An extension of the proximal point algorithm beyond convexity

Semicontinuous Function ◽

Proximal Point ◽

Firmly Nonexpansive ◽

Difference Of Convex ◽

The One

AbstractWe introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.

Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

Axioms ◽

10.3390/axioms11010021 ◽

2022 ◽

Vol 11 (1) ◽

pp. 21

Author(s):

Yasunori Kimura ◽

Keisuke Shindo

Keyword(s):

Asymptotic Behavior ◽

Convex Functions ◽

Lower Semicontinuous ◽

Convergent Sequence ◽

Semicontinuous Function ◽

Geodesic Space ◽

Proper Convex ◽

Convex Lower Semicontinuous Function ◽

Sequence Of Functions

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problems

Computational Optimization and Applications ◽

10.1007/s10589-019-00139-0 ◽

2019 ◽

Vol 75 (1) ◽

pp. 263-290 ◽

Cited By ~ 1

Author(s):

Glaydston de Carvalho Bento ◽

Sandro Dimy Barbosa Bitar ◽

João Xavier da Cruz Neto ◽

Antoine Soubeyran ◽

João Carlos de Oliveira Souza

Keyword(s):

Convex Functions ◽

Point Method ◽

Proximal Point Method ◽

Dynamic Problems ◽

Multi Objective Optimization ◽

Group Dynamic ◽

Proximal Point ◽

Multi Objective ◽

A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2017020 ◽

2018 ◽

Vol 24 (2) ◽

pp. 463-477 ◽

Cited By ~ 2

Author(s):

Radu Ioan Boţ ◽

Ernö Robert Csetnek

Keyword(s):

Dynamical System ◽

Smooth Function ◽

Convergence Rates ◽

Lower Semicontinuous ◽

Semicontinuous Function ◽

Proximal Point ◽

Nonconvex Function ◽

Proper Convex ◽

Point Operator

We address the minimization of the sum of a proper, convex and lower semicontinuous function with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The trajectory generated by the dynamical system is proved to asymptotically converge to a critical point of the objective, provided a regularization of the latter satisfies the Kurdyka−Łojasiewicz property. Convergence rates for the trajectory in terms of the Łojasiewicz exponent of the regularized objective function are also provided.

The Proximal Subgradient Formula in Banach Space

Canadian Mathematical Bulletin ◽

10.4153/cmb-1988-051-9 ◽

1988 ◽

Vol 31 (3) ◽

pp. 353-361 ◽

Cited By ~ 12

Author(s):

Philip D. Loewen

Keyword(s):

Banach Space ◽

Lower Semicontinuous ◽

Semicontinuous Function ◽

Generalized Gradient ◽

Infinite Dimensional ◽

Fréchet Differentiable ◽

The One ◽

The Given ◽

Normal Formula

AbstractThe proximal subgradient formula is a refinement due to Rockafellar of Clarke's fundamental proximal normal formula. It expresses Clarke's generalized gradient of a lower semicontinuous function in terms of analytically simpler proximal subgradients. We use the infinite-dimensional proximal normal formula recently given by Borwein and Strojwas to derive a new version of the proximal subgradient formula in a reflexive Banach space X with Frechet differentiable and locally uniformly convex norm. Our result improves on the one given by Borwein and Strojwas by referring only to the given norm on X.

The Boosted Difference of Convex Functions Algorithm for Nonsmooth Functions

SIAM Journal on Optimization ◽

10.1137/18m123339x ◽

2020 ◽

Vol 30 (1) ◽

pp. 980-1006 ◽

Cited By ~ 3

Author(s):

Francisco J. Aragón Artacho ◽

Phan T. Vuong

Keyword(s):

Convex Functions ◽

Nonsmooth Functions ◽

A Characterization of Proximal Subgradient Set-Valued Mappings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1993-017-4 ◽

1993 ◽

Vol 36 (1) ◽

pp. 116-122 ◽

Cited By ~ 9

Author(s):

R. A. Poliquin

Keyword(s):

Lower Semicontinuous ◽

Semicontinuous Function ◽

Set Valued Mapping ◽

Selection Property ◽

Set Valued Mappings ◽

Bounded Below

AbstractIn this paper we tackle the problem of identifying set-valued mappings that are subgradient set-valued mappings. We show that a set-valued mapping is the proximal subgradient mapping of a lower semicontinuous function bounded below by a quadratic if and only if it satisfies a monotone selection property.

A Difference of Convex Functions Approach to Large-Scale Log-Linear Model Estimation

IEEE Transactions on Audio Speech and Language Processing ◽

10.1109/tasl.2013.2271592 ◽

2013 ◽

Vol 21 (11) ◽

pp. 2255-2266 ◽

Cited By ~ 2

Author(s):

Theodoros Tsiligkaridis ◽

Etienne Marcheret ◽

Vaibhava Goel

Keyword(s):

Linear Model ◽

Convex Functions ◽

Large Scale ◽

Model Estimation ◽

Difference Of Convex ◽

Log Linear

A Difference of Convex Functions Algorithm for Switched Linear Regression

IEEE Transactions on Automatic Control ◽

10.1109/tac.2014.2301575 ◽

2014 ◽

Vol 59 (8) ◽

pp. 2277-2282 ◽

Cited By ~ 19

Author(s):

Tao Pham Dinh ◽

Hoai Minh Le ◽

Hoai An Le Thi ◽

Fabien Lauer

Keyword(s):

Linear Regression ◽

Convex Functions ◽

Difference of convex functions algorithms (DCA) for image restoration via a Markov random field model

Optimization and Engineering ◽

10.1007/s11081-017-9359-0 ◽

2017 ◽

Vol 18 (4) ◽

pp. 873-906 ◽

Cited By ~ 6

Author(s):

Hoai An Le Thi ◽

Tao Pham Dinh

Keyword(s):

Random Field ◽

Image Restoration ◽

Markov Random Field ◽

Convex Functions ◽

Field Model ◽

Random Field Model ◽

Markov Random Field Model ◽

Markov Random ◽

Smooth variational principles in Radon-Nikodým spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700033372 ◽

1999 ◽

Vol 60 (1) ◽

pp. 109-118 ◽

Cited By ~ 7

Author(s):

Robert Deville ◽

Abdelhakim Maaden

Keyword(s):

Banach Space ◽

Real Function ◽

Variational Principles ◽

Lower Semicontinuous ◽

Semicontinuous Function ◽

Weakly Continuous ◽

Fréchet Differentiable

We prove that if f is a real valued lower semicontinuous function on a Banach space X, for which there exist a > 0 and b ∈ ℝ such that f(x) ≥ 2a∥x∥ + b, x ∈ X, and if X has the Radon-Nikody´m property, then for every Ε > 0 there exists a real function φ X such that φ is Fréchet differentiable, ∥φ∥∞ < Ε, ∥φ′∥∞ < Ε, φ′ is weakly continuous and f + φ attains a minimum on X. In addition, if we assume that the norm in X is β-smooth, we can take the function φ = g1 + g2 where g1 is radial and β-smooth, g2 is Fréchet differentiable, ∥g1∥∞ < Ε, ∥g2∥∞ < Ε, ∥g′1∥∞ < Ε, ∥g′1∥∞ < Ε, g′2 is weakly continuous and f + g1 + g2 attains a minimum on X.