Joint model of probabilistic-robust (probust) constraints with application to gas network optimization

Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability

Optimal privatization portfolios in the presence of arbitrary risk aversion

We consider the global portfolio of privatized state assets from 1985 to 2012 in the non-parametric decision-making context of Stochastic Dominance Efficiency for broad classes of investor preferences. We estimate all possible portfolios in the context of Strategic vs. non-Strategic and Cyclical vs. non-Cyclical asset allocations that dominate the market benchmark and provide a complete efficiency ranking. The optimal solutions are computed using linear and mixed integer programming formulations. Dominant portfolios tend to overweight non-Cyclical and non-Strategic assets, while rotation may take place across business cycles. Bayesian investment style return attribution analysis, based on Monte Carlo Integration, suggests that Growth drives returns during the first business cycle, rotating to a balanced mix of styles with Size and Debt Leverage during the second business cycle and finally to Size during the last business cycle. Value is found to be the least influential style in all periods

An SDP approach for multiperiod mixed 0-1 linear programming models with stochastic dominance constraints for risk management *

Abstract In this paper we consider multiperiod mixed 0-1 linear programming models under uncertainty. We propose a risk averse strategy using stochastic dominance constraints (SDC) induced by mixed-integer linear recourse as the risk measure. The SDC strategy extends the existing literature to the multistage case and includes both first-order and second-order constraints. We propose a stochastic dynamic programming (SDP) solution approach, where one has to overcome the negative impact the cross-scenario constraints, due to SDC, have on the decomposability of the model. In our computational experience we compare our SDP against a commercial optimization package, in terms of solution accuracy and elapsed time. We use supply chain planning instances, where procurement, production, inventory, and distribution decisions need to be made under demand uncertainty. We confirm the hardness of the testbed, where the benchmark cannot find a feasible solution for half of the test instances while we always find one, and show the appealing tradeoff of SDP, in terms of solution accuracy and elapsed time, when solving medium-to-large instances

Advancing stability analysis of mean-risk stochastic programs: Bilevel and two-stage models

Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean-risk models are essentially defined by a parametric recourse problem and a quantification of risk. The thesis addresses sufficient conditions for weak continuity of the resulting objective functions with respect to perturbations of the underlying probability measure. The approach is based on so called psi-weak topologies that are finer than the topology of weak convergence and allows to unify and extend known results for a comprehensive class of risk measures and recourse problems. In particular, stability of mean-risk models with mixed-integer quadratic and general mixed-integer convex recourse problems is derived for any law-invariant, convex and nondecreasing quantification of risk. From a conceptual point of view, two-stage stochastic programs and bilevel problems under stochastic uncertainty are closely related. Assuming that only the follower can observe the realization of the randomness, the optimistic and pessimistic setting give rise to two-stage problems where only optimal solutions of the lower level are feasible for the recourse problem. So far, stability in stochastic bilevel programming has only been examined for a specific model based on a quantile criterion. The novel approach allows to identify sufficient conditions for stability of stochastic bilevel problems with quadratic lower level and is applicable for a comprehensive class of risk measures.Die Bewertung und das Management von Risken sind ein wesentlicher Aspekt von Entscheidungsproblemen unter stochastischer Unsicherheit. Zielfunktionsbasierte risikoaverse Modelle der zweistufigen stochastischen Optimierung lassen sich im Wesentlichen durch ihr parametrisches Zweitstufenproblem und das betrachtete Risikomaß charakterisieren. Die Arbeit befasst sich mit hinreichenden Bedingungen für Stetigkeit der resultierenden Zielfunktion unter Störungen des zu Grunde liegenden Wahrscheinlichkeitsmaßes bezüglich der Topologie schwacher Konvergenz. Der Ansatz basiert auf so genannten psi-schwachen Topologien, die feiner als die Topologie schwacher Konvergenz sind. Für eine umfassende Klasse von Risikomaßen und Zweitstufenproblemen werden so bestehende Resultate vereinheitlicht und erweitert. Insbesondere lassen sich für jedes verteilungsinvariante, konvexe und nichtfallende Risikomaß Stabilitätsaussagen für Aufgaben mit quadratischem oder konvexem gemischt-ganzzahligen Zweitstufenproblem treffen. Aus konzeptioneller Sicht sind zweistufige stochastische Programme und Bilevel Probleme unter stochastischer Unsicherheit eng miteinander verbunden. Unter der Annnahme, dass nur der Entscheider auf der unteren Ebene die Realisierung des Zufalls beobachten kann, führen sowohl der optimistische als auch der pessimistische Ansatz auf ein zweistufiges stochastisches Programm. Bei diesem sind nur die Optimallösungen der unteren Ebene zulässig für das Zweitstufenproblem. Bisher ist die Stabilität solcher Aufgaben nur für Modelle mit einem speziellen Quantilkriterium untersucht worden. Der neue Ansatz erlaubt es, hinreichende Bedingungen für die Stabilität von stochastischen Bilevel Problemen mit quadratischem Nachfolgerproblem zu identifizieren und ist auf eine reichhaltige Klasse von Risikomaßen anwendbar