Non-differentiable pseudo-convex functions and duality for minimax programming problems

Optimality results are derived for a general minimax programming problem under non-differentiable pseudo-convexity assumptions. A dual in terms of Dini derivatives is introduced and duality results are established. Finally, two duals again in terms of Dini derivatives are introduced for a generalized fractional minimax programming problem and corresponding results are studied.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

Lagrangian Approach to Quasiconvex Programing

For a mathematical programming problem, we consider a Lagrangian approach inspired by quasiconvex duality, but as close as possible to the usual convex Lagrangian. We focus our attention on the set of multipliers and we look for their interpretation as generalized derivatives of the performance function associated with a simple perturbation of the given problem. We do not use quasiconvex dualities, but simple direct arguments.     

Integer programming and pricing revisited

Three applications of duality are mentioned: mathematical, computational,and economic. One of the earliest attempts toproduce a dualof an integer programme with economic interpretations was byGomory & Baumol in 1960. This is describedtogether withits economic properties and some refinements and corrections.A more recent integer programming dual due to Chvátal,whose main use to date has been computational, is then described.It is shown that this can be given an economic interpretationas a generalization of Gomory & Baumol‘s dual whichrectifies some of the deficiencies of the latter. The computationalproblems of calculating Chvátal’s dual are remarkedon.     

Duality for Algebraic Linear Programming

A duality theory for algebraic linear (integer) programming (ALP) is developed which is of the same importance for linear (integer) programming with linear algebraic objectives as linear programming duality is for classical LP. In particular, optimality criteria for primal, primal-dual, and dual methods are given which generalize feasibility and complementarity criteria of classical LP. Strong duality results are given for special combinatorial problems. Further, the validity and finiteness of a primal simplex method based on a feasibility criterion are proved in the case of nondiscrete variables. In this case a strong duality result is shown.     

Understanding linear semi-infinite programming via linear programming over cones

In this article, the standard primal and dual linear semi-infinite programming (DLSIP) problems are reformulated as linear programming (LP) problems over cones. Therefore, the dual formulation via the minimal cone approach, which results in zero duality gap for the primal–dual pair for LP problems over cones, can be applied to linear semi-infinite programming (LSIP) problems. Results on the geometry of the set of the feasible solutions for the primal LSIP problem and the optimality criteria for the DLSIP problem are also discussed.     

On sufficiency and duality for a class of interval-valued programming problems

In this paper, we are concerned with an interval-valued programming problem. Sufficient optimality conditions are established under generalized convex functions for a feasible solution to be an efficient solution. Appropriate duality theorems for Mond-Weir and Wolfe type duals are discussed in order to relate the efficient solutions of primal and dual programs.     

凸二次-线性双层规划的共轭对偶及其性质

研究具有一般形式的凸二次-线性双层规划问题。讨论了这类双层规划问题的DC规划等价形式,利用DC规划共轭对偶理论,提出了凸二次-线性双层规划的共轭对偶规划,并给出相应的对偶性质。     

Some relationships between lagrangian and surrogate duality in integer programming

Lagrangian dual approaches have been employed successfully in a number of integer programming situations to provide bounds for branch-and-bound procedures. This paper investigates some relationship between bounds obtained from lagrangian duals and those derived from the lesser known, but theoretically more powerful surrogate duals. A generalization of Geoffrion's integrality property, some complementary slackness relationships between optimal solutions, and some empirical results are presented and used to argue for the relative value of surrogate duals in integer programming. These and other results are then shown to lead naturally to a two-phase algorithm which optimizes first the computationally easier lagrangian dual and then the surrogate dual.     

Multilinear programming: Duality theories

A type of nonlinear programming problem, called multilinear, whose objective function and constraints involve the variables through sums of products is treated. It is a rather straightforward generalization of the linear programming problem. This, and the fact that such problems have recently been encountered in several fields of application, suggested their study, with particular emphasis on the analogies between them and linear problems. This paper develops one such analogy, namely a duality concept which includes its linear counterpart as a special case and also retains essentially all of the desirable characteristics of linear duality theory. It is, however, found that a primal then has several duals. The duals are arrived at by way of a game which is closely associated with a multilinear programming problem, but which differs in some respects from those generally treated in game theory. Its generalizations may in fact be of interest in their own right.Professor J. Stoer and an anonymous reviewer made helpful comments on an earlier version of this paper. Those comments are greatly appreciated.     

An η-approximation approach to duality in mathematical programming problems involving r-invex functions

An η-approximation approach introduced by Antczak [T. Antczak, A new method of solving nonlinear mathematical programming problems involving r-invex functions, J. Math. Anal. Appl. 311 (2005) 313-323] is used to obtain a solution Mond-Weir dual problems involving r-invex functions. η-Approximated Mond-Weir dual problems are introduced for the η-approximated optimization problem constructed in this method associated with the original nonlinear mathematical programming problem. By the help of η-approximated dual problems various duality results are established for the original mathematical programming problem and its original Mond-Weir duals.     

Generalized concavity and duality with a square root term

Duality results for a class of nondifferentiable mathematical programming problems are given. These results allow for the weakening of the usual convexity conditions required for duality to hold. A pair of symmetric and self dual nondifferentiable programs under weaker convexity conditions are also given. A subgradient symmetric duality is proposed and its limitations discussed. Finally, a pair of nondifferentiable mathematical programs containing arbitrary norms is presented.     

