Common Mathematical Foundations of Expected Utility and Dual Utility Theories

We show that the main results of the expected utility and dual utility theories can be derived in a unified way from two fundamental mathematical ideas: the separation principle of convex analysis, and integral representations of continuous linear functionals from functional analysis. Our analysis reveals the dual character of utility functions. We also derive new integral representations of dual utility models

Portfolio Optimization With Stochastic Dominance Constraints

We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.portfolio optimization, stochastic dominance, risk, utility functions, duality

Convexification of Stochastic Ordering

We consider sets defined by the usual stochastic ordering relation and by the second order stochastic dominance relation. Under fairy general assumptions we prove that in the space of integrable random variables the closed convex hull of the first set is equal to the second set.Stochastic Dominance, Stochastic Ordering

Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems

We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit formulae for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest

On Risk Evaluation and Control of Distributed Multi-Agent Systems

In this paper, we deal with risk evaluation and risk-averse optimization of complex distributed systems with general risk functionals. We postulate a novel set of axioms for the functionals evaluating the total risk of the system. We derive a dual representation for the systemic risk measures and propose a way to construct non-trivial families of measures by using either a collection of linear scalarizations or non-linear risk aggregation. The new framework facilitates risk-averse sequential decision-making by distributed methods. The proposed approach is compared theoretically and numerically to some of the systemic risk measurements in the existing literature. We formulate a two-stage decision problem with monotropic structure and systemic measure of risk. The structure is typical for distributed systems arising in energy networks, robotics, and other practical situations. A distributed decomposition method for solving the two-stage problem is proposed and it is applied to a problem arising in communication networks. We have used this problem to compare the methods of systemic risk evaluation. We show that the proposed risk aggregation leads to less conservative risk evaluation and results in a substantially better solution of the problem at hand as compared to an aggregation of the risk of individual agents and other methods

Optimal Power Generation under Uncertainty via Stochastic Programming

A power generation system comprising thermal and pumped-storage hydro plants is considered. Two kinds of models for the cost-optimal generation of electric power under uncertain load are introduced: (i) a dynamic model for the short-term operation and (ii) a power production planning model. In both cases, the presence of stochastic data in the optimization model leads to multi-stage and two-stage stochastic programs, respectively. Both stochastic programming problems involve a large number of mixed-integer (stochastic) decisions, but their constraints are loosely coupled across operating power units. This is used to design Lagrangian relaxation methods for both models, which lead to a decomposition into stochastic single unit subproblems. For the dynamic model a Lagrangian decomposition based algorithm is described in more detail. Special emphasis is put on a discussion of the duality gap, the efficient solution of the multi-stage single unit subproblems and on solving the dual problem by bundle methods for convex nondifferentiable optimization

Stability and sensitivity of optimization problems with first order stochastic dominance constraints

We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type. Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the dominance constraints