We present a branch-and-bound method for multiobjective mixed-integer convex quadratic programs that computes a superset of efficient integer assignments and a coverage of the nondominated set. The method relies on outer approximations of the upper image set of continuous relaxations. These outer approximations are obtained addressing the dual formulations of specific subproblems where the values … Read more
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 and image reconstruction. We … Read more
In many applications, when building linear regression models, it is important to account for the presence of outliers, i.e., corrupted input data points. Such problems can be formulated as mixed-integer optimization problems involving cubic terms, each given by the product of a binary variable and a quadratic term of the continuous variables. Existing approaches in … Read more
\(\)In the academic literature and in expert testimony, the Polsby-Popper score is the most popular way to measure the compactness of a political district. Given a district with area \(A\) and perimeter \(P\), its Polsby-Popper score is given by \( (4 \pi A)/P^2\). This score takes values between zero and one, with circular districts achieving … Read more
This paper discusses MATRS, a new matrix adaptation trust region strategy for solving noisy derivative-free mixed-integer optimization problems with simple bounds. MATRS repeatedly cycles through five phases, mutation, selection, recombination, trust-region, and mixed-integer in this order. But if in the mutation phase a new best point (point with lowest inexact function value among all evaluated … Read more
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 suitable set of test instances, also known as benchmark problems, is a key ingredient to systematically evaluate numerical solution algorithms for a given class of optimization problems. While in recent years several solution algorithms for the class of multiobjective mixed-integer nonlinear optimization problems have been proposed, there is a lack of a well-established set … Read more
Among the most famous algorithms for solving classification problems are support vector machines (SVMs), which find a separating hyperplane for a set of labeled data points. In some applications, however, labels are only available for a subset of points. Furthermore, this subset can be non-representative, e.g., due to self-selection in a survey. Semi-supervised SVMs tackle … Read more
In this article we introduce an algorithm that approximates Pareto fronts of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches and is applicable to problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which … Read more
\(\) 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