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Variable aggregation for nonlinear optimization problems

Sakshi Naik, Lorenz Biegler, Russell Bent, Robert Parker

TL;DR

The paper investigates automatic variable aggregation as a nonlinear presolve to produce reduced-space formulations for constrained nonlinear programs. It introduces an approximate-maximum aggregation approach and several structure-preserving variants, analyzes their structural effects, solve-time implications, and convergence reliability across four challenging test problems. Key findings show that aggregation generally improves convergence reliability and can reduce solve time when the structure is preserved, but aggressive aggregation can inflate problem density and Hessian costs, potentially harming performance. The work provides practical guidance for implementing pre-solve reductions in nonlinear optimization environments and lays the groundwork for theoretical and empirical analysis of solver-aggregation interactions.

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.

Variable aggregation for nonlinear optimization problems

TL;DR

The paper investigates automatic variable aggregation as a nonlinear presolve to produce reduced-space formulations for constrained nonlinear programs. It introduces an approximate-maximum aggregation approach and several structure-preserving variants, analyzes their structural effects, solve-time implications, and convergence reliability across four challenging test problems. Key findings show that aggregation generally improves convergence reliability and can reduce solve time when the structure is preserved, but aggressive aggregation can inflate problem density and Hessian costs, potentially harming performance. The work provides practical guidance for implementing pre-solve reductions in nonlinear optimization environments and lays the groundwork for theoretical and empirical analysis of solver-aggregation interactions.

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.

Paper Structure

This paper contains 26 sections, 3 theorems, 15 equations, 12 figures, 7 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Nonlinear optimization
  4. Algebraic variables and constraints
  5. Bipartite graph of variables and constraints
  6. Variable aggregation
  7. Aggregation methods
  8. Approximate maximum elimination
  9. Greedy heuristic
  10. Matching-based algorithm
  11. Fixed-variable aggregation
  12. Structure-preserving aggregation
  13. Incidence-preserving aggregation
  14. Linearity-preserving elimination
  15. Equal coefficient degree-2 aggregation
  16. ...and 11 more sections

Key Result

lemma thmcounterlemma

Let be a subset of constraints in Equation eqn:reduced-nlopt, where $\nabla_v \tilde{g}^{\mathrm{def},T}$ is strictly lower triangular. Then $\nabla_v \tilde{g}^T$ is nonsingular everywhere and the subsets of variables and constraints $v$ and $\tilde{g}$ may be eliminated to form a reduced-space optimiz

Figures (12)

  • Figure 1: Algebraic expression tree for the expression $3x^2 -1$
  • Figure 2: Bipartite graph and incidence matrix corresponding to variables and equations with a maximum matching highlighted.
  • Figure 3: Block triangular form of an incidence matrix. Here, $T_1,\dots,T_{n_b}$ are the submatrices corresponding to subgraphs induced by subsets in the block triangular partition.
  • Figure 4: Illustration of the linear matching algorithm for the given equation system. Middle: A maximum matching of linear variable-constraint pairs ($\mathcal{M}_L$ in Algorithm \ref{['alg:matching']}). Right: A subset of the matching with a lower-triangular incidence matrix ($\mathcal{M}_T$ in Algorithm \ref{['alg:matching']}).
  • Figure 5: Percent of variables eliminated by each method for each model. Methods are in the same order as presented in Table \ref{['tab:structure']}
  • ...and 7 more figures

Theorems & Definitions (6)

  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof
  • theorem 2
  • proof