Robust Combinatorial Optimization with Locally Budgeted Uncertainty

Abstract

Budgeted uncertainty sets have been established as a major influence on uncertainty modeling for robust optimization problems. A drawback of such sets is that the budget constraint only restricts the global amount of cost increase that can be distributed by an adversary. Local restrictions, while being important for many applications, cannot be modeled this way.We introduce a new variant of budgeted uncertainty sets, called locally budgeted uncertainty. In this setting, the uncertain parameters are partitioned, such that a classic budgeted uncertainty set applies to each part of the partition, called region.In a theoretical analysis, we show that the robust counterpart of such problems for a constant number of regions remains solvable in polynomial time, if the underlying nominal problem can be solved in polynomial time as well. If the number of regions is unbounded, we show that the robust selection problem remains solvable in polynomial time, while also providing hardness results for other combinatorial problems.In computational experiments using both random and real-world data, we show that using locally budgeted uncertainty sets can have considerable advantages over classic budgeted uncertainty sets.

Multiparametric robust solutions for combinatorial problems with parameterized locally budgeted uncertainty

“…There exists an optimal solution for DW (x, Γ) with π ∈ {0, 1} K ( [5]), therefore there exists an optimal solution for DW (x, Γ) with (π, ρ) ∈ {0, 1} K ×{0, 1} n , hence R(Γ) may be rewritten as a 0-1-ILP problem in (π, ρ, x) as follows:…”

“…As before, dualizing the inner problem in R(Γ) lead us to a 0-1-MILP problem in (π, ρ, x) defined as follows ( [5]):…”

“…Appendix D: Remark about the computational complexity R(Γ) may be rewritten as follows ( [5]): min Γ t π + v(P (c(π))) R(Γ) s.t. π ∈ {0, 1} K Let X ⊆ X such that x ∈ X if and only there exists π ∈ {0, 1} K such that x is an optimal solution for P (c(π)).…”

“…Let Ω ⊆ R K + and let us suppose that Ω is 0-1-MILP-representable, let U ∈ R K + such that Γ ≤ U for all Γ ∈ Ω, let c, d ∈ R n + , let [q] = {1, • • • , q} for all q ∈ N and let {P 1 , • • • , P K } be a partition of [n]. In this paper we consider locally budgeted uncertainty sets parameterized with Γ ∈ Ω as follows ( [5]):…”

“…Computational experience is presented in section 4 for Shortest Path and (l, p)-Medians problems by using ILOG-Cplex 12.10 from a DOcplex Python code ([14]) . In section 5 we present a 0-1-MILP problem equivalent to R(Γ) for the variant for the uncertainty set defined as follows ( [5]):…”