A convex cone K is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD) matrices that follow a sparsity pattern given by a homogeneous chordal graph. Our goal in this paper is to elucidate the … Read more
Every symmetric cone K arises as the cone of squares in a Euclidean Jordan algebra V. As V is a real inner-product space, we may denote by Isom(V) its group of isometries. The groups JAut(V) of its Jordan-algebra automorphisms and Aut(K) of the linear cone automorphisms are then related. For certain inner products, JAut(V) = … Read more
For equality-constrained linear mixed-integer programs (MIP) defined by rational data, it is known that the subadditive dual is a strong dual and that there exists an optimal solution of a particular form, termed generator subadditive function. Motivated by these results, we explore the connection between Lagrangian duality, subadditive duality and generator subadditive functions for general … Read more
We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality constraints (semi-infinite QCQPs). Specifically, we present two sufficient conditions on the feasible region under which the QCQP, with any quadratic objective function over the … Read more
We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we … Read more
The Standard Quadratic optimization Problem (StQP), arguably the simplest among all classes of NP-hard optimization problems, consists of extremizing a quadratic form (the simplest nonlinear polynomial) over the standard simplex (the simplest polytope/compact feasible set). As a problem class, StQPs may be nonconvex with an exponential number of inefficient local solutions. StQPs arise in a … Read more
We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point … Read more
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Robust Optimization (RO) is an approach to tackle uncertainties in the parameters of an optimization problem. Constructing an uncertainty set is crucial for RO, as it determines the quality and the conservativeness of the solutions. In this paper, we introduce an approach for constructing a data-driven uncertainty set through volume-based clustering, which we call Minimum-Volume … Read more
We consider the problem of learning support vector machines robust to uncertainty. It has been established in the literature that typical loss functions, including the hinge loss, are sensible to data perturbations and outliers, thus performing poorly in the setting considered. In contrast, using the 0-1 loss or a suitable non-convex approximation results in robust … Read more