It has been known for almost 50 years that the discrete l_1 approximation problem can be solved effectively by linear programming. However, improved algorithms involve a step which can be interpreted as a line search, and which is not part of the standard LP solution procedures. l_1 provides the simplest example of a class of … Read more
It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least one of these problems is feasible. It is also well known that for linear conic problems this property of “no duality gap” may not hold. It is then natural to ask … Read more
Density estimation is a classical and important problem in statistics. The aim of this paper is to develop a new computational approach to density estimation based on semidefinite programming (SDP), a new technology developed in optimization in the last decade. We express a density as the product of a nonnegative polynomial and a base density … Read more
Primal-Dual Interior-Point Methods (IPMs) have shown their power in solving large classes of optimization problems. However, there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results with respect to the update strategies of the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best … Read more
We present results on global and polynomial-time convergence of infeasible-interior-point methods for self-scaled conic programming, which includes linear and semidefinite programming. First, we establish global convergence for an algorithm using a wide neighborhood. Next, we prove polynomial complexity for the algorithm with a slightly narrower neighborhood. Both neighborhoods are related to the wide (minus infinity) … Read more
A parallel computational method SDPARA-C is presented for SDPs (semidefinite programs). It combines two methods SDPARA and SDPA-C proposed by the authors who developed a software package SDPA. SDPARA is a parallel implementation of SDPA and it features parallel computation of the elements of the Schur complement equation system and a parallel Cholesky factorization of … Read more
The properties of the barrier F(x)=-log(det(x)), defined over the cone of squares of an Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, it is proved … Read more
A polynomial SDP (semidefinite program) minimizes a polynomial objective function over a feasible region described by a positive semidefinite constraint of a symmetric matrix whose components are multivariate polynomials. Sums of squares relaxations developed for polynomial optimization problems are extended to propose sums of squares relaxations for polynomial SDPs with an additional constraint for the … Read more
We provide two new, simple proofs of Afriat’s celebrated theorem stating that a finite set of price-quantity observations is consistent with utility maximization if, and only if, the observations satisfy a variation of the Strong Axiom of Revealed Preference known as the Generalized Axiom of Revealed Preference. Citation Technical Report No. 1381, School of Operations … Read more
In this paper, we consider a modified version of a well-known long-step primal-dual infeasible IP algorithm for solving the linear program $\min\{c^T x : Ax=b, \, x \ge 0\}$, $A \in \Re^{m \times n}$, where the search directions are computed by means of an iterative linear solver applied to a preconditioned normal system of equations. … Read more