The nonlinear bilevel programming problem:formulations,regularity and optimality conditions

In this paper, we study regularity and optimality conditions for the BLPP by using a marginal function formulation, where the marginal function is defined by the optimal value function of the lower problem. We address the regularity issue by exploring the structure of the tangent cones of the feasible set of the BLPP. These regularity results indicate that the nonlinear/nonlinear BLPP is most likely degenerate and a class of nonlinear/linear BLPP is regular in the conventional sense. Existence of exact penalty function is proved for a class of nonlinear/linear BLPP. Fritz-John type optimality conditions are derived for nonlinear BLPP, while KKT type conditions are obtained for a class of nonlinear/linear BLPP in the framework of nonsmooth analysis. A typical example is examined for these conditions and some applications of these conditions are pointed out     

Geometry of optimality conditions and constraint qualifications

Certain types of necessary optimality conditions for mathematical programming problems are equivalent to corresponding regularity conditions on the constraint set. For any problem, a certain natural optimality condition, dependent upon the particular constraint set, is always satisfied. This condition can be strengthened in numerous ways by invoking appropriate regularity assumptions on the constraint set. Results are presented for Euclidean spaces and some extensions to Banach spaces are given.This work was supported in part by the Office of Naval Research, Contract No. N00014-67-A-0321-0003 (NR-047-095).     

Geometry of optimality conditions and constraint qualifications: The convex case

The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem. In addition, the badly behaved set of constraints, i.e. the set of constraints which causes problems in the Kuhn—Tucker theory, is isolated and a computational procedure for checking whether a feasible point is regular or not is presented.This research was supported by the National Research Council of Canada and le Gouvernement du Quebec and is part of the author's Ph.D. Dissertation done at McGill University, Montreal, Que., under the guidance of Professor S. Zlobec.     

Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.     

Single-level reformulations of a specific non-smooth bilevel programming problem and their applications

Optimization Letters - We study a bilevel programming problem (BLPP) with a maximin lower level problem which is non-smooth and sometimes even non-convex. We reformulate such a non-smooth BLPP as a...     

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.     

Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications

We derive first- and second-order necessary optimality conditions for set-constrained optimization problems under the constraint qualification-type conditions significantly weaker than Robinson’s constraint qualification. Our development relies on the so-called 2-regularity concept, and unifies and extends the previous studies based on this concept. Specifically, in our setting constraints are given by an inclusion, with an arbitrary closed convex set on the right-hand side. Thus, for the second-order analysis, some curvature characterizations of this set near the reference point must be taken into account.      

New necessary and sufficient optimality conditions for strong bilevel programming problems

In this paper we are interested in a strong bilevel programming problem (S). For such a problem, we establish necessary and sufficient global optimality conditions. Our investigation is based on the use of a regularization of problem (S) and some well-known global optimization tools. These optimality conditions are new in the literature and are expressed in terms of \(\max \)–\(\min \) conditions with linked constraints.     

Nonsmooth multiobjective programming and constraint qualifications

We consider a nonsmooth multiobjective programming problem with inequality and set constraints. By using the notion of convexificator, we extend the Abadie constraint qualification, and derive the strong Kuhn-Tucker necessary optimality conditions. Some other constraint qualifications have been generalized and their interrelations are investigated.     

Necessary and sufficient conditions for some constraint qualifications in quasiconvex programming

In this paper, we investigate relations between constraint qualifications in quasiconvex programming. At first, we show a necessary and sufficient condition for the closed cone constraint qualification for quasiconvex programming (Q-CCCQ), and investigate some sufficient conditions for the Q-CCCQ. Also, we consider a relation between the Q-CCCQ and the basic constraint qualification for quasiconvex programming (Q-BCQ) and we compare the Q-BCQ with some constraint qualifications.     

Locally Lipschitz vector optimization problems: second-order constraint qualifications,regularity condition and KKT necessary optimality conditions

Positivity - In the present paper, we are concerned with a class of constrained vector optimization problems, where the objective functions and active constraint functions are locally Lipschitz at...     

均衡约束数学规划的约束规格和最优性条件综述

约束规格在约束优化问题的最优性条件中起着重要的作用,介绍了近几年国际上关于均衡约束数学规划(简记为MPEC)的约束规格以及最优性条件的研究成果, 包括以下主要内容: (1) MPEC常用的约束规格(如线性无关约束规格 (MPEC-LICQ)、Mangasarian-Fromovitz约束规格 (MPEC-MFCQ)等)和新的约束规格(如恒秩约束规格、常数正线性相关约束规格等), 以及它们之间的关系; (2) MPEC常用的稳定点; (3) MPEC的最优性条件. 最后还对MPEC的约束规格和最优性条件的研究前景进行了探讨.     

Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming

For an inequality system defined by an infinite family of proper convex functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for constrained minimization problems to have total Lagrangian dualities. Several known results in the conic programming problem are extended and improved.     

On constraint qualifications for second-order optimality conditions depending on a single Lagrange multiplier

We present new constraint qualifications (CQs) to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single Lagrange multiplier, instead of the whole set of Lagrange multipliers. For each condition, we characterize the weakest CQ that guarantees its fulfillment at local minimizers, while proposing new weak conditions implying them. Relations with other CQs are discussed.     

On constraint qualifications in nonlinear programming

In this paper we give conditions for deriving the inconsistency of an inequality system of positively homogeneous functions starting from the inconsistency of another one. When the impossibility of the starting system represents a necessary optimality condition for an inequality constrained extremum problem and the positively homogeneous functions involved have suitable properties of convexity, such conditions collapse into the well known constraint qualifications.     

General constraint qualifications in nondifferentiable programming

We show that a familiar constraint qualification of differentiable programming has nonsmooth counterparts. As a result, necessary optimality conditions of Kuhn—Tucker type can be established for inequality-constrained mathematical programs involving functions not assumed to be differentiable, convex, or locally Lipschitzian. These optimality conditions reduce to the usual Karush—Kuhn—Tucker conditions in the differentiable case and sharpen previous results in the locally Lipschitzian case.     

Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications

A survey is given of old and new results on the sensitivity of solutions to systems of optimality conditions with respect to parametric perturbations. Results of this kind play a key role in subtle convergence analysis of various constrained optimization algorithms. General systems of optimality conditions for problems with abstract constraints, Karush-Kuhn-Tucker systems for mathematical programs, and Lagrange systems for problems with equality constraints are examined. Special attention is given to the cases where the traditional constraint qualifications are violated.     

On sufficient optimality conditions for a quasiconvex programming problem

Some remarks are made on a paper by Bector, Chandra, and Bector (see Ref. 1) concerning the Fritz John and Kuhn-Tucker sufficient optmality conditions as well as duality theorems for a nonlinear programming problem with a quasiconvex objective function.This research was supported by the Italian Ministry of University Scientific and Technological Research.     

Sufficient optimality conditions and duality for a quasiconvex programming problem

Under differentiability assumptions, Fritz John Sufficient optimality conditions are proved for a nonlinear programming problem in which the objective function is assumed to be quasiconvex and the constraint functions are assumed to quasiconcave/strictly pseudoconcave. Duality theorems are proved for Mond-Weir type duality under the above generalized convexity assumptions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are thankful to Professor B. Mond for suggestions that improved the original draft of the paper.