# Oracle-Based Algorithms for Binary Two-Stage Robust Optimization

## TL;DR

This paper introduces efficient algorithms for binary two-stage robust optimization with objective uncertainty, including a branch & bound method and a column-and-constraint generation approach, tested on hub-location and capital budgeting problems.

## Contribution

It presents a novel branch & bound algorithm for lower bounds and applies a column-and-constraint generation method, advancing solution techniques for robust optimization with objective uncertainty.

## Key findings

- Branch & bound outperforms column-and-constraint generation
- Algorithms handle non-linear concave objectives in uncertain parameters
- Efficient lower bounds are computed via iterative deterministic and adversarial problem solving

## Abstract

In this work we study binary two-stage robust optimization problems with objective uncertainty. We present an algorithm to calculate efficiently lower bounds for the binary two-stage robust problem by solving alternately the underlying deterministic problem and an adversarial problem. For the deterministic problem any oracle can be used which returns an optimal solution for every possible scenario. We show that the latter lower bound can be implemented in a branch & bound procedure, where the branching is performed only over the first-stage decision variables. All results even hold for non-linear objective functions which are concave in the uncertain parameters. As an alternative solution method we apply a column-and-constraint generation algorithm to the binary two-stage robust problem with objective uncertainty. We test both algorithms on benchmark instances of the uncapacitated single-allocation hub-location problem and of the capital budgeting problem. Our results show that the branch & bound procedure outperforms the column-and-constraint generation algorithm.