A rational polyhedron has Chvátal rank at most one if a single round of Chvátal-Gomory cuts yields its integer hull. For such polyhedra, integer feasibility is in NP ∩ coNP by a result of Boyd and Pulleyblank from the early 1980s, so it is unlikely to be NP-hard. Whether it is polynomial has remained open … Read more
We give a deterministic polynomial-time algorithm for integer quadratic programming in every fixed dimension: it minimizes an arbitrary rational quadratic exactly over the integer points of a rational polyhedron, or certifies infeasibility or integer unboundedness. The core is a sign test that decides whether \(d^{\mathsf{T}}Qd\ge 0\) for every integer point \(d\) of a bounded symmetric … Read more
We study cutting-plane generation for binary polynomial optimization (BPO), whose feasible region is the multilinear set of a hypergraph. Strong inequalities for this set—such as two-links, flowers, and odd $\beta$-cycles—are classically hand-derived for fixed support patterns. Instead, we propose a decision-diagram (DD) approach: for any chosen support, it separates a facet-defining cut in the local … Read more
Many large-scale optimization problems decompose into a master problem and scenario subproblems, a structure that can be exploited by Benders decomposition. In Benders decomposition, each iteration may generate many cuts from scenario subproblems, and adding all of them as constraints then causes the master problem to grow rapidly. These are constraints that may need to … Read more
Algorithms based on polyhedral outer approximations provide a powerful approach to solving mixed-integer nonlinear optimization problems. An initial relaxation of the feasible set is strengthened by iteratively adding linear inequalities and separating infeasible points. However, when the constraints are nonconvex, computing such separating hyperplanes becomes challenging. In this article, the moment-/sums-of-squares hierarchy is used in … Read more
Screening is widely utilized for Mixed-Integer Programming (MIP) presolving. It aims to certify a priori whether one or multiple specific binary variables can be fixed to optimal values based on solutions from convex relaxations. This paper studies the challenge of solving Sparsity-constrained (strongly) Convex Quadratic Programming (SCQP) and proposes the Screening-based Cut Presolving Framework (SCPF). … Read more
We consider a two-stage optimization problem with sparsity constraints, motivated by a common challenge in packaging logistics: minimizing the volume of transported air by optimizing the size and number of available packaging boxes, given the demand for order items. In the first stage, we select the optimal dimensions of the boxes, while in the second … Read more
This paper investigates the a-posteriori analysis of Branch-and-Bound (BB) trees to extract structural information about the feasible region of mixed-binary linear programs. We introduce three novel outer approximations of the feasible region, systematically constructed from a BB tree. These are: a tight formulation based on disjunctive programming, a branching-based formulation derived from the tree’s branching … Read more
We study multi-follower bilevel optimization problems with binary linking variables where the second level consists of many independent integer-constrained subproblems. This problem class not only generalizes many classical interdiction problems but also arises naturally in many network design problems where the second-level subproblems involve complex routing decisions of the actors involved. We propose a novel … Read more
The increasing penetration of renewable energy and rising electricity demand are driving the need to integrate new buses and transmission lines into transmission grids. These trends are reshaping transmission expansion planning (TEP), motivating the development of effective methodologies to manage the resulting complexity. This paper introduces the longest shortest-path connection (LSPC) algorithm, a graph-based method … Read more