Berc Rustem Large scale nonsmooth convex optimization is a common problem for a range of computational areas including machine learning and computer vision. Problems in these areas contain special domain structures and characteristics. Special treatment of such problem domains, exploiting their structures, can significantly improve the the computational burden. We present a weighted Mirror Descent method to … Read more
Bi-level problems are optimisation problems in which some of the decision variables must optimise a subordinate (lower-level) problem. In general, the lower-level problem can possess multiple optimal solutions. One therefore distinguishes between optimistic formulations, which assume that the most favourable lower-level solution is implemented, and pessimistic formulations, in which the most adverse lower-level solution is … Read more
We propose an algorithm that computes the epsilon-correlated equilibria with global-optimal (i.e., maximum) expected social welfare for single stage polynomial games. We first derive an infinite-dimensional formulation of epsilon-correlated equilibria using Kantorovich polynomials and re-express it as a polynomial positivity constraint. In addition, we exploit polynomial sparsity to achieve a leaner problem formulation involving Sum-Of-Squares … Read more
We propose a tractable approximation scheme for convex (not necessarily linear) multi-stage robust optimization problems. We approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization. We obtain a-priori probabilistic guarantees on the feasibility properties of … Read more
We develop tractable semidefinite programming (SDP) based approximations for distributionally robust individual and joint chance constraints, assuming that only the first- and second-order moments as well as the support of the uncertain parameters are given. It is known that robust chance constraints can be conservatively approximated by Worst-Case Conditional Value-at-Risk (CVaR) constraints. We first prove … Read more
Markov decision processes (MDPs) are powerful tools for decision making in uncertain dynamic environments. However, the solutions of MDPs are of limited practical use due to their sensitivity to distributional model parameters, which are typically unknown and have to be estimated by the decision maker. To counter the detrimental effects of estimation errors, we consider … Read more
Portfolio optimization problems involving Value-at-Risk (VaR) are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are compounded when the portfolio contains derivatives. We develop two tractable conservative approximations for the VaR of a derivative portfolio by evaluating the worst-case VaR … Read more
In this chapter we present a primal-dual interior point algorithm for solving constrained nonlinear programming problems. Switching rules are implemented that aim at exploiting the merits and avoiding the drawbacks of three different merit functions. The penalty parameter is determined using an adaptive penalty strategy that ensures a descent property for the merit function. The … Read more
We present a primal-dual interior point method for constrained nonlinear, discrete minimax problems where the objective functions and constraints are not necessarily convex. The algorithm uses two merit functions to ensure progress towards the points satisfying the first order optimality conditions of the original problem. Convergence properties are described and numerical results provided. Citation Dept. … Read more
Robust portfolio optimization finds the worst-case portfolio return given that the asset returns are realized within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns … Read more