We consider mixed-integer convex optimization problems in which binary indicators control continuous variables. We introduce the \emph{Coordinate Optimality Reformulation} (CORe) framework, which augments standard indicator formulations by incorporating coordinate-wise optimality information. The resulting reformulations preserve global optimality while substantially improving branch-and-bound performance, particularly in sparse and structured settings where the coordinate-wise optimality conditions expose exploitable problem structure. We first develop the main components of CORe, including coordinate-wise optimality conditions, closed-form characterizations, and disjunctive reformulations. We then demonstrate the framework across multiple problem families, including quadratic problems and robust single-index models. Computational experiments show that CORe can substantially improve solver performance compared with standard big-$M$ formulations.
Tong Xu, S. Fattahi, Andrés Gómez et al.· 0 citations
We present an open-source software package that implements a provably convergent Benders-type decomposition algorithm for multistage stochastic integer programs. In addition to standard cut families, such as Benders, strengthened Benders, and Lagrangian cuts, the algorithm incorporates rectified linear unit (ReLU) cuts, which provide convergence guarantees for general mixed-integer state variables. However, the dual problems used to generate these cuts often admit multiple optimal solutions. Although each solution yields a valid cut that separates the incumbent, the resulting cuts can differ in how well they approximate the subproblem cost. To strengthen these cuts, our package implements and evaluates two cut-selection strategies based on normalization and regularization of the dual problem. We also incorporate an alternating-cut criterion that uses cheaper Benders cuts when they are effective and invokes more expensive tight cuts only when necessary. Computational experiments on four classes of multistage stochastic integer programs benchmark these methods and provide insights on how problem structure affects their practical performance.
Akul Bansal, Simge Küçükyavuz· 0 citations
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