Inexact Stabilized Benders' Decomposition Approaches, with Application to Chance-Constrained Problems with Finite Support

We explore modifications of the standard cutting-plane approach for minimizing a convex nondifferentiable function, given by an oracle, over a combinatorial set, which is the basis of the celebrated (generalized) Benders' decomposition approach. Specifically, we combine stabilization—in two ways: via a trust region in the L1 norm, or via a level constraint—and inexact function computation (solution of the subproblems). Managing both features simultaneously requires a nontrivial convergence analysis; we provide it under very weak assumptions on the handling of the two parameters (target and accuracy) controlling the informative on-demand inexact oracle corresponding to the subproblem, strengthening earlier know results. This yields new versions of Benders' decomposition, whose numerical performance are assessed on a class of hybrid robust and chance-constrained problems that involve a random variable with an underlying discrete distribution, are convex in the decision variable, but have neither separable nor linear probabilistic constraints. The numerical results show that the approach has potential, especially for instances that are difficult to solve with standard techniques.