We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. Interval linear programs with inequality constraints only can be processed in polynomial time.
This problem naturally emerged when dealing with various questions and problems regarding interval linear programming. A survey on results can be found in @cite_12 . Since that time, a partial characterization of the weakly optimal solution set was given in @cite_3 and an inner approximation was considered in @cite_7 . More general concepts of solutions, extending weak and strong solutions, were recently addressed in @cite_11 @cite_5 @cite_4 . Particular quantified solutions were also studied in @cite_8 @cite_2 . Duality in interval linear programming, which helps in charaterizing of weak optimality, among others, was studied in @cite_10 .
@cite_4 @cite_7 @cite_8 @cite_3 @cite_2 @cite_5 @cite_10 @cite_12 @cite_11
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This paper considers optimal solutions of interval linear programming problems, in a unified framework. Necessary and sufficient conditions for checking (A,b)-strong optimality and (A,b,c)-strong optimality of interval linear programming with inequality constraints are developed. The features of the proposed methods are illustrated by some examples. We consider a linear programming problem with interval data. We discuss the problem of checking whether a given solution is optimal for each realization of interval data. This problem was studied for particular forms of linear programming problems. Herein, we extend the results to a general model and simplify the overall approach. Moreover, we inspect computational complexity, too. Eventually, we investigate a related optimality concept of semi-strong optimality, showing its characterization and complexity. This paper deals with the problem of linear programming with inexact data represented by real closed intervals. Optimization problems with interval data arise in practical computations and they are of theoretical interest for more than forty years. We extend the concept of duality gap (DG), the difference between the primal and its dual optimal value, into interval linear programming. We consider two situations: First, DG is zero for every realization of interval parameters (the so called strongly zero DG) and, second, DG is zero for at least one realization of interval parameters (the so called weakly zero DG). We characterize strongly and weakly zero DG and its special case where the matrix of coefficients is real. We discuss computational complexity of testing weakly and strongly zero DG for commonly used types of interval linear programs and their variants with the real matrix of coefficients. We distinguish the NP-hard cases and the cases that are efficiently decidable. Based on DG conditions, we extend previous results about the bounds of the optimal value set given by Rohn. We provide equivalent statements for the bounds In this paper, theidentification problem, thetolerance problem, and thecontrol problem are treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution set Σ∃∃(A, b)= x ∈ ℝ n | (∃A ∈ A)(Ax ∈ b) , of thetolerable solution set Σ∀∃(A, b)= x ∈ ℝ n | (∀A ∈ A)(Ax ∈ b) , and of thecontrollable solution set Σ∃∀(A, b)= x ∈ ℝ n | (∀b ∈ b)(Ax ∈b) respectively. Analgebraic approach to their solution is developed in which the initial problem is replaced by that of finding analgebraic solution of some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its convergence and discuss numerical experiments.