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

Deviation measures in stochastic programming with mixed-integer recourse

Stochastic programming offers a way to treat uncertainty in decision problems. In particular, it allows the minimization of risk. We consider mean-risk models involving deviation measures, as for instance the standard deviation and the semideviation, and discuss these risk measures in the framework of stochastic dominance as well as in the framework of coherent risk measures. We derive statements concerning the structure and the stability of the resulting optimization problems whereby we emphasize on models including integrality requirements on some decision variables. Then we propose decomposition algorithms for the mean-risk models under consideration and present numerical results for two stochastic programming applications

Shape Optimization under Uncertainty from a Stochastic Programming Point of View

We consider an elastic body subjected to internal and external forces which are uncertain. Simply averaging the possible loadings will result in a structure that might not be robust for the individual loadings at all. Instead, we apply techniques from level set based shape optimization and two-stage stochastic programming: In the first stage, the non-anticipative decision on the shape has to be taken. Afterwards, the realizations of the random forces are observed, and the variational formulation of the elasticity system takes the role of the second-stage problem. Taking advantage of the PDE's linearity, we are able to compute solutions for an arbitrary number of scenarios without increasing the computational effort significantly. The deformations are described by PDEs that are solved efficiently by Composite Finite Elements. The objective is, e.g., to minimize the compliance. A gradient method using the shape derivative is used to solve the problem. Results for 2D instances are shown. The obtained solutions strongly depend on the initial guess, in particular its topology. To overcome this issue, we included the topological derivative into our algorithm as well. The stochastic programming perspective also allows us to incorporate risk measures into our model which might be a more appropriate objective in many practical applications