The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, … Read more
An appealing feature of interior methods for linear programming is that the number of iterations required to solve a problem tends to be relatively insensitive to the choice of initial point. This feature has the drawback that it is difficult to design interior methods that efficiently utilize information from an optimal solution to a “nearby” … Read more
This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed polytopic constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques … Read more
We introduce a framework in which updating rules for the barrier parameter in primal-dual interior-point methods become dynamic. The original primal-dual system is augmented to incorporate explicitly an updating function. A Newton step for the augmented system gives a primal-dual Newton step and also a step in the barrier parameter. Based on local information and … Read more
Given $\cA := \{a^1,\ldots,a^m\} \subset \R^d$ whose affine hull is $\R^d$, we study the problems of computing an approximate rounding of the convex hull of $\cA$ and an approximation to the minimum volume enclosing ellipsoid of $\cA$. In the case of centrally symmetric sets, we first establish that Khachiyan’s barycentric coordinate descent (BCD) method is … Read more
In Multi-Input Multi-Output (MIMO) systems, Maximum-Likelihood (ML) decoding is equivalent to finding the closest lattice point in an N-dimensional complex space. In general, this problem is known to be NP hard. In this paper, we propose a quasi-maximum likelihood algorithm based on Semi-Definite Programming (SDP). We introduce several SDP relaxation models for MIMO systems, with … Read more
The main goals of this paper are to: i) relate two iteration-complexity bounds associated with the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming (LP), and; ii) study the geometrical structure of the central path in the context of LP. The first forementioned iteration-complexity bound is expressed in terms of an integral introduced by Sonnevend, Stoer and … Read more
We present an optimization engine for large scale pattern recognition using Support Vector Machine (SVM). Our treatment is based on conversion of soft-margin SVM constrained optimization problem to an unconstrained form, and solving it using newly developed Sequential Subspace Optimization (SESOP) method. SESOP is a general tool for large-scale smooth unconstrained optimization. At each iteration … Read more
Let $V\subset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $\epsilon>0$, there exist nonnegative scalars $\{\lambda_j\}_{j\in J}$ such that, for all $r$ sufficiently large, $f+\epsilon\theta_r+\sum_{j\in J} \lambda_j g_j^2$ is a sum of squares. Here, … Read more
We consider the class of nonlinear optimal control problems with all data (differential equation, state and control constraints, cost) being polynomials. We provide a simple hierarchy of LMI-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Preliminary results show that good approximations are obtained with few moments. CitationLAAS report … Read more