Fragility of the Commons under Prospect-Theoretic Risk Attitudes

We study a common-pool resource game where the resource experiences failure with a probability that grows with the aggregate investment in the resource. To capture decision making under such uncertainty, we model each player's risk preference according to the value function from prospect theory. We show the existence and uniqueness of a pure Nash equilibrium when the players have heterogeneous risk preferences and under certain assumptions on the rate of return and failure probability of the resource. Greater competition, vis-a-vis the number of players, increases the failure probability at the Nash equilibrium; we quantify this effect by obtaining bounds on the ratio of the failure probability at the Nash equilibrium to the failure probability under investment by a single user. We further show that heterogeneity in attitudes towards loss aversion leads to higher failure probability of the resource at the equilibrium.Comment: Accepted for publication in Games and Economic Behavior, 201

Data-Driven Chance Constrained Optimization under Wasserstein Ambiguity Sets

We present a data-driven approach for distributionally robust chance constrained optimization problems (DRCCPs). We consider the case where the decision maker has access to a finite number of samples or realizations of the uncertainty. The chance constraint is then required to hold for all distributions that are close to the empirical distribution constructed from the samples (where the distance between two distributions is defined via the Wasserstein metric). We first reformulate DRCCPs under data-driven Wasserstein ambiguity sets and a general class of constraint functions. When the feasibility set of the chance constraint program is replaced by its convex inner approximation, we present a convex reformulation of the program and show its tractability when the constraint function is affine in both the decision variable and the uncertainty. For constraint functions concave in the uncertainty, we show that a cutting-surface algorithm converges to an approximate solution of the convex inner approximation of DRCCPs. Finally, for constraint functions convex in the uncertainty, we compare the feasibility set with other sample-based approaches for chance constrained programs.Comment: A shorter version is submitted to the American Control Conference, 201

Real forms of simple lie algebras

This thesis is about the real forms of simple Lie algebras. Firstly we start with some basic theory of Lie algebra and different types of Lie algebras with some examples in Chapter 2. Thereafter in Chapters 3 to 4 we give the complete classification of real forms of simple Lie algebras. Also we have discuss the Chevalley basis which is important for the Simple Lie algebras. This project introduces Lie groups and their associated Lie algebras. In this thesis we introduce various properties of real forms, conjugations, and automorphisms of complex simple Lie algebras. Finally, there is a complete classification of real forms of simple Lie algebras

Data-Driven Risk-sensitive Model Predictive Control for Safe Navigation in Multi-Robot Systems

Safe navigation is a fundamental challenge in multi-robot systems due to the uncertainty surrounding the future trajectory of the robots that act as obstacles for each other. In this work, we propose a principled data-driven approach where each robot repeatedly solves a finite horizon optimization problem subject to collision avoidance constraints with latter being formulated as distributionally robust conditional value-at-risk (CVaR) of the distance between the agent and a polyhedral obstacle geometry. Specifically, the CVaR constraints are required to hold for all distributions that are close to the empirical distribution constructed from observed samples of prediction error collected during execution. The generality of the approach allows us to robustify against prediction errors that arise under commonly imposed assumptions in both distributed and decentralized settings. We derive tractable finite-dimensional approximations of this class of constraints by leveraging convex and minmax duality results for Wasserstein distributionally robust optimization problems. The effectiveness of the proposed approach is illustrated in a multi-drone navigation setting implemented in Gazebo platform

Consistency of Distributionally Robust Risk-and Chance-Constrained Optimization under Wasserstein Ambiguity Sets

We study stochastic optimization problems with chance and risk constraints, where in the latter, risk is quantified in terms of the conditional value-at-risk (CVaR). We consider the distributionally robust versions of these problems, where the constraints are required to hold for a family of distributions constructed from the observed realizations of the uncertainty via the Wasserstein distance. Our main results establish that if the samples are drawn independently from an underlying distribution and the problems satisfy suitable technical assumptions, then the optimal value and optimizers of the distributionally robust versions of these problems converge to the respective quantities of the original problems, as the sample size increases