In this paper, we propose a semi-metric for Markov processes that allows to bound optimal values of linear Markov Decision Processes (MDPs). Similar to existing notions of distance for general stochastic processes our distance is based on transportation metrics. Apart from the specialization to MDPs, our contribution is to make the distance problem specific, i.e., … Read more
This paper proposes a multi-stage stochastic programming formulation based on affine decision rules for the reservoir management problem. Our approach seeks to find a release schedule that balances flood control and power generation objectives while considering realistic operating conditions as well as variable water head. To deal with the non-convexity introduced by the variable water … Read more
This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although some prior work has established convergence results for algorithms that integrate both descent and negative curvature directions, there has not yet been numerical evidence showing that such methods offer significant performance improvements. In … Read more
We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of … Read more
Trust-region algorithms have been proved to globally converge with probability one when the accuracy of the trust-region models is imposed with a certain probability conditioning on the iteration history. In this paper, we study their complexity, providing global rates and worst case complexity bounds on the number of iterations (with overwhelmingly high probability), for both … Read more
Direct search is a methodology for derivative-free optimization whose iterations are characterized by evaluating the objective function using a set of polling directions. In deterministic direct search applied to smooth objectives, these directions must somehow conform to the geometry of the feasible region and typically consist of positive generators of approximate tangent cones (which then … Read more
Natural gas storage valuation is an important problem in energy trading, yet most valuation approaches are based on heuristics or ignore that gas markets are incomplete. We propose an exact valuation model for incomplete gas markets based on multistage stochastic programming. Market incompleteness structurally changes the problem of storage valuation and asset backed trading and … Read more
A distributionally robust joint chance constraint involves a set of uncertain linear inequalities which can be violated up to a given probability threshold $\epsilon$, over a given family of probability distributions of the uncertain parameters. A conservative approximation of a joint chance constraint, often referred to as a Bonferroni approximation, uses the union bound to … Read more
A lower bound for a finite-scenario-based chance-constrained program is the quantile value corresponding to the sorted optimal objective values of scenario subproblems. This quantile bound can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. The quality of the bound depends on how the scenarios are grouped. In this paper, … Read more
In this paper, we introduce and study a two-stage distributionally robust mixed binary problem (TSDR-MBP) where the random parameters follow the worst-case distribution belonging to an uncertainty set of probability distributions. We present a decomposition algorithm, which utilizes distribution separation procedure and parametric cuts within Benders’ algorithm or L-shaped method, to solve TSDR-MBPs with binary … Read more