Duality for nonsmooth semi-infinite programming problems

This paper deals with nonsmooth semi-infinite programming problem which in recent years has become an important field of active research in mathematical programming. A semi-infinite programming problem is characterized by an infinite number of inequality constraints. We formulate Wolfe as well as Mond-Weir type duals for the nonsmooth semi-infinite programming problem and establish weak, strong and strict converse duality theorems relating the problem and the dual problems. To the best of our knowledge such results have not been done till now.     

The minimal cone for conic linear programming

It is known that the minimal cone for the constraint system of a conic linear programming problem is a key component in obtaining strong duality without any constraint qualification. For problems in either primal or dual form, the minimal cone can be written down explicitly in terms of the problem data. However, due to possible lack of closure, explicit expressions for the dual cone of the minimal cone cannot be obtained in general. In the particular case of semidefinite programming, an explicit expression for the dual cone of the minimal cone allows for a dual program of polynomial size that satisfies strong duality. In this paper we develop a recursive procedure to obtain the minimal cone and its dual cone. In particular, for conic problems with so-called nice cones, we obtain explicit expressions for the cones involved in the dual recursive procedure. As an example of this approach, the well-known duals that satisfy strong duality for semidefinite programming problems are obtained. The relation between this approach and a facial reduction algorithm is also discussed.     

一般形式多阶段有补偿问题的广义对偶理论

本文在文［１］的基础上，讨论一般形式多阶段有补偿非线性随机规划问题的广义对偶理论与最优化性条件．通过发掘凸规划对偶理论的本质，首先推广了与通常规划问题对偶理论有关的概念的含义，由此构造出所论问题在等价意义下的广义原始泛函与广义对偶泛函，进而得到其广义对偶理论，所得结论不仅能恰当合理地反映问题本身的属性，而且有关定理的表述形式简明、结论较强，可直接应用于多阶段有补偿问题的其它理论研究与数值求解算法的设计中去．上述结果与所用研究方法均推广和发展了通常的对偶理论     

On variational problems involving higher order derivatives

Fritz John, and Karush-Kuhn-Tucker type optimality conditions for a constrained variational problem invoving higher order derivatives are obtained. As an application of these Karush-Kuhn-Tucker type optimality conditions, Wolfe and Mond-Weir type duals are formulated, and various duality relationships between the primal problem and each of the duals are established under invexity and generalized invexity. It is also shown that our results can be viewed as dynamic generalizations of those of the mathematical programming already reported in the literature.     

Duality for quasi-concave programs with explicit constraints

The idea of duality is now well established in the theory of concave programming. The basis of this duality is the concave conjugate transform. This has been exemplified in the development of generalised geometric programming. Much of the current research in duality theory is focused on relaxing the requirement of concavity. Here we develop a duality theory for mathematical programs with a quasi concave objective function and explicit quasi concave constraints. Generalisations of the concave conjugate transform are introduced which pair quasi concave functions as the concave conjugate transform does for concave functions. Optimality conditions are derived relating the primal quasi concave program to its dual. This duality theory was motivated by and has implications in certain problems of mathematical economics. An application to economics is given.     

Implied constraints and LP duals of general nonlinear programming problems

A very surprising result is derived in this paper, that there exists a family of LP duals for general NLP problems. A general dual problem is first derived from implied constraints via a simple bounding technique. It is shown that the Lagrangian dual is a special case of this general dual and that other special cases turn out to be LP problems. The LP duals provide a very powerful computational device but are derived using fairly strict conditions. Hence, they can often be infeasible even if the primal NLP problem is feasible and bounded. Many directions for relaxing these conditions are outlined for future research. A concept of local duality is also introduced for the first time akin to the concept of local optimality.     

Efficiency Conditions and Duality for a Class of Multiobjective Fractional Programming Problems

A class of constrained multiobjective fractional programming problems is considered from a viewpoint of the generalized convexity. Some basic concepts about the generalized convexity of functions, including a unified formulation of generalized convexity, are presented. Based upon the concept of the generalized convexity, efficiency conditions and duality for a class of multiobjective fractional programming problems are obtained. For three types of duals of the multiobjective fractional programming problem, the corresponding duality theorems are also established.     

基于广义几何规划的最小广义信息密度区分问题及其对偶理论

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｑｕａｄｒａｔｉｃａｌｌｙｃｏｎｓｔｒａｉｎｅｄａｎｄｅｎｔｒｏｐｙｄｅｎｓｉｔｙｃｏｎｓｔｒａｉｎｅｄｑｕａｄｒａｔｉｃｐｒｏｇｒａｍｔｈａｔｉｓｇｏｉｎｇｔｏｂｅｓｔｕｄｉｅｄｉｎｔｈｉｓｐａｐｅｒｉｓｃｈａｒａｃｔｅｒｉｚｅｄａｓｔｈｅｆｏｌｌｏｗｉｎｇｆｏｒｍ :Ｐｒｏｇｒａｍ (Ｑ)(Ｑ)　　ｍｉｎ　Ｑ0 (ｚ)ｓ .ｔ .　Ｐｊ(ｚ)≤ 0 ,　ｊ =1 ,2 ,...,ｌ,Ｑｉ(ｚ) ≤ 0 ,　ｉ =1 ,2 ,...,ｒ ,ｚ=(ｚ1,...,ｚｎ) Ｔ ≥ 0 ,ｗｈｅｒｅＰｊ(ｚ) = ｎｋ =1ｚｋｌｏｇ ｚｋｅ…