Boundedness relations in linear semi-infinite programming (original) (raw)
On Duality in Semi-Infinite Programming and Existence Theorems for Linear Inequalities
Journal of Mathematical Analysis and Applications, 1999
Linear semi-infinite programming deals with the optimization of linear functionals on finite-dimensional spaces under infinitely many linear constraints. For such kind of programs, a positive duality gap can occur between them and their corresponding dual problems, which are linear programs posed on infinite-dimensional spaces. This paper exploits some recent existence theorems for systems of linear inequalities in order to obtain a complete classification of linear semi-infinite programming problems from the point of view of the duality gap and the viability of the discretization numerical approach. The elimination of the duality gap is also discussed.
Duality for nonsmooth semi-infinite programming problems
Optimization Letters, 2012
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.
Duality for inexact semi-infinite linear programming
Optimization, 2005
The aim of this work is to generalize strong duality theorems for inexact linear programming and to derive duality results for inexact semi-infinite programming problems. We give a detailed proof of the general result, using the Dubovitskii-Milyutin approach. The last section contains applications to inexact problems and a few comments for further developments.
A duality theorem for semi-infinite convex programs and their finite subprograms
Mathematical Programming, 1983
113 this paper, we first establish a general recession condition under which a semi-infinite convex program and its formal Lagrangian dual have the same value. We go on to show that, under this condition, the following hold. First, every finite subprogram, with 'enough' of the given constraints. has the same value as its Lagrangian dual. Second, the weak value of the primal program is equal to the optimal value of the primal.
Proceedings of MOL2NET 2018, International Conference on Multidisciplinary Sciences, 4th edition, 2018
We consider two partitions over the space of linear semi-infinite programming parameters with a fixed index set and bounded coefficients (the functions of the constraints are bounded). The first one is the primal-dual partition inspired by consistency and boundedness of the optimal value of the linear semiinfinite optimization problems. The second one is a refinement of the primal-dual partition that arises considering the boundedness of the optimal set. These two partitions have been studied in the continuous case, this is, the set of indices is a compact infinite compact Hausdorff topological space and the functions defining the constraints are continuous. In this work, we present an extension of this case. We study same topological properties of the cells generated by the primal-dual partitions and characterize their interior. Through examples, we show that the results characterizing the sets of the partitions in the continuous case are neither necessary nor sufficient in both refinements. In addition, a sufficient condition for the boundedness of the optimal set of the dual problem has been presented. .
Optimality, scalarization and duality in linear vector semi-infinite programming
Optimization, 2018
In this paper, we introduce some qualification conditions for a linear vector semi-infinite programming problem. The new qualification conditions are used for development of necessary conditions for weak efficient, isolated efficient and ε-efficient solutions of such a problem. Sufficient conditions for the existence of such solutions are also provided. In addition, a new general scalarization technique for solving the reference problem is presented. Finally, we propose a type of Wolf dual problem and examine weak\strong duality relations.
Clark's theorem for semi-infinite convex programs
Advances in Applied Mathematics, 1981
In l%l, Clark proved that if either the feasible region of a linear program or its dual is nonempty and bounded, then the other is unbounded. Recently, Duffin has extended this result to a convex program and its Lagrangian dual. Moreover, Duffin showed that under this boundedness assumption there is no duality gap. The purpose of this paper is to extend Duffm's results to semi-infinite programs.
Asymptotic optimality conditions for linear semi-infinite programming
Optimization, 2015
In this paper, the classical KKT, complementarity, and Lagrangian saddle point conditions are generalized to obtain equivalent conditions characterizing the optimality of a feasible solution to a general linear semi-infinite programming problem without constraint qualifications. The method of this paper differs from the usual convex analysis methods and its main idea is rooted in some fundamental properties of linear programming.
Stability and well-posedness in linear semi-infinite programming
This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.
2013
1≤i≤m yibi. Now let P = {x ∈ (R) | Ax ≤ b and x ≥ 0}. Suppose x is a feasible point for our primal linear programming problem, so x ∈ P , i.e., Ax ≤ b and x ≥ 0. Also suppose we can choose y1, . . . , ym defining an m-covector y such that (yA)i ≥ ci and yi ≥ 0 for i = 1, . . . , n. Then c x ≤ yAx, and we have yAx ≤ yb, so altogether we have cx ≤ yAx ≤ yb. We see that yAx and yb are both upper-bounds of our primal objective function ζ(x) = cx for admissible choices of x and y, and yb is independent of x, so for any admissible choice of y, yb is an upper-bound of cx for all feasible points x ∈ P .
