On the Complexity of Branching Proofs

We consider the task of proving integer infeasibility of a bounded convex set K in R^n using a general branching proof system. In a general branching proof, one constructs a branching tree by adding an integer disjunction at each node, such that the leaves of the tree correspond to empty sets (i.e., K together with … Read more

A Combinatorial Cut-and-Lift Procedure with an Application to 0-1 Chance Constraints

Cut generation and lifting are key components for the performance of state-of-the-art mathematical programming solvers. This work proposes a new general cut-and-lift procedure that exploits the combinatorial structure of 0-1 problems via a binary decision diagram (BDD) encoding of their constraints. We present a general framework that can be applied to a large range of … Read more

Complexity of cutting planes and branch-and-bound in mixed-integer optimization

We investigate the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization. In particular, we study the relative efficiency of BB and CP, when both are based on the same family of disjunctions. We extend a result of Dash to the nonlinear setting which shows that for convex 0/1 problems, CP … Read more

A Finitely Convergent Disjunctive Cutting Plane Algorithm for Bilinear Programming

\(\) In this paper we present and analyze a finitely-convergent disjunctive cutting plane algorithm to obtain an \(\epsilon\)-optimal solution or detect infeasibility of a general nonconvex continuous bilinear program. While the cutting planes are obtained in a manner similar to Saxena, Bonami, and Lee [Math. Prog. 130: 359–413, 2011] and Fampa and Lee [J. Global … Read more

Achieving Consistency with Cutting Planes

Cutting planes accelerate branch-and-bound search primarily by cutting off fractional solutions of the linear programming (LP) relaxation, resulting in tighter bounds for pruning the search tree. Yet cutting planes can also reduce backtracking by excluding inconsistent partial assignments that occur in the course of branching. A partial assignment is inconsistent with a constraint set when … Read more

On Polyhedral and Second-Order-Cone Decompositions of Semidefinite Optimization Problems

We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method’s convergence, under a boundedness assumption. By relating the method’s rate of convergence to an initial outer approximation’s diameter, we argue that the method performs well when initialized with a second-order-cone approximation, instead of a linear approximation. We invoke … Read more

Gaining traction – On the convergence of an inner approximation scheme for probability maximization

We analyze an inner approximation scheme for probability maximization. The approach was proposed in Fabian, Csizmas, Drenyovszki, Van Ackooij, Vajnai, Kovacs, Szantai (2018) Probability maximization by inner approximation, Acta Polytechnica Hungarica 15:105-125, as an analogue of a classic dual approach in the handling of probabilistic constraints. Even a basic implementation of the maximization scheme proved … Read more

Multi-Row Intersection Cuts based on the Infinity Norm

When generating multi-row intersection cuts for Mixed-Integer Linear Optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet-defining for the corner relaxation, the number of potential candidates is still very large, specially for instances of large size. In this paper, we introduce a subset … Read more

Generalized Chvatal-Gomory closures for integer programs with bounds on variables

Integer programming problems that arise in practice often involve decision variables with one or two sided bounds. In this paper, we consider a generalization of Chvatal-Gomory inequalities obtained by strengthening Chvatal-Gomory inequalities using the bounds on the variables. We prove that the closure of a rational polyhedron obtained after applying the generalized Chvatal-Gomory inequalities is … Read more