Stochastic Approximation approach to Stochastic Programming

In this paper we consider optimization problems where the objective function is given in a form of the expectation. A basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals (expectations) cannot be computed with high accuracy. The aim of this paper is to compare two computational approaches based on Monte … Read more

Large-Scale Semidefinite Programming via Saddle Point Mirror-Prox Algorithm

In this paper, we first develop “economical” representations for positive semidefinitness of well-structured sparse symmetric matrix. Using the representations, we then reformulate well-structured large-scale semidefinite problems into smooth convex-concave saddle point problems, which can be solved by a Prox-method with efficiency ${\cal O}(\epsilon^{-1})$ developed in \cite{Nem}. Some numerical implementations for large-scale Lovasz capacity and MAXCUT … Read more

Computing Mountain Passes

We propose the elastic string algorithm for computing mountain passes in finite-dimensional problems. We analyze the convergence properties and numerical performance of this algorithm for benchmark problems in chemistry and discretizations of infinite-dimensional variational problems. We show that any limit point of the elastic string algorithm is a path that crosses a critical point at … Read more

Fast iterative solution of saddle point problems in optimal control based on wavelets

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an … Read more