(19d) A Branch-and-Bound Scheme for K-Adaptability Problems in Robust Optimization
Amongst the various methodologies that have been proposed to address such problems, robust optimization  has received a lot of attention over the last decade since it avoids many of the shortcomings associated with traditional approaches. In particular, it replaces probability distributions with uncertainty sets as the fundamental models of uncertainty and seeks to determine decisions that remain feasible for any realization of the uncertainty from within the modeled uncertainty set.
While multi-stage robust optimization problems involving continuous recourse decisions have been well-studied [1, 2], the literature on problems involving discrete recourse decisions is relatively sparse. In the special case of two-stage problems in which the uncertainty appears only in the constraint âright-hand sides,â a nested column-and-constraint generation solution approach that converges in finite time is developed in . Other approaches are concerned with the design of conservative approximations of the true multi-stage problem and fall into one of three categories: (i) decision rule approaches, in which the recourse decisions are modeled as explicit functions of the uncertainty [4, 5]; (ii) uncertainty set partitioning approaches, in which the recourse nature of discrete decisions is simulated by designing separate sets of decisions for different, pre-specified subsets of the uncertainty set [6, 7]; and (iii) K-adaptability approaches, in which K sets of discrete recourse decisions are designed before the uncertainty is resolved and the best decision is implemented after observing the actual realization of the uncertainty [8, 9].
In this work, we focus on the K-adaptability approach to address two-stage robust optimization problems. In contrast to the other two approaches, K-adaptability has been shown to provide a remarkably good approximation of the true problem, both in theory and numerical experiments [8, 9]. Furthermore, rather than designing the optimal recourse decision for every possible realization of the uncertainty , it aims to design a small number of contingency plans, which may be preferable for operational or organizational reasons.
Existing solution approaches for K-adaptability problems are either heuristic  or consist of reformulating the problem as a finite-dimensional mixed-integer linear program under the additional assumption that both the first- and second-stage decisions are binary . In this work, we alleviate both of these shortcomings and devise a solution scheme that is both exact and applicable for problems featuring mixed-integer decisions and constraints with uncertainty affecting their âleft-hand sides.â By viewing the K-adaptability problem as a semi-infinite disjunctive program, our solution scheme uses a sampling-based disjunctive branch-and-bound search procedure to asymptotically converge to an optimal solution. For problems with uncertainty sets of finite cardinality, our solution scheme converges in finite time. In addition to our convergence proof, we also analyze the solvability, convexity and tractability properties of general K-adaptability problems, and present conditions under which they reduce to single-stage (also known as static) robust optimization problems, and under which their optimal values coincide with that of the true two-stage problem.
Extensive numerical experiments conducted on benchmark data from a number of applications demonstrate that: (a) we can significantly improve upon decisions determined via static robust optimization within small computing times, and (b) our solution scheme for K-adaptability problems is a practically tractable approach for addressing two-stage robust optimization problems that improves upon the current state-of-the-art.
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