Fast algorithms for approximately solving large convex problems and applications

Fast algorithms for approximately solving large convex problems and applications

We propose in this research to extend and to study theoretical and practical aspects of some fast optimization schemes recently proposed for large-scale linear optimization problems. In direct continuation to the work of Dr. Fabian Chudak and Vania Dos Santos Eleuterio, we plan to investigate extensions of these schemes to semidefinite optimization problems and to convex optimization problems with a stochastic component.

In order to apply these optimization algorithms, namely smoothing techniques, to semidefinite and symmetric optimization problems, several questions need to be explored. From a theoretical point of view, different smoothing strategies can be envisaged, from which we want to obtain convergence and speed guarantees. Of particular interest are issues involving the robustness of these schemes with respect to approximative solutions of subproblems, and/or to inexact oracle information. Also, several numerical questions need to be investigated: the advantage of dealing with almost low-rank matrices can be exploited, and the utilization of the possible sparsity of the data must be addressed. Our research is essentially applications oriented. For instance, we focus on semidefinite relaxations of hard combinatorial problems and largest eigenvalue minimization arising in network applications.

We study extensions of our methods to some structured stochastic optimization problems where the convex objective function includes the expected value of a random variable. Besides the convergence and speed aspects, we are highly interested in the probabilistic and statistical properties of the approximate solution delivered by our algorithm. For practical applications, we will focus on portfolio optimization problems involving the Conditional Value-at-Risk as risk measure.

We intend to contribute to other research efforts inside ETH related to the solution of large-scale structured convex optimization problems.