A hybrid branch-and-bound and interior-point algorithm for stochastic mixed-integer nonlinear second-order cone programming

One of the chief attractions of stochastic mixed-integer second-order cone programming is its diverse applications, especially in engineering (Alzalg and Alioui, {\em IEEE Access}, 10:3522-3547, 2022). The linear and nonlinear versions of this class of optimization problems are still unsolved yet. In this paper, we develop a hybrid optimization algorithm coupling branch-and-bound and primal-dual interior-point … Read more

Cutting planes from the simplex tableau for quadratically constrained optimization problems

We describe a method to generate cutting planes for quadratically constrained optimization problems. The method uses information from the simplex tableau of a linear relaxation of the problem in combination with McCormick estimators. The method is guaranteed to cut off a basic feasible solution of the linear relaxation that violates the quadratic constraints in the … Read more

Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications

Submodularity is an important concept in integer and combinatorial optimization. A classical submodular set function models the utility of selecting homogenous items from a single ground set, and such selections can be represented by binary variables. In practice, many problem contexts involve choosing heterogenous items from more than one ground set or selecting multiple copies … Read more

A Stochastic Benders Decomposition Scheme for Large-Scale Data-Driven Network Design

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a data-driven version where operational costs are uncertain and estimated on historical data. This problem is computationally challenging, and instances with as few as 50 nodes cannot be solved to optimality by current … Read more

Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator

\(\) We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász-Schrijver SDP operator \( \text{LS}_+\), with a particular focus on a search for relatively small graphs with high \( \text{LS}_+\)-rank (the least number of iterations of the \( \text{LS}_+\) operator on the fractional stable set polytope to compute … Read more

On Supervalid Inequalities for Binary Interdiction Games

Supervalid inequalities are a specific type of constraints often used within the branch-and-cut framework to strengthen the linear relaxation of mixed-integer programs. These inequalities share the particular characteristic of potentially removing feasible integer solutions as long as they are already dominated by an incumbent solution. This paper focuses on supervalid inequalities for solving binary interdiction … Read more

Submodular maximization and its generalization through an intersection cut lens

\(\) We study a mixed-integer set \(\mathcal{S}:=\{(x,t) \in \{0,1\}^n \times \mathbb{R}: f(x) \ge t\}\) arising in the submodular maximization problem, where \(f\) is a submodular function defined over \(\{0,1\}^n\). We use intersection cuts to tighten a polyhedral outer approximation of \(\mathcal{S}\). We construct a continuous extension \(\mathsf{F}\) of \(f\), which is convex and defined over … Read more

Monoidal Strengthening of Simple V-Polyhedral Disjunctive Cuts

Disjunctive cutting planes can tighten a relaxation of a mixed-integer linear program. Traditionally, such cuts are obtained by solving a higher-dimensional linear program, whose additional variables cause the procedure to be computationally prohibitive. Adopting a V-polyhedral perspective is a practical alternative that enables the separation of disjunctive cuts via a linear program with only as … Read more

Recovering Dantzig-Wolfe Bounds by Cutting Planes

Dantzig-Wolfe (DW) decomposition is a well-known technique in mixed-integer programming (MIP) for decomposing and convexifying constraints to obtain potentially strong dual bounds. We investigate cutting planes that can be derived using the DW decomposition algorithm and show that these cuts can provide the same dual bounds as DW decomposition. More precisely, we generate one cut … Read more

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations have been among the most popular convexification techniques for binary polynomial optimization problems. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal recursive sequence amounts to solving a difficult combinatorial optimization problem. In this paper, we prove that any … Read more