# Risk Minimization, Regret Minimization and Progressive Hedging Algorithms

## TL;DR

This paper explores the dual representations of risk and regret measures in multistage decision making, establishing a relationship that enables a modified progressive hedging algorithm to efficiently solve related optimization problems.

## Contribution

It introduces a novel relationship between risk and regret envelopes via duality, and develops a modified progressive hedging algorithm for complex multistage risk and regret minimization problems.

## Key findings

- Modified progressive hedging algorithm effectively solves multistage risk minimization problems.
- Numerical results demonstrate the efficiency of the proposed algorithms.
- The duality relationship enhances understanding of risk and regret measures in decision making.

## Abstract

This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. A relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. Such a relationship opens a door to a decomposition scheme, called progressive hedging, for solving multistage risk minimization and regret minimization problems. In particular, the classical progressive hedging algorithm is modified in order to handle a new class of linkage constraints that arises from reformulations and other applications of risk and regret minimization problems. Numerical results are provided to show the efficiency of the progressive hedging algorithms.