# Polytopic Superset Algorithm for Nonlinear Robust Optimization

## TL;DR

This paper introduces a novel polytopic superset algorithm for nonlinear robust optimization that guarantees feasible iterates and bounds on the optimal value, improving over traditional methods especially with many constraints.

## Contribution

The proposed algorithm generates feasible solutions and bounds for NRO problems using polytopic supersets and a cutting plane method, with proven convergence to the optimum.

## Key findings

- The superset algorithm converges to the optimal solution.
- It outperforms Polak's outer approximation method with many constraints.
- Numerical results demonstrate improved efficiency and feasibility.

## Abstract

Nonlinear robust optimization (NRO) is widely used in different applications, including energy, control, and economics, to make robust decisions under uncertainty. One of the classical solution methods in NRO is an outer approximation method that iteratively solves a sample-based nonlinear problem and updates the sample set by solving an auxiliary problem subject to the uncertainty set. Although it guarantees convergence under certain assumptions, its solution iterates are generally infeasible in the original NRO problem, and it provides only a lower bound on the optimal objective value. We propose a new algorithm for a class of NRO problems that generates feasible solution iterates and provides both lower and upper bounds to the optimal objective value. In each iteration, the algorithm solves the reformulation of an NRO subproblem with respect to the polytopic supersets of the original uncertainty set and uses a cutting plane method to improve the supersets over iteration. If the NRO subproblem is infeasible, we provide a feasibility restoration step to detect whether the original NRO problem is infeasible or construct a new superset to restore the feasibility of the NRO subproblem. Further, we prove that our superset algorithm converges to the optimal solution of the original NRO problem. In numerical studies, we use application instances from portfolio optimization and production cost minimization and compare the performance between the superset algorithm and an outer approximation method called Polak's algorithm. Our result shows that the superset algorithm is more advantageous than Polak's algorithm when the number of robust constraints is large.