Variable aggregation for nonlinear optimization problems

TL;DR

This paper explores variable aggregation as a pre-solve technique for constrained nonlinear programs, demonstrating its potential to improve convergence and reduce solve time, while highlighting computational trade-offs.

Contribution

It formalizes variable aggregation for nonlinear optimization, introduces a novel approximate maximum aggregation strategy, and compares different aggregation approaches.

Findings

Aggregation improves convergence reliability.

Variable aggregation reduces total solve time.

Hessian evaluation can become a bottleneck.

Abstract

Variable aggregation has been largely studied as an important pre-solve algorithm for optimization of linear and mixed-integer programs. Although some nonlinear solvers and algebraic modeling languages implement variable aggregation as a pre-solve, the impact it can have on constrained nonlinear programs is unexplored. In this work, we formalize variable aggregation as a pre-solve algorithm to develop reduced-space formulations of nonlinear programs. A novel approximate maximum variable aggregation strategy is developed to aggregate as many variables as possible. Furthermore, aggregation strategies that preserve the problem structure are compared against approximate maximum aggregation. Our results show that variable aggregation can generally help to improve the convergence reliability of nonlinear programs. It can also help in reducing total solve time. However, Hessian evaluation can become a bottleneck if aggregation significantly increases the number of variables appearing nonlinearly in many constraints.