A Proximal Bundle Method with Exact Penalty Technique and Bundle Modification Strategy for Nonconvex Nonsmooth Constrained Optimization

2021 ◽  
pp. 2150015
Author(s):  
Xiaoliang Wang ◽  
Liping Pang ◽  
Qi Wu
Keyword(s):  
Operation Research ◽  
Bundle Method ◽  
Exact Penalty ◽  
Constraint Function ◽  
Modified Model ◽  
Proximal Bundle Method ◽  
Unconstrained Problem ◽  
Unconstrained Problems ◽  
Penalty Strategy ◽  
Penalty Technique

The bundle modification strategy for the convex unconstrained problems was proposed by Alexey et al. [[2007] European Journal of Operation Research, 180(1), 38–47.] whose most interesting feature was the reduction of the calls for the quadratic programming solver. In this paper, we extend the bundle modification strategy to a class of nonconvex nonsmooth constraint problems. Concretely, we adopt the convexification technique to the objective function and constraint function, take the penalty strategy to transfer the modified model into an unconstrained optimization and focus on the unconstrained problem with proximal bundle method and the bundle modification strategies. The global convergence of the corresponding algorithm is proved. The primal numerical results show that the proposed algorithms are promising and effective.

scholarly journals An Approximate Proximal Bundle Method to Minimize a Class of Maximum Eigenvalue Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Wei Wang ◽  
Lingling Zhang ◽  
Miao Chen ◽  
Sida Lin
Keyword(s):  
Objective Function ◽  
Optimal Solution ◽  
Cutting Plane ◽  
Bundle Method ◽  
Maximum Eigenvalue ◽  
Plane Model ◽  
Essential Idea ◽  
Proximal Bundle Method ◽  
The Difference ◽  
And Function

We present an approximate nonsmooth algorithm to solve a minimization problem, in which the objective function is the sum of a maximum eigenvalue function of matrices and a convex function. The essential idea to solve the optimization problem in this paper is similar to the thought of proximal bundle method, but the difference is that we choose approximate subgradient and function value to construct approximate cutting-plane model to solve the above mentioned problem. An important advantage of the approximate cutting-plane model for objective function is that it is more stable than cutting-plane model. In addition, the approximate proximal bundle method algorithm can be given. Furthermore, the sequences generated by the algorithm converge to the optimal solution of the original problem.

scholarly journals AUV-Method for a Class of Constrained Minimized Problems of Maximum Eigenvalue Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Wei Wang ◽  
Ming Jin ◽  
Shanghua Li ◽  
Xinyu Cao
Keyword(s):  
Exact Penalty ◽  
Maximum Eigenvalue ◽  
Convex Approximation ◽  
Equivalent Problem ◽  
Composite Function ◽  
Primal Problem ◽  
Approximate Problem ◽  
Constrained Problem ◽  
Constrained Minimization Problem ◽  
Unconstrained Problem

In this paper, we apply theUV-algorithm to solve the constrained minimization problem of a maximum eigenvalue function which is the composite function of an affine matrix-valued mapping and its maximum eigenvalue. Here, we convert the constrained problem into its equivalent unconstrained problem by the exact penalty function. However, the equivalent problem involves the sum of two nonsmooth functions, which makes it difficult to applyUV-algorithm to get the solution of the problem. Hence, our strategy first applies the smooth convex approximation of maximum eigenvalue function to get the approximate problem of the equivalent problem. Then the approximate problem, the space decomposition, and theU-Lagrangian of the object function at a given point will be addressed particularly. Finally, theUV-algorithm will be presented to get the approximate solution of the primal problem by solving the approximate problem.

scholarly journals An analytical and numerical approach to a bilateral contact problem with nonmonotone friction

2013 ◽  
Vol 23 (2) ◽  
pp. 263-276 ◽  
Author(s):  
Mikaël Barboteu ◽  
Krzysztof Bartosz ◽  
Piotr Kalita
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Numerical Approach ◽  
Bundle Method ◽  
Nonconvex Problem ◽  
Proximal Bundle Method ◽  
Loss Of Contact ◽  
The Galerkin Method ◽  
Bilateral Contact ◽  
Numerical Approaches

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.

A Redistributed Proximal Bundle Method for Nonconvex Optimization

2010 ◽  
Vol 20 (5) ◽  
pp. 2442-2473 ◽  
Author(s):  
Warren Hare ◽  
Claudia Sagastizábal
Keyword(s):  
Nonconvex Optimization ◽  
Bundle Method ◽  
Proximal Bundle Method

scholarly journals A Nonmonotone Proximal Bundle Method with (Potentially) Continuous Step Decisions

2013 ◽  
Vol 23 (3) ◽  
pp. 1784-1809 ◽  
Author(s):  
A. Astorino ◽  
A. Frangioni ◽  
A. Fuduli ◽  
E. Gorgone
Keyword(s):  
Bundle Method ◽  
Proximal Bundle Method
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