Mixed-Integer Optimization for Responsible Machine Learning

In the last few decades, Machine Learning (ML) has achieved significant success across domains ranging from healthcare, sustainability, and the social sciences, to criminal justice and finance. But its deployment in increasingly sophisticated, critical, and sensitive areas affecting individuals, the groups they belong to, and society as a whole raises critical concerns around fairness, transparency, … Read more

Fast Stochastic Second-Order Adagrad for Nonconvex Bound-Constrained Optimization

ADAGB2, a generalization of the Adagrad algorithm for stochastic optimization is introduced, which is also applicable to bound-constrained problems and capable of using second-order information when available. It is shown that, given  delta in (0,1) and epsilon in (0,1], the ADAGB2 algorithm needs at most O(epsilon^{-2}) iterations to ensure an epsilon-approximate first-order critical point of … Read more

Deterministic global optimization with trained neural networks: Is the envelope of single neurons worth it?

Optimization problems containing trained neural networks remain challenging due to their nonconvexity. Deterministic global optimization relies on relaxations which should be tight, quickly convergent, and cheap to evaluate. While envelopes of common activation functions have been established for several years, the envelope of an entire neuron had not. Recently, Carrasco and Mu\~{n}oz (arXiv.2410.23362, 2024) proposed … Read more

Rank-one convexification for convex quadratic optimization with step function penalties

We investigate convexification in convex quadratic optimization with step function penalties. Such problems can be cast as mixed-integer quadratic optimization problems, where binary variables are used to encode the non-convex step function. First, we derive the convex hull for the epigraph of a quadratic function defined by a rank-one matrix. Using this rank-one convexification, we … Read more

A Foundational Perspective for Partitional Clustering on Networks

This study presents a theoretical analysis of partitional clustering on networks. Different versions of the problem are studied considering different assignment schemes (hard and soft) and different objective functions. Cluster centers are not restricted to only the set of nodes, it is assumed that centers can also be at the edges of the network. Four … Read more

Mathematical programs with complementarity constraints and application to hyperparameter tuning for nonlinear support vector machines

We consider the Mathematical Program with Complementarity Constraints (MPCC). One of the main challenges in solving this problem is the systematic failure of standard Constraint Qualifications (CQs). Carefully accounting for the combinatorial nature of the complementarity constraints, tractable versions of the Mangasarian Fromovitz Constraint Qualification (MFCQ) have been designed and widely studied in the literature. … Read more

Tight Semidefinite Relaxations for Verifying Robustness of Neural Networks

For verifying the safety of neural networks (NNs), Fazlyab et al. (2019) introduced a semidefinite programming (SDP) approach called DeepSDP. This formulation can be viewed as the dual of the SDP relaxation for a problem formulated as a quadratically constrained quadratic program (QCQP). While SDP relaxations of QCQPs generally provide approximate solutions with some gaps, … Read more

An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function

Nonlinear matrix decomposition (NMD) with the ReLU function, denoted ReLU-NMD, is the following problem: given a sparse, nonnegative matrix \(X\) and a factorization rank \(r\), identify a rank-\(r\) matrix \(\Theta\) such that \(X\approx \max(0,\Theta)\). This decomposition finds application in data compression, matrix completion with entries missing not at random, and manifold learning. The standard ReLU-NMD … Read more

A Rank-One-Update Method for the Training of Support Vector Machines

This paper considers convex quadratic programs associated with the training of support vector machines (SVM). Exploiting the special structure of the SVM problem a new type of active set method with long cycles and stable rank-one-updates is proposed and tested (CMU: cycling method with updates). The structure of the problem allows for a repeated simple … Read more

Spherical Support Vector Machine for Interval-Valued Data

In this work we propose a generalization of the Spherical Support Vector Machine method, in which the separator is a sphere, applied to Interval-valued data. This type of data belongs to a more general class, known as Symbolic Data, for which features are described by sets, intervals or histograms instead of classic arrays. This paradigm … Read more