Improper Linear Programming Problems Research Articles

Linear-quadratic programming and its application to data correction of improper linear programming problems

AbstractThe problem of maximizing a linear function with linear and quadratic constraints is considered. The solution of the problem is obtained in a constructive form using the Lagrange function and the optimality conditions. Many optimization problems can be reduced to the problem of this type. In this paper, as an application, we consider an improper linear programming problem formalized in the form of maximization of the initial linear criterion with a restriction to the Euclidean norm of the correction vector of the right-hand side of the constraints or the Frobenius norm of the correction matrix of both sides of the constraints.

Approximation of the Improper Linear Programming Problem with Restriction on the Norm of the Correction Matrix of the Left-Hand Side of the Constraints

Methods of correction (approximation) for inconsistent systems of linear algebraic equations and inequalities and improper linear programming problems have been widely used. In this paper, we consider an improper linear programming problem with an empty admissible set. It is formalized in the form of the problem of maximizing the initial criterion with the upper bound on the Frobenius norm of the correction matrix of the left-hand side of the constraints. A range of threshold values is found for which this problem has a solution, and its solution is obtained in an analytical form.

Duality and Correction of Inconsistent Constraints for Improper Linear Programming Problems

We continue the study of approximation properties of alternative duality schemes for improper problems of linear programming. The schemes are based on the use of the classical Lagrange function regularized simultaneously in primal and dual variables. The earlier results on the connection of its saddle points with the lexicographic correction of the right-hand sides of constraints in improper problems of the first and second kind are transferred to a more general type of improperness. Convergence theorems are presented and an informal interpretation of the obtained generalized solution is given.

Lexicographic regularization and duality for improper linear programming problems

A new approach to the optimal lexicographic correction of improper linear programming problems is proposed. The approach is based on the multistep regularization of the classical Lagrange function with respect to primal and dual variables simultaneously. The regularized function can be used as a basis for generating new duality schemes for problems of this kind. Theorems on the convergence and numerical stability of the method are presented, and an informal interpretation of the obtained generalized solution is given.

Dual approach to the application of barrier functions for the optimal correction of improper linear programming problems of the first kind

A novel dual approach to the problem of optimal correction of first-kind improper linear programming problems with respect to their right-hand sides is proposed. It is based on the extension of the traditional Lagrangian by introducing additional regularization and barrier components. Convergence theorems are given for methods based on the augmented Lagrangian, an informal interpretation of the obtained generalized solution is suggested, and results of numerical experiments are presented.

Correction of improper linear programming problems in canonical form by applying the minimax criterion

Given an inconsistent system of linear algebraic equations, necessary and sufficient conditions are established for the solvability of the problem of its matrix correction by applying the minimax criterion with the assumption that the solution is nonnegative. The form of the solution to the corrected system is presented. Two formulations of the problem are considered, specifically, the correction of both sides of the original system and correction with the right-hand-side vector being fixed. The minimax-criterion correction of an improper linear programming problem is reduced to a linear programming problem, which is solved numerically in MATLAB.

Matrix correction of a dual pair of improper linear programming problems with a block structure

The following problem is considered: how to modify the coefficient matrix of a dual pair of improper linear programs with a block structure so as to make these problems proper and minimize the sum of the squares of the Euclidean norms of the blocks in the correction matrix? Two variants of this problem are examined: (1) all the blocks in the coefficient matrix are modified, and (2) the upper block, which constraints all the primal variables, is left unchanged. Methods are presented for reducing these problems to minimizing quadratic fractional functions subject to linear equality and inequality constraints. The latter problem allows the use of conventional methods for constrained minimization. A numerical example is given.

Matrix correction of a dual pair of improper linear programming problems

A matrix is sought that solves a given dual pair of systems of linear algebraic equations. Necessary and sufficient conditions for the existence of solutions to this problem are obtained, and the form of the solutions is found. The form of the solution with the minimal Euclidean norm is indicated. Conditions for this solution to be a rank one matrix are examined. On the basis of these results, an analysis is performed for the following two problems: modifying the coefficient matrix for a dual pair of linear programs (which can be improper) to ensure the existence of given solutions for these programs, and modifying the coefficient matrix for a dual pair of improper linear programs to minimize its Euclidean norm. Necessary and sufficient conditions for the solvability of the first problem are given, and the form of its solutions is described. For the second problem, a method for the reduction to a nonlinear constrained minimization problem is indicated, necessary conditions for the existence of solutions are found, and the form of solutions is described. Numerical results are presented.

Symmetric duality for lexicographic linear programming problems

A symmetric pair of lexicographic linear-programming (LP) problems, connected by regular relations, is formulated for problems of multicriteria linear optimization. A duality theorem for improper linear-programming problems (ILPP) is constructed in terms of lexicographic optimization.

Deterministic and stochastic approximating problems with applications to production control

Improper mathematical programming problems are analyzed and deterministic and stochastic approaches to correcting these problems are suggested. Numerical experiments with test examples are presented. The paper focuses on numerical analysis of improper linear programming problems [1–4], which arise in the context of scarce resources in economics [5]. Parametrization is applied to examine one of the possible approaches to approximation of improper LP problems under deterministic and stochastic conditions. Although the main focus is on improper problems of the 1st kind, we also touch upon some issues connected with improper problems of 2nd and 3rd kind [1]. The analysis of improper LP problems is based on duality theory [2]. Some results specialize the ideas previously presented in [1, 6]. The present paper is a continuation of [4].

Duality for A class of improper linear programming problems and their applications

Duality for A class of improper linear programming problems and their applications