In this study, we explore how the domain of objective function values for challenging integer programs can be reduced and whether such a reduction can improve the solution process. Our work is motivated by binary search, a technique that efficiently narrows a search space by repeatedly halving it through feasibility checks. Building on this idea, … Read more
We propose an exact binary search-based branch-and-bound algorithm, termed the Quadtree Search Method, for solving bi-objective integer programs. The existing literature on criterion space search methods for multi-objective optimization predominantly assumes that subproblems can be solved to optimality, an assumption that becomes computationally prohibitive for hard instances. In contrast, our approach departs from this assumption … Read more
We study computational aspects of a key problem in robust statistics—the penalized least trimmed squares (LTS) regression problem, a robust estimator that mitigates the influence of outliers in data by capping residuals with large magnitudes. Although statistically attractive, penalized LTS is NP-hard, and existing mixed-integer optimization (MIO) formulations scale poorly due to weak relaxations and … Read more
Machine learning is increasingly used to improve decisions within branch-and-bound algorithms for mixed-integer programming. Many existing approaches rely on deep learning, which often requires very large training datasets and substantial computational resources for both training and deployment, typically with GPU parallelization. In this work, we take a different path by developing interpretable models that are … Read more
We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in {0,±1}. While semidefinite programming (SDP) techniques are well established for {0,1}- and {±1}-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation … Read more
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This work proposes a combinatorial branch-and-bound (B&B) algorithm for the capacitated facility location problem under strict customer preferences (CFLP-SCP). We use combinatorial insights into the problem structure to do preprocessing, model branching implications, enforce feasibility or prove infeasibility in each node, select variables and derive primal and dual bounds in each node of the B&B … Read more
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
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 the … Read more
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 problem … Read more