On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization

• Published in: Journal of global optimization Vol. 84; no. 3; pp. 527 - 561
• Main Authors: Amaral, V. S., Andreani, R., Birgin, E. G., Marcondes, D. S., Martínez, J. M.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applied mathematics; Complexity; Computer Science; Convergence; Global optimization; Mathematics; Mathematics and Statistics; Multidimensional methods; Neighborhoods; Operations Research/Decision Theory; Optimization; Ordinary differential equations; Polynomials; Real Functions; Regularization methods; Variables; 68Q25; Bound-constrained minimization; 90C30; 65K05; Coordinate descent methods; 49M37; 90C60; Worst-case evaluation complexity
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-022-01168-6

Coordinate descent methods have considerable impact in global optimization because global (or, at least, almost global) minimization is affordable for low-dimensional problems. Coordinate descent methods with high-order regularized models for smooth nonconvex box-constrained minimization are introduced in this work. High-order stationarity asymptotic convergence and first-order stationarity worst-case evaluation complexity bounds are established. The computer work that is necessary for obtaining first-order ε -stationarity with respect to the variables of each coordinate-descent block is O ( ε - ( p + 1 ) / p ) whereas the computer work for getting first-order ε -stationarity with respect to all the variables simultaneously is O ( ε - ( p + 1 ) ) . Numerical examples involving multidimensional scaling problems are presented. The numerical performance of the methods is enhanced by means of coordinate-descent strategies for choosing initial points.