Lagrange duality in canonical DC programming

We study Lagrange duality theorems for canonical DC programming problems. We show two types Lagrange duality results by using a decomposition method to infinite convex programming problems and by using a previous result by Lemaire (1998)  [6]. Also we observe these constraint qualifications for the duality theorems.     

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.     

Stable zero duality gaps in convex programming: Complete dual characterisations with applications to semidefinite programs

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality.     

Symmetric duality for a class of nondifferentiable multi-objective fractional variational problems

We introduce a symmetric dual pair for a class of nondifferentiable multi-objective fractional variational problems. Weak, strong, converse and self duality relations are established under certain invexity assumptions. The paper includes extensions of previous symmetric duality results for multi-objective fractional variational problems obtained by Kim, Lee and Schaible [D.S. Kim, W.J. Lee, S. Schaible, Symmetric duality for invex multiobjective fractional variational problems, J. Math. Anal. Appl. 289 (2004) 505-521] and symmetric duality results for the static case obtained by Yang, Wang and Deng [X.M. Yang, S.Y. Wang, X.T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl. 274 (2002) 279-295] to the dynamic case.     

Constrained duality via unconstrained duality in generalized geometric programming

A specialization of unconstrained duality (involving problems without explicit constraints) to constrained duality (involving problems with explicit constraints) provides an efficient mechanism for extending to the latter many important theorems that were previously established for the former.This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-73-2516.     

A Dual Parametrization Method for Convex Semi-Infinite Programming

We formulate convex semi-infinite programming problems in a functional analytic setting and derive optimality conditions and several duality results, based on which we develop a computational framework for solving convex semi-infinite programs.     

多目标分式规划逆对偶研究

考虑了一类可微多目标分式规划问题.首先,建立原问题的两个对偶模型.随后,在相关文献的弱对偶定理基础上,利用Fritz John型必要条件,证明了相应的逆对偶定理.     

Duality for nonconvex absolute value programming and a characterization of linear max-min programs

In this paper a dual problem for nonconvex linear programs with absolute value functionals is constructed by means of a max-min problem involving bivalent variables. A relationship between the classical linear max-min problem and a linear program with absolute value functionals is developed. This program is then used to compute the duality gap between some max-min and min-max linear problems.     

Duality in Fuzzy Linear Programming: Some New Concepts and Results

A class of fuzzy linear programming (FLP) problems based on fuzzy relations is introduced, the concepts of feasible and -efficient solutions are defined. The class of crisp (classical) LP problems and interval LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems a new concept of duality is introduced and the weak and strong duality theorems are derived. The previous results are applied to the special case of interval LP and compared to the existing literature.     

On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems. The authors like to thank the comments of the anonymous referees for their remarks, which greatly improved the presentation of this paper.     

New strong duality results for convex programs with separable constraints

It is known that convex programming problems with separable inequality constraints do not have duality gaps. However, strong duality may fail for these programs because the dual programs may not attain their maximum. In this paper, we establish conditions characterizing strong duality for convex programs with separable constraints. We also obtain a sub-differential formula characterizing strong duality for convex programs with separable constraints whenever the primal problems attain their minimum. Examples are given to illustrate our results.     

Duality in disjunctive linear fractional programming

In this note a dual problem is formulated for a given class of disjunctive linear fractional programming problems. This result generalizes to fractional programming the duality theorem of disjunctive linear programming originated by Balas. Two examples are given to illustrate the result.     

Vector-valued lagrangian and multiobjective fractional programming duality

A class of multiobjective fractional programming problems is considered and duality results are established in terms of properly efficient solutions of the primal and dual programs. Further a vector-valued ratio type Lagrangian is introduced and certain vector saddlepoint results are presented.     

The gap function of a convex program

The gap function expresses the duality gap of a convex program as a function of the primal variables only. Differentiability and convexity properties are derived, and a convergent minimization algorithm is given. An example gives a simple one-variable interpretation of weak and strong duality. Application to user-equilibrium traffic assignment yields an appealing alternative optimization problem.     

Some Results on Mathematical Programming with Generalized Ratio Invexity

In this paper, a generalized ratio invexity concept has been applied for single objective fractional programming problems. A concept which has been invoked seems to be more general than the one used earlier by Khan and Hanson in such contexts. Further, duality results for fractional programs have also been obtained.     

A note on duality in disjunctive programming

We state a duality theorem for disjunctive programming, which generalizes to this class of problems the corresponding result for linear programming.This work was supported by the National Science Foundation under Grant No. MPS73-08534 A02 and by the US Office of Naval Research under Contract No. N00014-75-C-0621-NR047-048.     

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.     

L-shaped decomposition of two-stage stochastic programs with integer recourse

We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.     

On perturbations of certain nonconvex optimization problems

Using the concept of -conjugate functions, a wide class of nonconvex optimization problems can be investigated. By generalized Lagrangians, the problems indicated and a generalized notion of stability can be treated. It is possible to give duality theorems for these problems such that the approach given by Rockafellar (1967) for convex problems can be extended to the nonconvex case.     

On semi-infinite minmax programming with generalized invexity

Here, we consider the minmax programming problem with a set of restrictions indexed in a compact. As a novelty, we obtain optimality criteria of the Kuhn--Tucker type involving a limited number of restrictions and prove both necessity and sufficiency under new weaker invexity assumptions. Also some dual problems are introduced and it is proved that the weak and strong duality properties hold within the same environment.