Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution

A viability theory approach to a two-stage optimal control problem of technology adoption

A new technology adoption problem can be modelled as a two-stage control problem, in which model parameters ("technology") might be altered at some time. An optimal solution to utility maximisation for this class of problems needs to contain information on the time, at which the change will take place (0, finite or never), along with the optimal control strategies before and after the change. For the change, or switch, to occur the "new technology" value function needs to dominate the "old technology" value function, after the switch. We charaterise the value function using the fact that its hypograph is a viability kernel of an auxiliary problem and we study when the graphs can intersect. If they do not, the switch cannot occur at a positive time. Using this characterisation we analyse a technology adoption problem and showmodels, for which the switch will occur at time zero or never.technology adoption, value function, viability kernel, viscosity solutions

Kontinuierliche Optimierung und Industrieanwendungen

[no abstract available

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

Markov Equilibrium in Models of Dynamic Endogenous Political Institutions

This paper examines existence of Markov equilibria in the class of dynamic political games (DPGs). DPGs are dynamic games in which political institutions are endogenously determined each period. The process of change is both recursive and instrumental: the rules for political aggregation at date t+1 are decided by the rules at date t, and the resulting institutional choices do not affect payoffs or technology directly. Equilibrium existence in dynamic political games requires a resolution to a “political fixed point problem” in which a current political rule (e.g., majority voting) admits a solution only if all feasible political rules in the future admit solutions in all states. If the class of political rules is dynamically consistent, then DPGs are shown to admit political fixed points. This result is used to prove two equilibrium existence theorems, one of which implies that equilibrium strategies, public and private, are smooth functions of the economic state. We discuss practical applications that require existence of smooth equilibria.Recursive, dynamic political games, political fixed points, dynamically consistent rules.

ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models

Deep neural networks (DNNs) are one of the most prominent technologies of our time, as they achieve state-of-the-art performance in many machine learning tasks, including but not limited to image classification, text mining, and speech processing. However, recent research on DNNs has indicated ever-increasing concern on the robustness to adversarial examples, especially for security-critical tasks such as traffic sign identification for autonomous driving. Studies have unveiled the vulnerability of a well-trained DNN by demonstrating the ability of generating barely noticeable (to both human and machines) adversarial images that lead to misclassification. Furthermore, researchers have shown that these adversarial images are highly transferable by simply training and attacking a substitute model built upon the target model, known as a black-box attack to DNNs. Similar to the setting of training substitute models, in this paper we propose an effective black-box attack that also only has access to the input (images) and the output (confidence scores) of a targeted DNN. However, different from leveraging attack transferability from substitute models, we propose zeroth order optimization (ZOO) based attacks to directly estimate the gradients of the targeted DNN for generating adversarial examples. We use zeroth order stochastic coordinate descent along with dimension reduction, hierarchical attack and importance sampling techniques to efficiently attack black-box models. By exploiting zeroth order optimization, improved attacks to the targeted DNN can be accomplished, sparing the need for training substitute models and avoiding the loss in attack transferability. Experimental results on MNIST, CIFAR10 and ImageNet show that the proposed ZOO attack is as effective as the state-of-the-art white-box attack and significantly outperforms existing black-box attacks via substitute models.Comment: Accepted by 10th ACM Workshop on Artificial Intelligence and Security (AISEC) with the 24th ACM Conference on Computer and Communications Security (CCS

The Hybridization of Branch and Bound with Metaheuristics for Nonconvex Multiobjective Optimization

A hybrid framework combining the branch and bound method with multiobjective evolutionary algorithms is proposed for nonconvex multiobjective optimization. The hybridization exploits the complementary character of the two optimization strategies. A multiobjective evolutionary algorithm is intended for inducing tight lower and upper bounds during the branch and bound procedure. Tight bounds such as the ones derived in this way can reduce the number of subproblems that have to be solved. The branch and bound method guarantees the global convergence of the framework and improves the search capability of the multiobjective evolutionary algorithm. An implementation of the hybrid framework considering NSGA-II and MOEA/D-DE as multiobjective evolutionary algorithms is presented. Numerical experiments verify the hybrid algorithms benefit from synergy of the branch and bound method and multiobjective evolutionary algorithms

Discrete-time controlled markov processes with average cost criterion: a survey

This work is a survey of the average cost control problem for discrete-time Markov processes. The authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades. The exposition ranges from finite to Borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies. The authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography. The authors have also identified several important questions that are still open to investigation