Convex Predictor–Nonconvex Corrector Optimization Strategy with Application to Signal Decomposition

• Published in: Journal of optimization theory and applications Vol. 202; no. 3; pp. 1286 - 1325
• Main Authors: Girometti, Laura, Huska, Martin, Lanza, Alessandro, Morigi, Serena
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2024; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Approximation; Calculus of Variations and Optimal Control; Optimization; Cost function; Engineering; Fines & penalties; Mathematics; Mathematics and Statistics; Numerical methods; Operations Research/Decision Theory; Optimization; Ordinary differential equations; Parameter identification; Predictor-corrector methods; Sparsity; Theory of Computation; 68U99; Nonconvex optimization; Predictor–corrector; 65K10; Multi-parameter selection; Signal decomposition; Cross-correlation
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-024-02479-2

Many tasks in real life scenarios can be naturally formulated as nonconvex optimization problems. Unfortunately, to date, the iterative numerical methods to find even only the local minima of these nonconvex cost functions are extremely slow and strongly affected by the initialization chosen. We devise a predictor–corrector strategy that efficiently computes locally optimal solutions to these problems. An initialization-free convex minimization allows to predict a global good preliminary candidate, which is then corrected by solving a parameter-free nonconvex minimization. A simple algorithm, such as alternating direction method of multipliers works surprisingly well in producing good solutions. This strategy is applied to the challenging problem of decomposing a 1D signal into semantically distinct components mathematically identified by smooth, piecewise-constant, oscillatory structured and unstructured (noise) parts.