On inexact generalized proximal methods with a weakened error tolerance criterion

An inexact proximal method for quasiconvex minimization

“…Observe that the above conditions (1.5)-(1.6) have been used in convex proximal methods, see for example Auslender, Teboulle, and Ben-Tiba (1999), Kaplan and Tichatschke (2004), Eckstein (1998), Xu, Bingsheng, andXiaoming (2006), Solodov and Svaiter (2000). We also get to rid the assumption (1.6) for a class of induced proximal distances which includes Bregman distances given by the standard entropy kernel and all strongly convex Bregman functions.…”

Perturbation resilience of proximal gradient algorithm for composite objectives

“…The bounded perturbation resilience is also concluded. Equation (1.1) is in fact a kind of inexact iterative scheme, which was developed in many literatures over the past two decades for the reason that solving the problems is either expensive or impossible, see [11,13,19,23,24] and references therein for instance. Motivated by [10] and [22], we consider the non-smooth composite optimization problem of the form…”

Some Results About Proximal-Like Methods

“…fc=l fc=l Eckstein explains that the relations (28), (29) are easier to check than in other inexact schemes. At the same time, using an element q € Q{u^^^)'> which transforms (28) into an equality, we obviously get…”

“…Let us recall that B8 does not cause any complication for the choice of an appropriate Bregman function. Moreover, in [29] the validity of condition B8 is proved for entropy-like and logarithmic-quadratic distance functions which are not of Bregman type. We also show there that the GPM in form (28) with these distance functions converges under the weaker error criterion ^lle'^ll < oo, than in former papers [2,41].…”

“…But this corresponds to scheme (2) with Q^ = Q and Sk := Xfclk^^^ll-^^^ ^^ the additional assumption B8 on Bregman function, our convergence analysis in [29] ensures convergence of the iterates (28) exclusively under e^ll <oo.…”