Nonlinear Analysis: Theory, Methods & Applications, 2009
In this paper, we present new dual constraint qualifications which completely characterize the zero duality gap property for convex programming problems in general Banach spaces. As an application, we derive constraint qualifications, characterizing zero duality gaps of semidefinite programs. Our approach makes use of convex majorants of support functions together with conjugate analysis and approximate subdifferentials of convex functions.
On Implicit Active Constraints in Linear Semi-Infinite Programs with Unbounded Coefficients
Applied Mathematics & Optimization, 2010
The concept of implicit active constraints at a given point provides useful local information about the solution set of linear semi-in…nite systems and about the optimal set in linear semi-in…nite programming provided the set of gradient vectors of the constraints is bounded, commonly under the additional assumption that there exists some strong Slater point. This paper shows that the mentioned global boundedness condition can be replaced by a weaker local condition (LUB) based on locally active constraints (active in a ball of small radius whose center is some nominal point), providing geometric information about the solution set and Karush-Kuhn-Tucker type conditions for the optimal solution to be strongly unique. The maintaining of the latter property under su¢ ciently small perturbations of all the data is also analyzed, giving a characterization of its stability with respect to these perturbations in terms of the strong Slater condition, the so-called Extended-Nürnberger condition, and the LUB condition.
Convex Generalized Semi-Infinite Programming Problems with Constraint Sets: Necessary Conditions
2012
We consider 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 convex. Considering a lower level constraint qualification, we derive a formula for estimating the subdifferential of the value function. Finally, we establish the Fritz-John necessary optimality conditions for the problem.
Mathematical Programming, 2013
The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for efficient computing the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.
Solving Strategies and Well-Posedness in Linear Semi-Infinite Programming
Annals of Operations Research, 2001
In this paper we introduce the concept of solving strategy for a linear semi-infinite programming problem, whose index set is arbitrary and whose coefficient functions have no special property at all. In particular, we consider two strategies which either approximately solve or exactly solve the approximating problems, respectively. Our principal aim is to establish a global framework to cope with different concepts of well-posedness spread out in the literature. Any concept of well-posedness should entail different properties of these strategies, even in the case that we are not assuming the boundedness of the optimal set. In the paper we consider three desirable properties, leading to an exhaustive study of them in relation to both strategies. The more significant results are summarized in a table, which allows us to show the double goal of the paper. On the one hand, we characterize the main features of each strategy, in terms of certain stability properties (lower and upper semicontinuity) of the feasible set mapping, optimal value function and optimal set mapping. On the other hand, and associated with some cells of the table, we recognize different notions of Hadamard well-posedness. We also provide an application to the analysis of the Hadamard well-posedness for a linear semi-infinite formulation of the Lagrangian dual of a nonlinear programming problem.
Optimality conditions for convex semi-infinite programming problems
Naval Research Logistics Quarterly, 1980
This paper gives characterizations of optilDal soiutions for convex I semi-infinite programming problems. These characterizations are free of I' theories, which give only necessary or sufficient conditions for optimality, but not both•. An a~plieation to the problem of best iinear Chebyshev approxiaation with constraints is delDOll8trated.
Computación y Sistemas, 2018
Different partitions of the parameter space of all linear semi-infinite programming problems with a fixed compact set of indices and continuous right and left hand side coefficients have been considered in this paper. The optimization problems are classified in a different manner, e.g., consistent and inconsistent, solvable (with bounded optimal value and nonempty optimal set), unsolvable (with bounded optimal value and empty optimal set) and unbounded (with infinite optimal value). The classification we propose generates a partition of the parameter space, called second general primal-dual partition. We characterize each cell of the partition by means of necessary and sufficient, and in some cases only necessary or sufficient conditions, assuring that the pair of problems (primal and dual), belongs to that cell. In addition, we show non emptiness of each cell of the partition and with plenty of examples we demonstrate that some of the conditions are only necessary or sufficient. Finally, we investigate various questions of stability of the presented partition.