Robust Contextual Portfolio Optimization with Gaussian Mixture Models

We consider the portfolio optimization problem with contextual information that is available to better quantify and predict the uncertain returns of assets. Motivated by the regime modeling techniques for the finance market, we consider the setting where both the uncertain returns and the contextual information follow a Gaussian Mixture (GM) distribution. This problem is shown … Read more

Robust Phi-Divergence MDPs

In recent years, robust Markov decision processes (MDPs) have emerged as a prominent modeling framework for dynamic decision problems affected by uncertainty. In contrast to classical MDPs, which only account for stochasticity by modeling the dynamics through a stochastic process with a known transition kernel, robust MDPs additionally account for ambiguity by optimizing in view … Read more

A Unifying Framework for the Capacitated Vehicle Routing Problem under Risk and Ambiguity

We propose a generic model for the capacitated vehicle routing problem (CVRP) under demand uncertainty. By combining risk measures or disutility functions with complete or partial characterizations of the probability distribution governing the demands, our formulation bridges the popular but often independently studied paradigms of stochastic programming and distributionally robust optimization. We characterize when an … Read more

Adjustable Distributionally Robust Optimization with Infinitely Constrained Ambiguity Sets

We study adjustable distributionally robust optimization problems where their ambiguity sets can potentially encompass an infinite number of expectation constraints. Although such an ambiguity set has great modeling flexibility in characterizing uncertain probability distributions, the corresponding adjustable problems remain computationally intractable and challenging. To overcome this issue, we propose a greedy improvement procedure that consists … Read more

Partial Policy Iteration for L1-Robust Markov Decision Processes

Robust Markov decision processes (MDPs) allow to compute reliable solutions for dynamic decision problems whose evolution is modeled by rewards and partially-known transition probabilities. Unfortunately, accounting for uncertainty in the transition probabilities significantly increases the computational complexity of solving robust MDPs, which severely limits their scalability. This paper describes new efficient algorithms for solving the … Read more

On Data-Driven Prescriptive Analytics with Side Information: A Regularized Nadaraya-Watson Approach

We consider generic stochastic optimization problems in the presence of side information which enables a more insightful decision. The side information constitutes observable exogenous covariates that alter the conditional probability distribution of the random problem parameters. A decision maker who adapts her decisions according to the observed side information solves an optimization problem where the … Read more

Multilevel Optimization Methods: Convergence and Problem Structure

Building upon multigrid methods, the framework of multilevel optimization methods was developed to solve structured optimization problems, including problems in optimal control, image processing, etc. In this paper, we give a broader view of the multilevel framework and establish some connections between multilevel algorithms and the other approaches. An interesting case of the so called … Read more

Empirical Risk Minimization: Probabilistic Complexity and Stepsize Strategy

Empirical risk minimization (ERM) is recognized as a special form in standard convex optimization. When using a first order method, the Lipschitz constant of the empirical risk plays a crucial role in the convergence analysis and stepsize strategies for these problems. We derive the probabilistic bounds for such Lipschitz constants using random matrix theory. We … Read more

Singularly Perturbed Markov Decision Processes: A Multiresolution Algorithm

Singular perturbation techniques allow the derivation of an aggregate model whose solution is asymptotically optimal for Markov Decision Processes with strong and weak interactions. We develop an algorithm that takes advantage of the asymptotic optimality of the aggregate model in order to compute the solution of the original model with theoretically better complexity than conventional … Read more