We introduce two-stage stochastic min-max and min-min integer programs with bi-parameterized recourse (BTSPs), where the first-stage decisions affect both the objective function and the feasible region of the second-stage problem. To solve these programs efficiently, we introduce Lagrangian-integrated \(L\)-shaped (\(L^2\)) methods, which guarantee exact solutions when the first-stage decisions are pure binary. For mixed-binary first-stage programs, we present a regularization-augmented variant of this method. Our computational results for a stochastic network interdiction problem show that the \(L^2\) method outperforms a benchmark method, solving all instances in 23 seconds on average, while the benchmark method failed to solve any instance within 3600 seconds. The \(L^2\) method also achieves optimal solutions, on average, 18.4 times faster for a stochastic facility location problem. Furthermore, we show that the \(L^2\) method can effectively address distributionally robust optimization problems with decision-dependent ambiguity sets that may be empty for some first-stage decisions, achieving optimal solutions, on average, 5.3 times faster than existing methods.
Bi-Parameterized Two-Stage Stochastic Min-Max and Min-Min Mixed Integer Programs
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