Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subdifferentials of Convex Operators Valued in the Space of Integrable Functions with Application to Risk-Averse Optimization

Darinka Dentcheva, Andrzej Ruszczynski

TL;DR

The paper develops a subdifferential calculus for convex operators valued in $L_p$ spaces and their compositions with convex risk functionals, introducing a vector-measure Radon–Nikodym approach that extends beyond normal integrands. It provides explicit subgradient structures for $\varrho\circ F$ and $\varrho\circ G$ when $F$ is either a general convex operator or a convex integrand, including extensions to partial information. A comprehensive treatment of stochastic-dominance operators via Lorenz functions yields dual characterizations and subdifferential representations for dominance constraints. The authors derive subdifferential-optimality conditions for risk-averse stochastic optimization problems with inverse stochastic-dominance constraints, linking to AVaR and spectral risk measures. The results have broad applicability to risk-averse optimization, with potential extensions to other non-linear risk-functionals and information structures.

Abstract

We study differentiability properties of convex operators defined on a Banach space with values in an $\Lc_p$ space and of their compositions with monotonic convex functionals on this space. We develop new tools for operators enjoying an additional feature known as the local property. The new approach and results go beyond the classical theory of normal integrands and lattice-valued operators. We further describe the subdifferentials of compositions of such operators with convex monotonic functionals. The new results are applied to obtain novel optimality conditions in the subdifferential form for a broad class of risk-averse stochastic optimization problems with risk functionals as objectives, with partial information, and with stochastic dominance constraints. While our analysis is motivated by the theory and methods of risk-averse optimization, it addresses problems of a more general structure and has potential for further applications.

Subdifferentials of Convex Operators Valued in the Space of Integrable Functions with Application to Risk-Averse Optimization

TL;DR

The paper develops a subdifferential calculus for convex operators valued in spaces and their compositions with convex risk functionals, introducing a vector-measure Radon–Nikodym approach that extends beyond normal integrands. It provides explicit subgradient structures for and when is either a general convex operator or a convex integrand, including extensions to partial information. A comprehensive treatment of stochastic-dominance operators via Lorenz functions yields dual characterizations and subdifferential representations for dominance constraints. The authors derive subdifferential-optimality conditions for risk-averse stochastic optimization problems with inverse stochastic-dominance constraints, linking to AVaR and spectral risk measures. The results have broad applicability to risk-averse optimization, with potential extensions to other non-linear risk-functionals and information structures.

Abstract

We study differentiability properties of convex operators defined on a Banach space with values in an space and of their compositions with monotonic convex functionals on this space. We develop new tools for operators enjoying an additional feature known as the local property. The new approach and results go beyond the classical theory of normal integrands and lattice-valued operators. We further describe the subdifferentials of compositions of such operators with convex monotonic functionals. The new results are applied to obtain novel optimality conditions in the subdifferential form for a broad class of risk-averse stochastic optimization problems with risk functionals as objectives, with partial information, and with stochastic dominance constraints. While our analysis is motivated by the theory and methods of risk-averse optimization, it addresses problems of a more general structure and has potential for further applications.

Paper Structure

This paper contains 5 sections, 14 theorems, 77 equations.

Table of Contents

  1. Composition of a Risk Functional with a Convex Operator
  2. Composition of a Risk Functional with a Convex Integrand
  3. The Case of Partial Information
  4. Stochastic Dominance Operators
  5. Optimization with Stochastic Dominance Constraints

Key Result

Theorem 1.1

Suppose $F:\mathfrak{X}\to \mathcal{L}_p(\Omega,\mathcal{F},P)$ is convex and continuous at $x$. Then $\partial F(x)\ne \emptyset$ and for every $h\in \mathfrak{X}$ we can find ${S}\in \partial F(x)$ such that $F'(x;h) = {S}h$.

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 1.2
  • proof
  • Lemma 2.1
  • proof
  • ...and 17 more