An Introduction to Decision Diagrams for Optimization

This tutorial provides an introduction to the use of decision diagrams for solving discrete optimization problems. A decision diagram is a graphical representation of the solution space, representing decisions sequentially as paths from a root node to a target node. By merging isomorphic subgraphs (or equivalent subproblems), decision diagrams can compactly represent an exponential solution … Read more

Probing-Enhanced Stochastic Programming

\(\) We consider a two-stage stochastic decision problem where the decision-maker has the opportunity to obtain information about the distribution of the random variables $\xi$ that appear in the problem through a set of discrete actions that we refer to as probing. Probing components of a random vector $\eta$ that is jointly-distributed with $\xi$ allows … Read more

Interdiction of minimum spanning trees and other matroid bases

\(\) In the minimum spanning tree (MST) interdiction problem, we are given a graph \(G=(V,E)\) with edge weights, and want to find some \(X\subseteq E\) satisfying a knapsack constraint such that the MST weight in \((V,E\setminus X)\) is maximized. Since MSTs of \(G\) are the minimum weight bases in the graphic matroid of \(G\), this … Read more

Branch-and-Bound versus Lift-and-Project Relaxations in Combinatorial Optimization

In this paper, we consider a theoretical framework for comparing branch-and-bound with classical lift-and-project hierarchies. We simplify our analysis of streamlining the definition of branch-and-bound. We introduce “skewed $k$-trees” which give a hierarchy of relaxations that is incomparable to that of Sherali-Adams, and we show that it is much better for some instances. We also … Read more

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

A novel algorithm for a broad class of nonconvex optimization problems

In this paper, we propose a new global optimization approach for solving nonconvex optimization problems in which the nonconvex components are sums of products of convex functions. A broad class of nonconvex problems can be written in this way, such as concave minimization problems, difference of convex problems, and fractional optimization problems. Our approach exploits … Read more

Compressed Sensing: A Discrete Optimization Approach

We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression, image reconstruction, and multi-label … Read more

Handling Symmetries in Mixed-Integer Semidefinite Programs

Symmetry handling is a key technique for reducing the running time of branch-and-bound methods for solving mixed-integer linear programs. In this paper, we generalize the notion of (permutation) symmetries to mixed-integer semidefinite programs (MISDPs). We first discuss how symmetries of MISDPs can be automatically detected by finding automorphisms of a suitably colored auxiliary graph. Then … Read more

A Unified Framework for Symmetry Handling

Handling symmetries in optimization problems is essential for devising efficient solution methods. In this article, we present a general framework that captures many of the already existing symmetry handling methods. While these methods are mostly discussed independently from each other, our framework allows to apply different methods simultaneously and thus outperforming their individual effect. Moreover, … Read more

A Fast Combinatorial Algorithm for the Bilevel Knapsack Problem with Interdiction Constraints

\(\) We consider the bilevel knapsack problem with interdiction constraints, a fundamental bilevel integer programming problem which generalizes the 0-1 knapsack problem. In this problem, there are two knapsacks and \(n\) items. The objective is to select some items to pack into the first knapsack such that the maximum profit attainable from packing some of … Read more