Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.     

线性约束三次规划问题的全局最优性必要条件和最优化算法

讨论了带线性不等式约束三次规划问题的最优性条件和最优化算法. 首先, 讨论了带有线性不等式约束三次规划问题的 全局最优性必要条件. 然后, 利用全局最优性必要条件, 设计了解线性约束三次规划问题的一个新的局部最优化算法(强局部最优化算法). 再利用辅助函数和所给出的新的局部最优化算法, 设计了带有线性不等式约束三 规划问题的全局最优化算法. 最后, 数值算例说明给出的最优化算法是可行的、有效的.     

Necessary optimality conditions for abnormal problems with geometric constraints

The abnormal minimization problem with a finite-dimensional image and geometric constraints is examined. In particular, inequality constraints are included. Second-order necessary conditions for this problem are established that strengthen previously known results.     

Necessary optimality conditions for nonsmooth semi-infinite programming problems

In this paper, for a nonsmooth semi-infinite programming problem where the objective and constraint functions are locally Lipschitz, analogues of the Guignard, Kuhn-Tucker, and Cottle constraint qualifications are given. Pshenichnyi-Levin-Valadire property is introduced, and Karush-Kuhn-Tucker type necessary optimality conditions are derived.     

Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality

In this paper, we propose two sets of theoretically filtered bound-factor constraints for constructing reformulation-linearization technique (RLT)-based linear programming (LP) relaxations for solving polynomial programming problems. We establish related theoretical results for convergence to a global optimum for these reduced sized relaxations, and provide insights into their relative sizes and tightness. Extensive computational results are provided to demonstrate the relative effectiveness of the proposed theoretical filtering strategies in comparison to the standard RLT and a prior heuristic filtering technique using problems from the literature as well as randomly generated test cases.     

Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems

This paper is devoted to the study of nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. We introduce a constraint qualification which is based on the Mordukhovich subdifferential. Then, we derive a Fritz–John type necessary optimality condition. Finally, interrelations between the new and the existing constraint qualifications such as the Mangasarian–Fromovitz, linear independent, and the Slater are investigated.     

Second-order sufficient optimality conditions for local and global nonlinear programming

This paper presents a new approach to the sufficient conditions of nonlinear programming. Main result is a sufficient condition for the global optimality of a Kuhn-Tucker point. This condition can be verified constructively, using a novel convexity test based on interval analysis, and is guaranteed to prove global optimality of strong local minimizers for sufficiently narrow bounds. Hence it is expected to be a useful tool within branch and bound algorithms for global optimization.     

Necessary optimality conditions for Stackelberg problems

First-order necessary optimality conditions are derived for a class of two-level Stackelberg problems in which the followers' lower-level problems are convex programs with unique solutions. To this purpose, generalized Jacobians of the marginal maps corresponding to followers' problems are estimated. As illustrative examples, two discretized optimum design problems with elliptic variational inequalities are investigated. The theoretical results may be used also for the numerical solution of the Stackelberg problems considered by nondifferentiable optimization methods.Communicated by M. Simaan     

Necessary optimality conditions for differential-difference inclusions with state constraints

We consider the Mayer optimal control problem with dynamics given by a nonconvex differential-difference inclusion, whose trajectories are constrained to a closed set. Necessary optimality conditions in the form of the maximum principle are obtained.     

On optimality conditions for structured families of nonlinear programming problems

Optimality conditions for families of nonlinear programming problems inR ⁿ are studied from a generic point of view. The objective function and some of the constraints are assumed to depend on a parameter, while others are held fixed. Techniques of differential topology are used to show that under suitable conditions, certain strong second-order conditions are necessary for optimality except possibly for parameter values lying in a negligible set.Research sponsored, in part, by the Air Force Office of Scientific Research, under grants number 77-3204 and 79-0120.     

New approach to optimality conditions for degenerate nonlinear programming problems

In the work, a new approach to constructing optimality conditions for degenerate smooth optimization problems with inequality constraints is proposed. The approach is based on the theory of p-regularity. A special case of degeneracy, when the first derivatives of some function-constraints are equal to zero up to some order, is considered. Optimality conditions for the general case of degeneracy with p = 2 are presented. Proposed constructions and optimality conditions are illustrated by an example. A general case of degeneracy is considered and optimality conditions for the case of p ≥ 2 are proposed.     

Necessary optimality conditions for a set-valued fractional extremal programming problem under inclusion constraints

In this paper, we are concerned with a set-valued fractional extremal programming problem under inclusion constraints. Our approach consists of using the extremal principle (an approach initiated by Mordukhovich, which does not involve any convex approximations and convex separation arguments) for the study of necessary optimality conditions.     

Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints

This paper studies multiobjective optimal control problems in presence of constraints in the discrete time framework. Both the finite- and infinite-horizon settings are considered. The paper provides necessary conditions of Pareto optimality under lighter smoothness assumptions compared to the previously obtained results. These conditions are given in the form of weak and strong Pontryagin principles which generalize the existing ones. To obtain some of these results, we provide new multiplier rules for multiobjective static optimization problems and new Pontryagin principles for the finite horizon multiobjective optimal control problems.     

Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions

In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. The work of Z. Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.     

Necessary conditions and duality for inexact nonlinear semi-infinite programming problems

First order necessary conditions and duality results for general inexact nonlinear programming problems formulated in nonreflexive spaces are obtained. The Dubovitskii–Milyutin approach is the main tool used. Particular cases of linear and convex programs are also analyzed and some comments about a comparison of the obtained results with those existing in the literature are given.     

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.