Abstract
In this work we study binary two-stage robust optimization problems with objective uncertainty. We present an algorithm to calculate efficiently lower bounds for the binary two-stage robust problem by solving alternately the underlying deterministic problem and an adversarial problem. For the deterministic problem any oracle can be used which returns an optimal solution for every possible scenario. We show that the latter lower bound can be implemented in a branch and bound procedure, where the branching is performed only over the first-stage decision variables. All results even hold for non-linear objective functions which are concave in the uncertain parameters. As an alternative solution method we apply a column-and-constraint 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 and of the capital budgeting problem. Our results show that the branch and bound procedure outperforms the column-and-constraint generation algorithm.
Highlights
The concept of robust optimization was created to tackle optimization problems with uncertain parameters
The oracle-based branch and bound algorithm is exemplarily applied to the single-allocation hub location problem which can be naturally defined as a two-stage problem
Each row shows the average over all 20 instances of the following values from left to right: The number of projects n; the number of risk factors m; the percentage of instances which could be solved to optimality during the timelimit of 7200 s, the optimality gap of the column-andconstraint generation algorithm (CCG) after 7200 s; the total solution time t in seconds; the average solution time tub to calculate the upper bounds; the average solution time tlb to calculate the lower bounds; the total number of iterations; the average percental difference Δ between the best solution in Z′ and the deterministic optimal solution in each scenario
Summary
The concept of robust optimization was created to tackle optimization problems with uncertain parameters. As in the classical robust framework it is assumed that all uncertain scenarios are contained in a known uncertainty set and the worst-case objective value is optimized. We assume that the vector c is uncertain and all possible realizations c are contained in a convex uncertainty set U ⊂ Rm. The binary two-stage robust problem is defined by min max min f (x, y, c) x∈X c∈U y∈Y(x). The calculation of the lower bound can be applied to the common convex uncertainty sets and is done by alternately calling an adversarial problem over U and an oracle which returns an optimal solution of Problem (CP) for a given scenario c ∈ U. We apply the branch and bound procedure and the CCG algorithm to the uncapacitated single-allocation hub-location problem and the capital budgeting problem and show that the branch and bound procedure outperforms the CCG algorithm
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