Robust optimization programs are hard to solve even when the constraints are convex. In previous contributions, it has been shown that approximately robust solutions (i.e. solutions feasible for all constraints but a small fraction of them) to convex programs can be obtained at low computational cost through constraints randomization. In this paper, we establish new … Read more
In this paper we study the hybrid extragradient method coupled with approximation and penalty schemes for minimization problems. Under certain hypotheses, that include for example the case of Tikhonov regularization, we prove convergence of the method to the solution set of our minimization problem. When we use schemes of penalization or barrier we can show … Read more
We consider solving minimization problems with $\ell_1$-regularization: $$\min \|x\|_1 + \mu f(x),$$ particularly for $f(x) = \frac{1}{2}\|Ax-b\|_M^2$ where $A \in \R^{m \times n}$ with $m < n$. Our goal is to construct efficient and robust algorithms for solving large-scale problems with dense data, and our approach is based on two powerful algorithmic ideas, operator-splitting and ... Read more
We propose a new gradient based scheme to approximate Nash equilibria of large sequential two-player, zero-sum games. The algorithm uses modern smoothing techniques for saddle-point problems tailored specifically for the polytopes used in the Nash equilibrium problem. Citation Working Paper, Tepper School of Business, Carnegie Mellon University Article Download View A gradient-based approach for computing … Read more
The basis pursuit technique is used to find a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise fits the least-squares problem only approximately, and a single parameter determines a curve that traces the trade-off between the least-squares fit and the one-norm of the solution. We show that the function that describes this … Read more
Let $S =\{x\in \re^n:\, g_1(x)\geq 0, \cdots, g_m(x)\geq 0\}$ be a semialgebraic set defined by multivariate polynomials $g_i(x)$. Assume $S$ is convex, compact and has nonempty interior. Let $S_i =\{x\in \re^n:\, g_i(x)\geq 0\}$, and $\bdS$ (resp. $\bdS_i$) be the boundary of $S$ (resp. $S_i$). This paper, as does the subject of semidefinite programming (SDP), concerns … Read more
Aiming at a fast and robust simulation of large multibody systems with contacts and friction, this work presents a novel method for solving large cone complementarity problems by means of a fixed-point iteration. The method is an extension of the Gauss-Seidel and Gauss-Jacobi method with overrelaxation for symmetric convex linear complementarity problems. The method is … Read more
A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the … Read more
We provide and analyze a based average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also … Read more
Given $\cA := \{a^1,\ldots,a^m\} \subset \R^n$ and $\eps > 0$, we propose and analyze two algorithms for the problem of computing a $(1 + \eps)$-approximation to the radius of the minimum enclosing ball of $\cA$. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of … Read more