We propose a combinatorial algorithm to compute the Hoffman constant of a system of linear equations and inequalities. The algorithm is based on a characterization of the Hoffman constant as the largest of a finite canonical collection of easy-to-compute Hoffman constants. Our algorithm and characterization extend to the more general context where some of the … Read more
We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by D the largest absolute value of the subdeterminants of the constraint matrix. In this paper we give an algorithm that finds an epsilon-approximate solution for this problem by solving a number of … Read more
We consider a system with an evolving state that can be stopped at any time by a decision maker (DM), yielding a state-dependent reward. The DM does not observe the state except for a limited number of monitoring times, which he must choose, in conjunction with a suitable stopping policy, to maximize his reward. Dealing … Read more
Given $X\subset R^n$, $\varepsilon \in (0,1)$, a parametrized family of probability distributions $(\mu_{a})_{a\in A}$ on $\Omega\subset R^p$, we consider the feasible set $X^*_\varepsilon\subset X$ associated with the {\em distributionally robust} chance-constraint \[X^*_\varepsilon\,=\,\{x\in X:\:{\rm Prob}_\mu[f(x,\omega)\,>\,0]> 1-\varepsilon,\,\forall\mu\in\mathscr{M}_a\},\] where $\mathscr{M}_a$ is the set of all possibles mixtures of distributions $\mu_a$, $a\in A$. For instance and typically, the family … Read more
Partial outer convexification is a relaxation technique for MIOCPs being constrained by time-dependent differential equations. Sum-Up-Rounding algorithms allow to approximate feasible points of the relaxed, convexified continuous problem with binary ones that are feasible up to an arbitrarily small $\delta > 0$. We show that this approximation property holds for ODEs and semilinear PDEs under … Read more
Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak$^*$ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, … Read more
This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a \emph{linear} equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is … Read more
Recovering a low-rank matrix from highly corrupted measurements arises in compressed sensing of structured high-dimensional signals (e.g., videos and hyperspectral images among others). Robust principal component analysis (RPCA), solved via principal component pursuit (PCP), recovers a low-rank matrix from sparse corruptions that are of unknown value and support by decomposing the observation matrix into two … Read more
We study infeasible-start primal-dual interior-point methods for convex optimization problems given in a typically natural form we denote as Domain-Driven formulation. Our algorithms extend many advantages of primal-dual interior-point techniques available for conic formulations, such as the current best complexity bounds, and more robust certificates of approximate optimality, unboundedness, and infeasibility, to Domain-Driven formulations. The … Read more
A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper we analyze the rate of convergence of gradient descent for smooth unconstrained multiobjective … Read more