# K-Adaptability in Two-Stage Mixed-Integer Robust Optimization

## TL;DR

This paper introduces a K-adaptability approach for two-stage mixed-integer robust optimization, enabling finite-dimensional approximation and efficient solution despite the challenges posed by discrete recourse decisions.

## Contribution

It proposes a K-adaptability formulation combined with a branch-and-bound algorithm for solving complex two-stage robust optimization problems with mixed decisions.

## Key findings

- Algorithm demonstrates asymptotic convergence.
- Finite convergence achieved under certain conditions.
- Performs well on benchmark datasets.

## Abstract

We study two-stage robust optimization problems with mixed discrete-continuous decisions in both stages. Despite their broad range of applications, these problems pose two fundamental challenges: (i) they constitute infinite-dimensional problems that require a finite-dimensional approximation, and (ii) the presence of discrete recourse decisions typically prohibits duality-based solution schemes. We address the first challenge by studying a $K$-adaptability formulation that selects $K$ candidate recourse policies before observing the realization of the uncertain parameters and that implements the best of these policies after the realization is known. We address the second challenge through a branch-and-bound scheme that enjoys asymptotic convergence in general and finite convergence under specific conditions. We illustrate the performance of our algorithm in numerical experiments involving benchmark data from several application domains.