On Linear Convergence of Non-Euclidean Gradient Methods without Strong Convexity and Lipschitz Gradient Continuity

• Published in: Journal of optimization theory and applications Vol. 182; no. 3; pp. 1068 - 1087
• Main Authors: Bauschke, Heinz H., Bolte, Jérôme, Chen, Jiawei, Teboulle, Marc, Wang, Xianfu
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Convergence; Convexity; Engineering; Inequalities; Mathematics; Mathematics and Statistics; Nonlinear programming; Operations Research/Decision Theory; Optimization; Theory of Computation; 65K05; Nonconvex minimization; 90C26; Bregman distance; Non-Euclidean gradient methods; Gradient dominated inequality; 49M10; Descent lemma without Lipschitz gradient; 90C30; Linear rate of convergence; Lipschitz-like convexity condition; 65K10; Łojasiewicz gradient inequality
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-019-01516-9

The gradient method is well known to globally converge linearly when the objective function is strongly convex and admits a Lipschitz continuous gradient. In many applications, both assumptions are often too stringent, precluding the use of gradient methods. In the early 1960s, after the amazing breakthrough of Łojasiewicz on gradient inequalities, it was observed that uniform convexity assumptions could be relaxed and replaced by these inequalities. On the other hand, very recently, it has been shown that the Lipschitz gradient continuity can be lifted and replaced by a class of functions satisfying a non-Euclidean descent property expressed in terms of a Bregman distance. In this note, we combine these two ideas to introduce a class of non-Euclidean gradient-like inequalities, allowing to prove linear convergence of a Bregman gradient method for nonconvex minimization, even when neither strong convexity nor Lipschitz gradient continuity holds.