Two-stage combinatorial optimization problems under risk

Abstract

In this paper a class of robust two-stage combinatorial optimization problems is discussed. It is assumed that the uncertain second stage costs are specified in the form of a convex uncertainty set, in particular polyhedral or ellipsoidal ones. It is shown that the robust two-stage versions of basic network and selection problems are NP-hard, even in a very restrictive cases. Some exact and approximation algorithms for the general problem are constructed. Polynomial and approximation algorithms for the robust two-stage versions of basic problems, such as the selection and shortest path problems, are also provided.Typically, the first stage costs are known while the second stage costs can only be predicted to belong to an uncertainty set U . First such models were discussed in [16,18,26,23], where the robust two-stage spanning tree and perfect matching problems were considered. In these papers, the uncertainty set U contains K explicitly listed scenarios. Several negative and positive complexity results for this uncertainty representation were established. Some of them have been recently extended in [19], where also the robust two-stage shortest path problem has been investigated. In [25] and [13] the robust two-stage selection problem has been explored. The problem is NP-hard for discrete uncertainty representation but it is polynomially solvable under a special case of polyhedral uncertainty set, called continuous budgeted uncertainty (see [13]).Robust two-stage problems belong to the class of three-level, min-max-min optimization problems. In mathematical programming, this approach is also called adjustable robustness (see, e.g. [5,31]). Namely, some variables must be determined before the realization of the uncertain parameters, while the other part are variables that can be chosen after the realization. Several such models have been recently considered in combinatorial optimization, which can be represented as a 0-1 programming problem. Among them there is the robust two-stage problem discussed in this paper, but also the robust recoverable models [11,12] and the k-adaptability approach [9]. In general, problems of this type can be hard to solve exactly. A standard approach is to apply row and column generation techniques, which consists in solving a sequence of MIP formulations (see, e.g., [32]). However, this method can be inefficient for larger problems, especially when the underlying deterministic problem is already NP-hard. Therefore, some faster approximation algorithms can be useful in this case.In this paper we consider the class of robust two-stage combinatorial problems under convex uncertainty, i.e. when the uncertainty set U is convex. Important special cases are polyhedral and ellipsoidal uncertainty, which are widely used in single-stage robust optimization. Notice that in the problems discussed in [16,18,26,23], U contains a fixed number of scenarios, so it is not a convex set. The problem formulation and description of the uncertainty sets are provided in Section 2. The complexity status of ba...

Two-Stage Robust Optimization Problems with Two-Stage Uncertainty

“…It is also a diagonal combinatorial optimization problem with interaction costs [L ĆP19], see also [IT21]. Furthermore, it is a two-stage combinatorial optimization problem with two scenarios under the expected value criterion [GKZ20]. Based on what is already known about problems (1) in [GKZ20], we can conclude the following complexity results.…”

“…Our main result is that for any nominal problem where solving the linear relaxation gives an integer solution, we can decompose the robust problem into O(n 2 ) nominal problems and O(n 3 ) two-stage problems of a special structure. This special structure (and different generalizations of it) has been considered in the literature before, under the names coordination problem [CG20], optimization with interaction costs [L ĆP19] and two-stage optimization under the expected value criterion [GKZ20]. As a consequence, the robust counterparts of many typical combinatorial optimization problems such as selection remain solvable in polynomial time.…”

Robust two-stage combinatorial optimization problems under convex second-stage cost uncertainty

“…Theorem 1 ( Kasperski and Zieliński 2017;Goerigk et al 2020) The RTSt 1 1 1 problem with U = {c c c 1 ,c c c 2 }⊂ R n + is NP-hard. Furthermore, if U = {c c c 1 , .…”

“…Several negative and positive complexity results for this uncertainty representation were established. Some of them have been recently extended in Goerigk et al (2020), where also the robust two-stage shortest path problem has been investigated. In Kasperski and Zieliński (2017) and Chassein et al (2018) the robust two-stage selection problem has been explored.…”

Solution methods for complex supply chain network optimization problems

“…Combinatorial Optimization (CO) refers to discrete optimization problems where the discrete set of solution space is finite, commonly huge and contains combinatorial structures, such as permutations or assignments (Goerigk et al, 2020). Typically, a CO problem is mathematically formulated as an MILP model, which means that the model includes both variables receiving values in an integer and a continuous domain (Della Croce, 2014).…”