article · Electronic Research Archive
A matrix-free adaptation of the Davidon-Fletcher-Powell (DFP) optimisation method has been developed for unconstrained optimisation challenges, targeting compressive sensing and image restoration. The technique introduces a novel search direction that integrates a scaling parameter, guaranteeing that the sufficient descent condition is met regardless of line search conditions. Theoretical convergence analysis confirms both the boundedness and mathematical validity of the approach. When evaluated across benchmark unconstrained optimisation and compressive sensing problems, the algorithm demonstrated high efficiency and robustness. In image restoration evaluations, the method achieved a 100 per cent success rate, outperforming established algorithms such as CG-DESCENT, MDL, and NSMA, which attained rates of 95.8, 84.5, and 53.5 per cent, respectively. Comparative measurements of computational time, relative error, and peak signal-to-noise ratio further demonstrate its viability for large-scale optimisation tasks.
Reconstructing high-quality images from limited data is a core mathematical challenge in modern signal processing. Many existing optimisation algorithms struggle with computational burden or fail to converge reliably on large-scale problems. By eliminating the need to store large matrices and guaranteeing stable descent, this algorithm provides a faster, more dependable tool for handling complex data reconstruction tasks.
The method could enhance image processing software and hardware in sectors like medical imaging or remote sensing that rely on compressive sensing. Target users include algorithm engineers and software developers handling large-scale computational reconstruction. Currently at an applied and tested stage in laboratory benchmarks, the algorithm demonstrates clear performance advantages but requires integration and testing in commercial software pipelines before near-market adoption.
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This paper introduces a matrix-free variant of the Davidon-Fletcher-Powell (DFP) method for unconstrained optimization problems with applications in compressive sensing and image restoration. The main contribution lies in the new search direction incorporating a scaling parameter that ensures the satisfaction of the sufficient descent condition, independent of the line search conditions. A rigorous convergence analysis guarantees the boundedness and theoretical validity of the proposed method. Comprehensive numerical experiments on benchmark unconstrained optimization test problems and compressive sensing problems demonstrate the efficiency and robustness of the algorithm. Specifically, in image restoration tasks, our method outperforms CG-DESCENT, MDL, and NSMA, achieving a 100% success rate compared to 95.8%, 84.5%, and 53.5%, respectively. Additionally, results on computational time, relative error, and PSNR confirm the superior performance of the proposed approach. These findings establish the proposed method as a competitive alternative for large-scale optimization problems.
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DOI: 10.3934/era.2025183
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