Robust combinatorial optimization under convex and discrete cost uncertainty

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Abstract

In this survey, we discuss the state of the art of robust combinatorial optimization under uncertain cost functions. We summarize complexity results presented in the literature for various underlying problems, with the aim of pointing out the connections between the different results and approaches, and with a special emphasis on the role of the chosen uncertainty sets. Moreover, we give an overview over exact solution methods for NP-hard cases. While mostly concentrating on the classical concept of strict robustness, we also cover more recent two-stage optimization paradigms.

Introduction

Combinatorial optimization problems arise in many real-world applications, e.g., in the fields of economy, industry, or transport logistics. For many such problems, theoretically (or practically) fast algorithms have been developed under the assumption that all problem data is known precisely. However, the situation becomes more complex when considering uncertainty in the problem parameters. For example, the travel times for the shortest path problem or the vehicle routing problem can be subject to uncertainty, since we cannot predict the exact traffic situation in the future. One successful approach to tackle uncertainty in the input data is robust optimization: for a given set U containing all relevant scenarios, i.e., all sufficiently likely realizations of the uncertain parameters, a solution is sought that is feasible for every scenario in U and that is worst-case optimal under this constraint. This idea was first introduced by Soyster in Soyster (1973). The approach received increasing attention in the late 1990s. Kouvelis and Yu studied finite uncertainty sets U for several combinatorial optimization problems in Kouvelis and Yu (1996). Almost at the same time, Ben-Tal and Nemirovski (1998, 1999) studied robust convex problems with conic or ellipsoidal uncertainty sets. Furthermore, El Ghaoui et al. applied the idea to semi-definite problems and least squares problems (Ghaoui et al. 1998; Ghaoui and Lebret 1997). Later, Bertsimas and Sim introduced budgeted uncertainty sets to reduce what they call the Price of Robustness (Bertsimas and Sim 2004a). A survey over robust optimization approaches for discrete and interval uncertainty can be found in Aissi et al. (2009). The different uncertainty sets and their robust counterparts are intensively studied in Li et al. (2011).

Subsequently, new robust optimization paradigms were presented and studied in the literature, with the main objective of making the approach better applicable to practical problems. Besides various two-stage approaches (Ben-Tal et al. 2004; Liebchen et al. 2009; Adjiashvili et al. 2015), which we will discuss in detail in Sect. 4, several other paradigms have been investigated, e.g., min–max regret robustness (Averbakh and Lebedev 2005; Inuiguchi and Sakawa 1995; Chassein and Goerigk 2016; Kouvelis and Yu 1996; Averbakh and Lebedev 2004; Aissi et al. 2005a, b, c) or the light robustness approach (Fischetti et al. 2009; Fischetti and Monaci 2009; Schöbel 2014). Surveys studying several of the different approaches can be found in Aissi et al. (2009), Bertsimas et al. (2011), Gabrel et al. (2014), Kasperski and Zieliński (2016), Ben-Tal and Nemirovski (2002), Gorissen et al. (2015) and Beyer and Sendhoff (2007); they also cover distributional robustness, which forms a connection between robust and stochastic optimization.

In the present survey, we consider general combinatorial optimization problems of the formminxXcxwhere X{0,1}n describes the certain set of feasible solutions and where only the cost vector cRn is subject to uncertainty. In particular, we assume that an uncertainty set URn is given which contains all possible cost vectors c. The classical robust counterpart of Problem (P) is then given by ProblemminxXmaxcUcx.In contrast to other surveys on this topic, we aim at pointing out the differences between several common classes of uncertainty sets, with a focus on ellipsoidal uncertainty; see Sect. 2. In Sect. 3, we will sort and structure the complexity results for Problem (RP) achieved in the literature for several underlying combinatorial problems, again with a focus on the role of the chosen class of uncertainty set. Typical complexity results for Problem (RP) are illustrated for the most elementary case X={0,1}n, including sketches of the main proofs. Furthermore, we will discuss exact methods to solve Problem (RP) for the NP-hard cases, covering IP-based methods as well as oracle-based algorithms, which can be applied to every combinatorial problem (P) given by an optimization oracle. Finally, in Sect. 4, we will give an overview over various robust two-stage approaches presented in the literature and point out the connections between them.

Section snippets

Common uncertainty sets

The choice of the uncertainty set U is crucial in the design of the robust counterpart (RP). On the one hand, this choice should reflect the situation given in the application and lead to a realistic model of the given uncertainty, including the user’s attitude toward risk. On the other hand, the choice of U influences the tractability of the resulting problem (RP). For this reason, many different types of uncertainty sets have been investigated in the literature and are still being proposed.

Strictly robust optimization

We consider the strictly robust counterpart (RP) of the underlying problem (P). We are mostly interested in the complexity of (RP), which of course depends both on the feasible set X and the uncertainty set U. We start by reviewing the complexity results for general discrete, polyhedral, and ellipsoidal uncertainty sets in Sect. 3.1. In Sect. 3.2, we will focus on uncertainty sets that often lead to tractable robust counterparts. In Sect. 3.3, we will survey possible solution approaches for

Robust two-stage problems

A general robust two-stage problem can be formulated asminxXmaxξUminyY(x,y)Zξfξ(x,y)where xX are the first-stage decisions which have to be taken before the scenario is known. After the scenario ξU materializes, we choose the best possible second-stage decisions yY, such that the pair (xy) is feasible for the actual scenario, i.e., (x,y)Zξ. As common in robust optimization, we optimize the worst case objective value of fξ(x,y) over all scenarios ξU. As before, we will concentrate on

Conclusion

Considering all classical types of uncertainty sets discussed above, the main dividing line between hard and easy cases seems to be the inclusion of correlations: in the case of interval uncertainty, where all cost coefficients can vary independently, the robust counterpart inherits the complexity of the underlying problem. In the case of uncorrelated ellipsoidal uncertainty, it is not known yet whether the same is true, but positive general results exist. On the other hand, uncertainty sets

Acknowledgements

The second author has been supported by the German Research Foundation (DFG) within the Research Training Group 1855 and under Grant BU 2313/2.

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