Canonical dual methods for general box constrained nonconvex and nonsmooth optimization problems

2019 ◽  
Vol 19 (3) ◽  
pp. 765-777
Author(s):  
Huicheng C. Liu ◽  
Jianwei Gao ◽  
Jing Liu
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Dual Methods ◽  
Nonconvex And Nonsmooth Optimization

scholarly journals Canonical Duality for Box Constrained Nonconvex and Nonsmooth Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Jing Liu ◽  
Huicheng Liu
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Transformation Method ◽  
Constrained Optimization Problems ◽  
Canonical Duality Theory ◽  
Canonical Duality ◽  
Local Extrema ◽  
Triality Theory ◽  
Nonconvex And Nonsmooth Optimization ◽  
Global And Local

This paper presents an application of the canonical duality theory for box constrained nonconvex and nonsmooth optimization problems. By use of the canonical dual transformation method, which is developed recently, these very difficult constrained optimization problems inRncan be converted into the canonical dual problems, which can be solved by deterministic methods. The global and local extrema can be identified by the triality theory. Some examples are listed to illustrate the applications of the theory presented in the paper.

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

scholarly journals Asynchronous Delay-Aware Accelerated Proximal Coordinate Descent for Nonconvex Nonsmooth Problems

2019 ◽  
Vol 33 ◽  
pp. 1528-1535
Author(s):  
Ehsan Kazemi ◽  
Liqiang Wang
Keyword(s):  
Large Scale ◽  
Convergence Rates ◽  
Optimization Problems ◽  
Coordinate Descent ◽  
Performance Guarantee ◽  
Nonsmooth Problems ◽  
Descent Property ◽  
Large Scale Problems ◽  
Nonconvex And Nonsmooth Optimization ◽  
Sufficient Descent

Nonconvex and nonsmooth problems have recently attracted considerable attention in machine learning. However, developing efficient methods for the nonconvex and nonsmooth optimization problems with certain performance guarantee remains a challenge. Proximal coordinate descent (PCD) has been widely used for solving optimization problems, but the knowledge of PCD methods in the nonconvex setting is very limited. On the other hand, the asynchronous proximal coordinate descent (APCD) recently have received much attention in order to solve large-scale problems. However, the accelerated variants of APCD algorithms are rarely studied. In this paper, we extend APCD method to the accelerated algorithm (AAPCD) for nonsmooth and nonconvex problems that satisfies the sufficient descent property, by comparing between the function values at proximal update and a linear extrapolated point using a delay-aware momentum value. To the best of our knowledge, we are the first to provide stochastic and deterministic accelerated extension of APCD algorithms for general nonconvex and nonsmooth problems ensuring that for both bounded delays and unbounded delays every limit point is a critical point. By leveraging Kurdyka-Łojasiewicz property, we will show linear and sublinear convergence rates for the deterministic AAPCD with bounded delays. Numerical results demonstrate the practical efficiency of our algorithm in speed.

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