On the Properties of Interval Linear Programs with a Fixed Coefficient Matrix

Garajová, Elif; Hladík, Milan; Rada, Miroslav

doi:10.1007/978-3-319-67308-0_40

Cited by 9 publications

(7 citation statements)

References 7 publications

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“…We weaken assumptions of finality in both theorems and extend them to equivalent statements (Thm. 8,9). Table 3: Computational complexity of both strongly and weakly zero DG with degenerated matrix.…”

Section: Our Resultsmentioning

confidence: 99%

“…The primal ILP is infeasible for each scenario. Thus f (A, b, c) = ∞ which is the left hand side of (9). The dual ILP is weakly feasible but not strongly feasible.…”

Section: Strongly Zero Dgmentioning

confidence: 99%

“…We construct a polynomial-time reduction from the problem of testing strong feasibility for type (A) ILP. This problem is known to be co-NP-hard [24] even with degenerated matrix [9].…”

Section: Strongly Zero Dgmentioning

confidence: 99%

“…Proof. Since testing weak feasibility of types (A), (B) and (C) ILP with degenerated matrix is polynomial [9] the testing weakly zero DG is according to Proposition 3 polynomial, too.…”

Section: Weakly Zero Dgmentioning

confidence: 99%

“…Weakly zero DG is not sufficient to fulfill the equation (9), see the Example 4. The primal ILP is infeasible for each scenario.…”

Section: Strongly Zero Dgmentioning

confidence: 99%

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Duality Gap in Interval Linear Programming

Novotná

Hladík

Masařík

2019

J Optim Theory Appl

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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.

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Section: Our Resultsmentioning

confidence: 99%

“…The primal ILP is infeasible for each scenario. Thus f (A, b, c) = ∞ which is the left hand side of (9). The dual ILP is weakly feasible but not strongly feasible.…”

Section: Strongly Zero Dgmentioning

confidence: 99%

“…We construct a polynomial-time reduction from the problem of testing strong feasibility for type (A) ILP. This problem is known to be co-NP-hard [24] even with degenerated matrix [9].…”

Section: Strongly Zero Dgmentioning

confidence: 99%

“…Proof. Since testing weak feasibility of types (A), (B) and (C) ILP with degenerated matrix is polynomial [9] the testing weakly zero DG is according to Proposition 3 polynomial, too.…”

Section: Weakly Zero Dgmentioning

confidence: 99%

“…Weakly zero DG is not sufficient to fulfill the equation (9), see the Example 4. The primal ILP is infeasible for each scenario.…”

Section: Strongly Zero Dgmentioning

confidence: 99%

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