scholarly journals Nonsmooth Problems of Calculus of VariationsviaCodifferentiation

2014 ◽  
Vol 20 (4) ◽  
pp. 1153-1180 ◽  
Author(s):  
Maxim Dolgopolik
Keyword(s):  
Nonsmooth Problems

Theory of Boundary Eigensolutions in Engineering Mechanics

2000 ◽  
Vol 68 (1) ◽  
pp. 101-108 ◽  
Author(s):  
A. R. Hadjesfandiari ◽  
G. F. Dargush
Keyword(s):  
Weight Function ◽  
Mixed Boundary ◽  
Traditional Approach ◽  
Mixed Boundary Conditions ◽  
Boundary Element Methods ◽  
Positive Weight ◽  
Nonsmooth Problems ◽  
Integral Equation Methods ◽  
Potential Problems ◽  
Engineering Mechanics

A theory of boundary eigensolutions is presented for boundary value problems in engineering mechanics. While the theory is quite general, the presentation here is restricted to potential problems. Contrary to the traditional approach, the eigenproblem is formed by inserting the eigenparameter, along with a positive weight function, into the boundary condition. The resulting spectra are real and the eigenfunctions are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of nonsmooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the convergence characteristics are also introduced. Furthermore, the connection is made to integral equation methods and variational methods. This paves the way toward the development of new computational formulations for finite element and boundary element methods. Two numerical examples are included to illustrate the applicability.

scholarly journals A CARTOPT METHOD FOR BOUND-CONSTRAINED GLOBAL OPTIMIZATION

2013 ◽  
Vol 55 (2) ◽  
pp. 109-128 ◽  
Author(s):  
B. L. ROBERTSON ◽  
C. J. PRICE ◽  
M. REALE
Keyword(s):  
Global Optimization ◽  
Random Sampling ◽  
Global Minimizer ◽  
Stochastic Algorithm ◽  
Constrained Global Optimization ◽  
Objective Functions ◽  
Search Region ◽  
Mild Conditions ◽  
Nonsmooth Problems ◽  
Classification And Regression

AbstractA stochastic algorithm for bound-constrained global optimization is described. The method can be applied to objective functions that are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines a region where the objective function is relatively low. Further points are drawn directly from the low region before a new partition is formed. Alternating between partition and sampling phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to an essential global minimizer with probability one under mild conditions. Nonprobabilistic results are also given when random sampling is replaced with points taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice.

Spectral methods for nonsmooth problems

2009 ◽  
pp. 153-186
Author(s):  
Jan S. Hesthaven ◽  
Sigal Gottlieb ◽  
David Gottlieb
Keyword(s):  
Spectral Methods ◽  
Nonsmooth Problems

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.

Block coordinate descentwith time perturbation for nonconvex nonsmooth problems in real-world studies

2019 ◽  
Vol 20 (10) ◽  
pp. 1390-1403
Author(s):  
Rui Liu ◽  
Wei-Chu Sun ◽  
Tao Hou ◽  
Chun-Hong Hu ◽  
Lin-Bo Qiao
Keyword(s):  
Real World ◽  
Nonsmooth Problems

scholarly journals Intelligent Optimization Algorithms: A Stochastic Closed-Loop Supply Chain Network Problem Involving Oligopolistic Competition for Multiproducts and Their Product Flow Routings

2015 ◽  
Vol 2015 ◽  
pp. 1-22 ◽  
Author(s):  
Yan Zhou ◽  
Chi Kin Chan ◽  
Kar Hung Wong ◽  
Y. C. E. Lee
Keyword(s):  
Supply Chain ◽  
Nash Equilibrium ◽  
Closed Loop ◽  
Real Life ◽  
Supply Chain Network ◽  
Oligopolistic Competition ◽  
Objective Functions ◽  
Closed Loop Supply Chain ◽  
Product Flow ◽  
Nonsmooth Problems

Recently, the first oligopolistic competition model of the closed-loop supply chain network involving uncertain demand and return has been established. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framework. In this paper, we modify the above model in two different directions. (i) For each returned product from demand market to firm in the reverse logistics, we calculate the percentage of its optimal product flows in each individual path connecting the demand market to the firm. This modification provides the optimal product flow routings for each product in the supply chain and increases the optimal profit of each firm at the Cournot-Nash equilibrium. (ii) Our model extends the method of finding the Cournot-Nash equilibrium involving smooth objective functions to problems involving nondifferentiable objective functions. This modification caters for more real-life applications as a lot of supply chain problems involve nonsmooth functions. Existence of the Cournot-Nash equilibrium is established without the assumption of differentiability of the given functions. Intelligent algorithms, such as the particle swarm optimization algorithm and the genetic algorithm, are applied to find the Cournot-Nash equilibrium for such nonsmooth problems. Numerical examples are solved to illustrate the efficiency of these algorithms.

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