Journal of Global Optimization· 2026Q1
Relaxation via separable estimators: arithmetic and implementation
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- Q1SCImago
- 2026year
Short summary
A new 'superposition relaxation' arithmetic tightens bounds on factorable functions by using separable under- and overestimating functions, outperforming McCormick relaxations for neural networks but with higher computational cost.
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Key points
- Introduces 'superposition relaxation' arithmetic for bracketing factorable functions.
- Uses separable under- and overestimating functions for tighter bounds.
- Demonstrates superior tightness compared to McCormick relaxations, especially for ANNs.
- Notes higher computational cost as a drawback.
- Analyzes local convergence properties, including quadratic convergence propagation.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in the composition. The local convergence properties of this arithmetic are also analyzed in both the pointwise and Hausdorff sense, including conditions under which quadratic pointwise convergence propagates through composition. Parameterizations of the univariate summands in a superposition relaxation either as piecewise-constant or continuous piecewise-linear functions are discussed for a practical implementation. It is shown through numerical case studies that superposition relaxations can be consistently tighter than McCormick relaxations, including for the relaxation of artificial neural networks. But superposition relaxations also incur a higher computational cost than McCormick relaxations. Further investigations are thus warranted as applications in global optimization seek to balance a relaxation’s tightness with its computational cost.
The authors' abstract, as published at the source. Journal of Global Optimization, 2026 · DOI ↗
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Field: Numerical Analysis
Numerical AnalysisMathematics