Recursive Optimization of Convex Risk Measures: Mean-Semideviation Models

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We develop recursive, data-driven, stochastic subgradient methods for optimizing a new, versatile, and application-driven class of convex risk measures, termed here as mean-semideviations, strictly generalizing the well-known and popular mean-upper-semideviation.

• We introduce the MESSAGEp algorithm, which is an efficient compositional subgradient procedure for iteratively solving convex mean-semideviation risk-averse problems to optimality.
• We analyze the asymptotic behavior of the MESSAGEp algorithm under a flexible and structure-exploiting set of problem assumptions.
• In particular: 1) Under appropriate stepsize rules, we establish pathwise convergence of the MESSAGEp algorithm in a strong technical sense, confirming its asymptotic consistency.