Garud Iyengar

Ambiguous chance constrained problems and robust optimization

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set of the distributions is of the form where ρp denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice.

Robust convex quadratically constrained programs

Abstract.In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. In contrast to [4], where it is shown that such robust problems can be formulated as semidefinite programs, our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second-order cone programs (SOCP). We propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit SOCP and present examples where these classes of uncertainty sets are natural.

Cuts for mixed 0-1 conic programming

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

An Axiomatic Approach to Systemic Risk

Systemic risk refers to the risk of collapse of an entire complex system as a result of the actions taken by the individual component entities or agents that comprise the system. Systemic risk is an issue of great concern in modern financial markets as well as, more broadly, in the management of complex business and engineering systems. We propose an axiomatic framework for the measurement and management of systemic risk based on the simultaneous analysis of outcomes across agents in the system and over scenarios of nature. Our framework defines a broad class of systemic risk measures that accomodate a rich set of regulatory preferences. This general class of systemic risk measures captures many specific measures of systemic risk that have recently been proposed as special cases and highlights their implicit assumptions. Moreover, the systemic risk measures that satisfy our conditions yield decentralized decompositions; i.e., the systemic risk can be decomposed into risk due to individual agents. Furthermore, one can associate a shadow price for systemic risk to each agent that correctly accounts for the externalities of the agents individual decision making on the entire system. This paper was accepted by Gerard P. Cachon, stochastic models and simulation.

Inverse conic programming with applications

Linear programming duality yields efficient algorithms for solving inverse linear programs. We show that special classes of conic programs admit a similar duality and, as a consequence, establish that the corresponding inverse programs are efficiently solvable. We discuss applications of inverse conic programming in portfolio optimization and utility function identification.

Inequality and Network Structure

This paper explores the manner in which the structure of a social network constrains the level of inequality that can be sustained among its members. We assume that any distribution of value across the network must be stable with respect to coalitional deviations, and that players can form a deviating coalition only if they constitute a clique in the network. We show that if the network is bipartite, there is a unique stable payoff distribution that is maximally unequal in that it does not Lorenz dominate any other stable distribution. We obtain a complete ordering of the class of bipartite networks and show that those with larger maximum independent sets can sustain greater levels of inequality. The intuition behind this result is that networks with larger maximum independent sets are more sparse and hence offer fewer opportunities for coalitional deviations. We also demonstrate that standard centrality measures do not consistently predict inequality. We extend our framework by allowing a group of players to deviate if they are all within distance k of each other, and show that the ranking of networks by the extent of extremal inequality is not invariant in k.

On two-stage convex chance constrained problems

In this paper we develop approximation algorithms for two-stage convex chance constrained problems. Nemirovski and Shapiro (Probab Randomized Methods Des Uncertain 2004) formulated this class of problems and proposed an ellipsoid-like iterative algorithm for the special case where the impact function f (x, h) is bi-affine. We show that this algorithm extends to bi-convex f (x, h) in a fairly straightforward fashion. The complexity of the solution algorithm as well as the quality of its output are functions of the radius r of the largest Euclidean ball that can be inscribed in the polytope defined by a random set of linear inequalities generated by the algorithm (Nemirovski and Shapiro in Probab Randomized Methods Des Uncertain 2004). Since the polytope determining r is random, computing r is difficult. Yet, the solution algorithm requires r as an input. In this paper we provide some guidance for selecting r. We show that the largest value of r is determined by the degree of robust feasibility of the two-stage chance constrained problem—the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous two-stage chance constrained problems. In this formulation, the random variables defining the chance constraint are known to have a fixed distribution; however, the decision maker is only able to estimate this distribution to within some error. We construct an algorithm that solves the ambiguous two-stage chance constrained problem when the impact function f (x, h) is bi-affine and the extreme points of a certain “dual” polytope are known explicitly.

Fast gradient descent method for Mean-CVaR optimization

We propose an iterative gradient descent algorithm for solving scenario-based Mean-CVaR portfolio selection problem. The algorithm is fast and does not require any LP solver. It also has efficiency advantage over the LP approach for large scenario size.

Approximating Semidefinite Packing Programs

In this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems. Semidefinite packing programs arise in many applications such as semidefinite programming relaxations for combinatorial optimization problems, sparse principal component analysis, and sparse variance unfolding techniques for dimension reduction. Our algorithm exploits the structural similarity between semidefinite packing programs and linear packing programs.

Growth optimal investment in horse race markets with costs

We formulate the problem of growth optimal investment in horse race markets with proportional costs and study growth optimal strategies both for stochastic horse races as well as races where one does not make any distributional assumptions. Our results extend all known results for frictionless horse race markets to their natural analog in markets with costs.