2015/11/05 by Hui Zhang, Zhang, Hui · 5 citations
Computer Science · Engineering · Mathematics · #Advanced Optimization Algorithms Research #Affine transformation #Applied mathematics #Computer science #Convex analysis #Convex function #Convex optimization #Convexity #Differentiable function #Equivalence (formal languages) #FOS: Mathematics #Geometry #Gradient descent #Mathematical optimization #Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Pseudoconvex function #Pure mathematics #Quadratic equation #Regular polygon #Sparse and Compressive Sensing Techniques #Subderivative #Variational analysis #math.OC
paper · pdf · doi:10.48550/arxiv.1511.01635
published in arXiv (Cornell University) (Cornell University) · 15 pages; accepted in Optimization Letter
openalex publication_date 2015/11/05 · arxiv created 2016/06/19 · arxiv updated 2016/06/21 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
The restricted strong convexity is an effective tool for deriving globally linear convergence rates of descent methods in convex minimization. Recently, the global error bound and quadratic growth properties appeared as new competitors. In this paper, with the help of Ekeland's variational principle, we show the equivalence between these three notions. To deal with convex minimization over a closed convex set and structured convex optimization, we propose a group of modified versions and a group of extended versions of these three notions by using gradient mapping and proximal gradient mapping separately, and prove that the equivalence for the modified and extended versions still holds. Based on these equivalence notions, we establish new asymptotically linear convergence results for the proximal gradient method. Finally, we revisit the problem of minimizing the composition of an affine mapping with a strongly convex differentiable function over a polyhedral set, and obtain a strengthened property of the restricted strong convex type under mild assumptions.