Abstract
We propose a new class of convex approximations for two-stage mixed-integer recourse models, the so-called generalized alpha-approximations. The advantage of these convex approximations over existing ones is that they are more suitable for efficient computations. Indeed, we construct a loose Benders decomposition algorithm that solves large problem instances in reasonable time. To guarantee the performance of the resulting solution, we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model. The error bounds converge to zero if these total variations converge to zero. We empirically assess our solution method on several test instances, including the SIZES and SSLP instances from SIPLIB. We show that our method finds near-optimal solutions if the variability of the random parameters in the model is large. Moreover, our method outperforms existing methods in terms of computation time, especially for large problem instances.
Highlights
Consider the two-stage mixed-integer recourse model with random right-hand side η∗ := min cx + Q(x) : xAx = b, x ∈ X ⊆ Rn+1, (1)N. van der Laan, W
That is why we propose an alternative class of convex approximations for general two-stage mixedinteger recourse (MIR) models, the so-called generalized α-approximations
– We propose a new class of convex approximations for general two-stage MIR models, which are based on Gomory relaxations of the second-stage problems
Summary
Consider the two-stage mixed-integer recourse model with random right-hand side η∗ := min cx + Q(x) : x. If they are exact on an entire grid of first-stage solutions Both the shifted LP-relaxation of [23] and the cutting plane framework of [21], cannot be applied directly to efficiently solve MIR models in general. That is why we propose an alternative class of convex approximations for general two-stage MIR models, the so-called generalized α-approximations They are derived by exploiting properties of Gomory relaxations [11] of the second-stage mixed-integer programming problems. – We propose a new class of convex approximations for general two-stage MIR models, which are based on Gomory relaxations of the second-stage problems These generalized α-approximations can be solved efficiently, and a similar error bound as for the shifted LP-relaxation applies to the generalized α-approximations.
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