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.      

The generalized optimality conditions of multiobjective programming problem in topological vector space

In this study, an alternative theorem for the subconvexlike mapping in topological vector space is established. With this alternative theorem as an aid, the generalized Fritz John conditions and the generalized Kuhn-Tucker conditions in terms of Gâteaux derivatives of multiobjective programming problem in the ordered topological vector space are given.     

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.     

Test problem construction for linear bilevel programming problems

A method of constructing test problems for linear bilevel programming problems is presented. The method selects a vertex of the feasible region, far away from the solution of the relaxed linear programming problem, as the global solution of the bilevel problem. A predetermined number of constraints are systematically selected to be assigned to the lower problem. The proposed method requires only local vertex search and solutions to linear programs.     

On bilevel programming,Part I: General nonlinear cases

This paper is concerned with general nonlinear nonconvex bilevel programming problems (BLPP). We derive necessary and sufficient conditions at a local solution and investigate the stability and sensitivity analysis at a local solution in the BLPP. We then explore an approach in which a bundle method is used in the upper-level problem with subgradient information from the lower-level problem. Two algorithms are proposed to solve the general nonlinear BLPP and are shown to converge to regular points of the BLPP under appropriate conditions. The theoretical analysis conducted in this paper seems to indicate that a sensitivity-based approach is rather promising for solving general nonlinear BLPP.This research is sponsored by the Office of Naval Research under contract N00014-89-J-1537.     

Necessary optimality conditions and regularity results for state and adjoint state of nonlinear control problems

For optimal control problems of nonlinear evolution equations of first order we derive regularity properties for the corresponding adjoint states where the proof is based on the necessary optimal conditions and smoothness properites of the optimal states. These results are used for the characterization of the optimal solution and the corresponding adjoint state of the periodic three-dimensionel NAVIER-STOKES system.     

Interactive fuzzy goal programming approach for bilevel programming problem

This paper presents an interactive fuzzy goal programming (FGP) approach for bilevel programming problems with the characteristics of dynamic programming (DP).     

An inexact-restoration method for nonlinear bilevel programming problems

We present a new algorithm for solving bilevel programming problems without reformulating them as single-level nonlinear programming problems. This strategy allows one to take profit of the structure of the lower level optimization problems without using non-differentiable methods. The algorithm is based on the inexact-restoration technique. Under some assumptions on the problem we prove global convergence to feasible points that satisfy the approximate gradient projection (AGP) optimality condition. Computational experiments are presented that encourage the use of this method for general bilevel problems. This work was supported by PRONEX-Optimization (PRONEX—CNPq/FAPERJ E-26/171.164/2003—APQ1), FAPESP (Grants 06/53768-0 and 05-56773-1) and CNPq.     

A dynamic programming algorithm for the bilevel knapsack problem

We propose an efficient dynamic programming algorithm for solving a bilevel program where the leader controls the capacity of a knapsack, and the follower solves the resulting knapsack problem. We propose new recursive rules and show how to solve the problem as a sequence of two standard knapsack problems.     

Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution

We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.     

Solving nonlinear principal-agent problems using bilevel programming

While significant progress has been made, analytic research on principal-agent problems that seek closed-form solutions faces limitations due to tractability issues that arise because of the mathematical complexity of the problem. The principal must maximize expected utility subject to the agent’s participation and incentive compatibility constraints. Linearity of performance measures is often assumed and the Linear, Exponential, Normal (LEN) model is often used to deal with this complexity. These assumptions may be too restrictive for researchers to explore the variety of relationships between compensation contracts offered by the principal and the effort of the agent. In this paper we show how to numerically solve principal-agent problems with nonlinear contracts. In our procedure, we deal directly with the agent’s incentive compatibility constraint. We illustrate our solution procedure with numerical examples and use optimization methods to make the problem tractable without using the simplifying assumptions of a LEN model. We also show that using linear contracts to approximate nonlinear contracts leads to solutions that are far from the optimal solutions obtained using nonlinear contracts. A principal-agent problem is a special instance of a bilevel nonlinear programming problem. We show how to solve principal-agent problems by solving bilevel programming problems using the ellipsoid algorithm. The approach we present can give researchers new insights into the relationships between nonlinear compensation schemes and employee effort.     

Some properties of the bilevel programming problem

The purpose of this paper is to elaborate on the difficulties accompanying the development of efficient algorithms for solving the bilevel programming problem (BLPP). We begin with a pair of examples showing that, even under the best of circumstances, solutions may not exist. This is followed by a proof that the BLPP is NP-hard.This work was partially supported by a grant from the Advanced Research Program of the Texas Higher Education Coordinating Board.     

The bilevel programming problem: reformulations, constraint qualifications and optimality conditions

We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.     

DC programming techniques for solving a class of nonlinear bilevel programs

We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a quadratic programming problem. This method is a combination of the local algorithm DCA in DC programming with a branch and bound scheme well known in discrete and global optimization. Computational results on a class of quadratic bilevel programs are reported.     

A bilevel programming approach to the travelling salesman problem

We show that the travelling salesman problem is polynomially reducible to a bilevel toll optimization program. Based on natural bilevel programming techniques, we recover the lifted Miller-Tucker-Zemlin constraints. Next, we derive an O(n²) multi-commodity extension whose LP relaxation is comparable to the exponential formulation of Dantzig, Fulkerson and Johnson.     

Descent approaches for quadratic bilevel programming

The bilevel programming problem involves two optimization problems where the data of the first one is implicitly determined by the solution of the second. In this paper, we introduce two descent methods for a special instance of bilevel programs where the inner problem is strictly convex quadratic. The first algorithm is based on pivot steps and may not guarantee local optimality. A modified steepest descent algorithm is presented to overcome this drawback. New rules for computing exact stepsizes are introduced and a hybrid approach that combines both strategies is discussed. It is proved that checking local optimality in bilevel programming is a NP-hard problem.Support of this work has been provided by INIC (Portugal) under Contract 89/EXA/5, by FCAR (Québec), and by NSERC and DND-ARP (Canada).     

Second order optimality conditions for generalized semi-infinite programming problems

We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditionsfor local optimality are given. The conditions are derived under assumptions such that the feasible set can be described by means of a finite number of optimal value functions. Since we do not require a strict complementary condition for the local reduction these functions are only of class C¹ A sufficient condition for optimality is proven under much weaker assumptions.     

Interactive fuzzy decision making method for solving bilevel programming problem

In this paper, an interactive fuzzy decision making method is proposed for solving bilevel programming problem. Introducing a new balance function, we consider the overall satisfactory balance between the leader and the follower. Then, a satisfactory solution can be obtained by the proposed method. Finally, numerical examples are reported to illustrate the feasibility of the proposed method.     

New formulations and valid inequalities for a bilevel pricing problem

Consider the problem of maximizing the toll revenue collected on a multi-commodity transportation network. This fits a bilevel framework where a leader sets tolls, while users respond by selecting cheapest paths to their destination. We propose novel formulations of the problem, together with valid inequalities yielding improved algorithms.