Local Linear Convergence for Alternating and Averaged Nonconvex Projections

Abstract

The idea of a finite collection of closed sets having "linearly regular intersection" at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness for example), we prove that von Neumann's method of "alternating projections" converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of "averaged projections" converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly.

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

“…As we shall see, our abstract framework (Theorem 2.9) allows to recover some of the results of [1]. In order to illustrate the versatility of our algorithmic framework, we also consider a fairly general semi-algebraic feasibility problem, and we provide, in the line of [42], a local convergence proof for an inexact averaged projection method.…”

“…Such examples are abundantly commented in [5], and they strongly motivate the present study. Other types of examples based on more analytical assumptions like uniform convexity, transversality or metric regularity can be found in [5], inequality (8.7) of [42], and Remark 3.6.…”

“…quadratic functions, polynomial functions, real analytic functions, but it can also be captured by adequate analytic assumptions, e.g. metric regularity [2,41,42], cohypomonotonicity [51,36], selfconcordance [49], partial smoothness [40,59]. In this paper, our central assumption for the study of such algorithms is that the function f satisfies the (nonsmooth) Kurdyka-Lojasiewicz inequality, which means, roughly speaking, that the functions under consideration are sharp up to a reparametrization (see Section 2.2).…”

“…We do not give a precise definition of definability in this work, but the flexibility of this concept is briefly illustrated in Example 5.4(b). Functions that are not necessarily tame but that satisfy Lojasiewicz inequality are given in [5], basic assumptions involve metric-regularity and transversality (see also [41,42] and Example 5.5).…”

Generalized Newton’s method based on graphical derivatives

“…Note metric regularity and related concepts of variational analysis has been employed in the analysis and justification of numerical algorithms starting with Robinson's seminal contribution; see, e.g., [1,18,25] and their references for the recent account. However, we are not familiar with any usage of graphical derivatives and coderivatives for these purposes.…”