K-Adaptability in Two-Stage Mixed-Integer Robust Optimization

Subramanyam, Anirudh; Gounaris, Chrysanthos E.; Wiesemann, Wolfram

doi:10.48550/arxiv.1706.07097

Cited by 3 publications

(6 citation statements)

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“…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.…”

Section: Discussionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, in applications that require preparing the ground, it 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.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Chassein

Goerigk

Kurtz

et al. 2019

European Journal of Operational Research

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We consider robust combinatorial optimization problems where the decision maker can react to a scenario by choosing from a finite set of k solutions. This approach is appropriate for decision problems under uncertainty where the implementation of decisions requires preparing the ground. We focus on the case that the set of possible scenarios is described through a budgeted uncertainty set and provide three algorithms for the problem. The first algorithm solves heuristically the dualized problem, a nonconvex mixed-integer non-linear program (MINLP), via an alternating optimization approach. The second algorithm solves the MINLP exactly for k = 2 through a dedicated spatial branch-and-bound algorithm. The third approach enumerates k-tuples, relying on strong bounds to avoid a complete enumeration. We test our methods on shortest path instances that were used in the previous literature and on randomly generated knapsack instances, and find that our methods considerably outperform previous approaches. Many instances that were previously not solved within hours can now be solved within few minutes, often even faster.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Chassein

Goerigk

Kurtz

et al. 2019

European Journal of Operational Research

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“…Examples are numerous, e.g. transporting relief supplies or evacuating citizens in case an uncertain disaster arises [CTC07, LPdB + 13], hub locations [ANSdG12], or parcel deliveries [EKBC19,SGW17];…”

Section: Introductionmentioning

confidence: 99%

Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty

Goerigk¹,

Kurtz²,

Poss³

2019

Preprint

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We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min-max-min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage problems. The uncertainty set assumed in this paper considers that in any scenario, at most Γ of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set.While the classical min-max problem with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min-max-min problem is N P-hard for many easy combinatorial optimization problems, and not approximable in general. We thus present an integer programming formulation for solving the problem through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on knapsack and shortest path instances previously

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Oracle-Based Algorithms for Binary Two-Stage Robust Optimization

Kämmerling¹,

Kurtz²

2019

Preprint

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In this work we study binary two-stage robust optimization problems with objective uncertainty. The concept of two-stage robustness is tailored for problems under uncertainty which have two different kinds of decision variables, first-stage decisions which have to be made here-and-now and second-stage decisions which can be determined each time after an uncertain scenario occured.We adapt an oracle-based algorithm, originally introduced to solve binary min-max-min robust optimization problems, to efficiently calculate lower bounds for the binary two-stage robust problem by using an oracle for the underlying deterministic problem. We show that the latter lower bound can be implemented in a branch & bound procedure, where the branching is performed only over the first-stage decision variables. Additionally we show that the same procedure can be extended for non-linear binary problems which can be linearized. As an alternative solution method we apply a famous column-andconstraint generation algorithm to the binary two-stage robust problem with objective uncertainty.We test both algorithms on benchmark instances of the uncapacitated single-allocation hub-location problem, which is classically modeled by a quadratic binary formulation and show that the branch & bound procedure outperforms the column-and-constraint generation algorithm.

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K-Adaptability in Two-Stage Mixed-Integer Robust Optimization

Cited by 3 publications

References 0 publications

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

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

Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty

Oracle-Based Algorithms for Binary Two-Stage Robust Optimization

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