K-adaptability in two-stage mixed-integer robust optimization

Abstract

We study two-stage robust optimization problems with mixed discrete-continuous decisions in both stages. Despite their broad range of applications, these problems pose two fundamental challenges: (i) they constitute infinite-dimensional problems that require a finite-dimensional approximation, and (ii) the presence of discrete recourse decisions typically prohibits duality-based solution schemes. We address the first challenge by studying a K-adaptability formulation that selects K candidate recourse policies before observing the realization of the uncertain parameters and that implements the best of these policies after the realization is known. We address the second challenge through a branch-and-bound scheme that enjoys asymptotic convergence in general and finite convergence under specific conditions. We illustrate the performance of our algorithm in numerical experiments involving benchmark data from several application domains.

Faster algorithms for min-max-min robustness for combinatorial problems with budgeted uncertainty

“…In this work we derived three fast algorithms to solve these problems exactly and heuristically, two of them based on a mixedinteger non-linear reformulation while the third is a discrete enumeration scheme involving ad-hoc dominance rules. The proposed exact algorithms have overcome the difficulties encountered by the previous literature (Hanasusanto et al, 2015;Subramanyam et al, 2017) for k = 2, by solving problems in a couple of minutes (often seconds) that were unsolvable in one or two hours using the previous approaches. We also found encouraging results for k = 3, solving many unsolved instances to optimality in a short amount of time.…”

“…Our focus in this paper is the alternative robust model introduced by Kurtz (2017, 2018a) which is based on the idea of k-adaptability first introduced by Bertsimas and Caramanis (2010) for general robust two-stage problems. Later this idea was studied for robust twostage problems with binary recourse in Hanasusanto et al (2015) and for the same problems with mixed-integer recourse in Subramanyam et al (2017). In the latter publications both cases, uncertainty only affecting the objective function and uncertainty affecting the constraint coefficients, are studied.…”

“…Using the min-max-min approach a small number of transportation-plans can be calculated in advance to decide which long-term decision have to be made and to prepare all employees. Another example, taken from Eufinger et al (2018); Subramanyam et al (2017), considers a parcel service delivering to the same customers every day, i.e. X is the set of feasible solutions of the vehicle routing problem.…”

“…is not practical to have too many alternatives, limiting the interest of the approach from Buchheim and Kurtz (2017), which requires k ≥ n + 1. Alternative solutions to (M 3 ) have also been proposed in a much more general context where there are also decisions that must be taken before the uncertainty is revealed, falling into the framework of two-stage robust optimization, see Hanasusanto et al (2015) and Subramanyam et al (2017). The former provides a MILP reformulation involving the linearization of products between binary and real variables, while the latter studies an ad-hoc branch-and-bound algorithm that branches over the assignments of solutions to scenarios.…”

Joint Pricing and Production: A Fusion of Machine Learning and Robust Optimization

“…Remark 1. The K-means adaptive markdown policy described in this section is related to the K-adaptive policy presented by Bertsimas and Caramanis (2010), Hanasusanto et al (2015bHanasusanto et al ( , 2016, and Subramanyam et al (2017). Although both policies allow for discrete recourse, the clusters in K-adaptive models are determined via optimization whereas the clusters in K-means adaptive models are predetermined by clustering on data.…”