An active set algorithm for robust combinatorial optimization based on separation oracles
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We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurobi.
KeywordsRobust optimization Active set methods SOCP
Mathematics Subject Classification90C25 90C27 90C57
The first author acknowledges support within the project “Mixed-Integer Non Linear Optimisation: Algorithms and Applications”, which has received funding from the Europeans Union’s EU Framework Programme for Research and Innovation Horizon 2020 under the Marie Skłodowska-Curie Actions Grant Agreement No 764759. The second author acknowledges support within the project “Nonlinear Approaches for the Solution of Hard Optimization Problems with Integer Variables”(No RP11715C7D8537BA) which has received funding from Sapienza, University of Rome. Moreover, the authors are grateful to Philipp Speckenmeyer for developing and implementing the special treatment of equations in our code (and for the careful reading of the manuscript).
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