Optimality conditions and strong duality in abstract and continuous-time linear programming

In this work, optimality conditions for infinite-dimensional linear programs are considered. Strong duality as an optimality condition is investigated. A new approach to duality in the form of positive extendability of linear functionals is proposed. A necessary and sufficient condition for duality in the form of a boundedness test of a related linear program is developed. Elaborating on the continuous time framework, counter cases where duality is not valid are given. In lieu of duality, other generalized duality conditions are proposed for the purpose of testing the optimality of a solution.     

Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.The authors are thankful to the referees and Professor P. L. Yu for their many useful comments and suggestions which have improved the presentation of the paper.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 also thankful to the Dean's Office, Faculty of Management, University of Manitoba, for the financial support provided for the third author's visit to the Faculty.     

Optimality conditions in smooth nonlinear programming

This survey is concerned with necessary and sufficient optimality conditions for smooth nonlinear programming problems with inequality and equality constraints. These conditions deal with strict local minimizers of order one and two and with isolated minimizers. In most results, no constraint qualification is required. The optimality conditions are formulated in such a way that the gaps between the necessary and sufficient conditions are small and even vanish completely under mild constraint qualifications.This paper is dedicated to the memory of W. Wetterling.The authors would like to thank Wolfgang Wetterling and Frank Twilt for fruitful discussions and an anonymous referee for many valuable comments.     

Optimality conditions for sparse nonlinear programming

The sparse nonlinear programming(SNP) is to minimize a general continuously differentiable function subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Fréchet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-KuhnTucker(KKT) conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.     

Optimality and duality in nonlinear programming involving semilocally B-preinvex and related functions

A nonlinear programming problem is considered where the functions involved are η-semidifferentiable. Fritz John and Karush–Kuhn–Tucker types necessary optimality conditions are obtained. Moreover, a result concerning sufficiency of optimality conditions is given. Wolfe and Mond–Weir types duality results are formulated in terms of η-semidifferentials. The duality results are given using concepts of generalized semilocally B-preinvex functions.     

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.     

Optimality conditions for piecewiseC 2 nonlinear programming

Several types of finite-dimensional nonlinear programming models are considered in this article. Second-order optimality conditions are derived for these models, under the assumption that the functions involved are piecewiseC ². In rough terms, a real-valued function defined on an open subsetW orR ⁿ is said to be piecewiseC ^k onW if it is continuous onW and if it can be constructed by piecing together onW a finite number of functions of classC ^k .     

Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions

In this paper, a class of functions called B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions are introduced by relaxing definitions of arcwise connected function (CN) and B-vex function. The differential properties of B-arcwise connected function (BCN) are studied. Their two extreme properties are proved. The necessary and sufficient optimality conditions are obtained for the nondifferentiable nonlinear semi-infinite programming involving B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions. Mond-Weir type duality results have also been established.     

极小扰动约束非线性规划的最优性条件

考虑当目标函数在约束条件下的最优值作扰动时，使各约束作极小扰动的非线性规划问题．文中引进了极小扰动约束规划的极小扰动有效解概念．利用把问题归为一个相应的多目标规划问题，给出了极小扰动约束有效解的最优性条件．     

Optimality conditions and global convergence for nonlinear semidefinite programming

Mathematical Programming - Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear...     

Optimality conditions and duality for a minimax fractional programming with generalized convexity

We establish the sufficient conditions for generalized fractional programming from a viewpoint of the generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for two types of duals of the generalized fractional programming. We extend the corresponding results of several authors.     

Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity

In this paper, we consider nondifferentiable multiobjective fractional programming problems. A concept of generalized convexity, which is called (C,α,ρ,d)-convexity, is first discussed. Based on this generalized convexity, we obtain efficiency conditions for multiobjective fractional programming (MFP). Furthermore, we establish duality results for three types of dual problems of (MFP) and present the corresponding duality theorems.     

Optimality conditions and duality in continuous programming II. The linear problem revisited

This is Part II of a two-part paper; the purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of infinite-dimensional programming, and (b) to obtain fruitful applications in continuous time programming. In Part II the continuous time version of Farkas' theorem developed in Part I serves as the foundation for the duality theory for a broad class of linear continuous time programming problems distinct from those previously examined. In particular, we establish duality under analytic conditions, e.g., whether the given functions are measurable or continuous, that are weaker, and algebraic conditions that are more general, than those previously imposed. The new class of problems arising from these conditions allows for several important resource allocation problems previously excluded from consideration. In addition, an assumption needed to prove the Kuhn-Tucker theorem for the nonlinear problem of Part I is shown in the linear case to be completely analogous to the well-known Slater condition utilized in finite-dimensional programming theory. An example is given that exhibits the essential role of the constraint qualification in linear continuous time programming, a result at variance with the theory in finite dimensions but consistent with other results concerning linear programs in infinite-dimensional spaces.     

The Fritz John and Kuhn-Tucker optimality conditions in continuous-time nonlinear programming

First-order stationary-point necessary optimality criteria of both the Fritz John and Kuhn-Tucker type are established for continuous-time nonlinear programming problems. Furthermore, the relationship between these criteria and saddlepoint optimality conditions is also discussed. The main auxiliary result employed in the derivation of the principal optimality criteria is a continuous-time version of Gordan's transposition theorem.     

Optimality conditions and duality in multiple objective programming involving semilocally convex and related functions

A nonlinear multiple objective programming problem is considered where the functions involved are nondifferentiable. By considering the concept of weak minima, the Fritz John type and Karush-Kuhn- Tucker type necessary optimality conditions and Wolfe and Mond-Weir type duality results are given in terms of the right differentials of the functions. The duality results are stated by using the concepts of generalized semilocally convex functions     

Optimality conditions and duality for a class of nondifferentiable multi-objective fractional programming problems

A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,θ,ρ,d)-convex class about the Clarke’s generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.      

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.     

Generalized fractional programming: Optimality and duality theory

The generalized fractional programming problem with a finite number of ratios in the objective is studied. Optimality and duality results are established, some with the help of an auxiliary problem and some directly. Convexity and stability of the auxiliary problem play a key role in the latter part of the paper.The authors are grateful to an unknown referee for suggesting the statement of Theorem 3.3.     

Pseudo-invexity and duality in nonlinear programming

The purpose of this paper is to study various duality results in nonlinear programming for pseudo-invex functions. Such results were known in the literature for invex functions.