using convexificators

In this paper, we consider a nonsmooth optimization problem with a convex feasible set described by constraint functions which are neither convex nor differentiable nor locally Lipschitz necessarily. Utilizing upper regular convexificators, we characterize the normal cone of the feasible set and derive KKT type necessary and sufficient optimality conditions. Under some assumptions, we show that the set of KKT multipliers is bounded. We also characterize the set of optimal solutions and introduce a linear approximation corresponding to the original problem which is useful in checking optimality. The obtained outcomes extend various results existing in the literature to a more general setting.     

Optimality conditions for vector equilibrium problems in normed spaces

By using the concepts of contingent epiderivative, radial epiderivative, Clarke tangent epiderivative and Y-epiderivative, we present necessary and sufficient conditions for the weakly efficient solution, the Henig efficient solution, and the globally proper efficient solution, respectively, to vector equilibrium problems with constraints.     

Optimality conditions of higher order for abnormal minimization problems

Moscow. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 15–23, July–August, 1992.     

Optimality conditions using sum-convex approximations

The purpose of this paper is to give necessary and sufficient conditions of optimality for a general mathematical programming problem, using not a linear approximation to the constraint function but an approximation possessing certain convexity properties. Such approximations are called sum-convex. Theorems of the alternative involving sum-convex functions are also presented as part of the proof.This work is part of the author's PhD Thesis under the supervision of Professor S. Zlobec at McGill University.