# Two-stage Combinatorial Optimization Problems under Risk

## TL;DR

This paper introduces a two-stage approach for combinatorial optimization under uncertainty, optimizing solutions based on the CVaR risk measure, which generalizes stochastic and robust methods, with new theoretical insights and results.

## Contribution

It presents a novel two-stage combinatorial optimization model using CVaR, unifying stochastic and robust approaches, and provides new theoretical results for basic problems.

## Key findings

- Model generalizes stochastic and robust approaches
- Provides new theoretical results for selection and network problems
- Analyzes both positive and negative aspects of the approach

## Abstract

In this paper a class of combinatorial optimization problems is discussed. It is assumed that a solution can be constructed in two stages. The current first-stage costs are precisely known, while the future second-stage costs are only known to belong to an uncertainty set, which contains a finite number of scenarios with known probability distribution. A partial solution, chosen in the first stage, can be completed by performing an optimal recourse action, after the true second-stage scenario is revealed. A solution minimizing the Conditional Value at Risk (CVaR) measure is computed. Since expectation and maximum are boundary cases of CVaR, the model generalizes the traditional stochastic and robust two-stage approaches, previously discussed in the existing literature. In this paper some new negative and positive results are provided for basic combinatorial optimization problems such as the selection or network problems.