Checking weak optimality and strong boundedness in interval linear programming
Interval programming provides a mathematical tool for dealing with uncertainty in optimization problems. In this paper, we study two properties of interval linear programs: weak optimality and strong boundedness. The former property refers to the existence of a scenario possessing an optimal solution, or the problem of deciding non-emptiness of the optimal set. We investigate the problem from a complexity-theoretic point of view and prove that checking weak optimality is NP-hard for all types of programs, even if the variables are restricted to a single orthant. The property of strong boundedness implies that each feasible scenario of the program has a bounded objective function. It is co-NP-hard to decide for inequality-constrained interval linear programs. For this class of programs, we derive a sufficient and necessary condition for testing strong boundedness using the orthant decomposition method. We also discuss the open problem of checking strong boundedness of programs described by equations with nonnegative variables.
KeywordsInterval linear programming Weak optimality Strong boundedness Computational complexity
This study was funded by the Czech Science Foundation (Grant P403-18-04735S) and by the Charles University (Project GA UK No. 156317).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
This article does not contain any studies with human participants or animals performed by any of the authors.
- Cerulli R, D’Ambrosio C, Gentili M (2017) Best and worst values of the optimal cost of the interval transportation problem. In: Sforza A, Sterle C (eds) Optimization and decision science: methodologies and applications: ODS, Sorrento, Italy, 4–7 September 2017. Springer International Publishing, Cham, pp 367–374Google Scholar
- Garajová E, Hladík M, Rada M (2017) On the properties of interval linear programs with a fixed coefficient matrix. In: Sforza A, Sterle C (eds) Optimization and decision science: methodologies and applications: ODS, Sorrento, Italy, 4–7 September 2017. Springer International Publishing, Cham, pp 393–401Google Scholar
- Hladík M (2012) Interval linear programming: a survey. Chapter 2. In: Mann ZA (ed) Linear programming—new frontiers in theory and applications. Nova Science, New York, pp 85–120Google Scholar
- Koníčková J (2006) Strong unboundedness of interval linear programming problems. In: 12th GAMM–IMACS international symposium on scientific computing, computer arithmetic and validated numerics (SCAN 2006), p 26Google Scholar
- Li DF (2016) Interval-valued matrix games. Springer, Berlin, pp 3–63Google Scholar