scholarly journals An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 874
Author(s):  
Xiaoliang Wang ◽  
Liping Pang ◽  
Qi Wu ◽  
Mingkun Zhang
Keyword(s):  
Convex Function ◽  
Cutting Plane ◽  
Bundle Method ◽  
Composite Optimization ◽  
Nonconvex Function ◽  
Inexact Information ◽  
Proximal Bundle Method ◽  
Approximate Function ◽  
Linearization Errors ◽  
Dc Problems

In this paper, an adaptive proximal bundle method is proposed for a class of nonconvex and nonsmooth composite problems with inexact information. The composite problems are the sum of a finite convex function with inexact information and a nonconvex function. For the nonconvex function, we design the convexification technique and ensure the linearization errors of its augment function to be nonnegative. Then, the sum of the convex function and the augment function is regarded as an approximate function to the primal problem. For the approximate function, we adopt a disaggregate strategy and regard the sum of cutting plane models of the convex function and the augment function as a cutting plane model for the approximate function. Then, we give the adaptive nonconvex proximal bundle method. Meanwhile, for the convex function with inexact information, we utilize the noise management strategy and update the proximal parameter to reduce the influence of inexact information. The method can obtain an approximate solution. Two polynomial functions and six DC problems are referred to in the numerical experiment. The preliminary numerical results show that our algorithm is effective and reliable.

scholarly journals An Approximate Redistributed Proximal Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Jie Shen ◽  
Xiao-Qian Liu ◽  
Fang-Fang Guo ◽  
Shu-Xin Wang
Keyword(s):  
Stationary Point ◽  
Cutting Planes ◽  
Bundle Method ◽  
Nonconvex Functions ◽  
Nonconvex Function ◽  
Inexact Data ◽  
Proximal Bundle Method ◽  
Approximate Function ◽  
Local Convexification ◽  
Linearization Errors

We describe an extension of the redistributed technique form classical proximal bundle method to the inexact situation for minimizing nonsmooth nonconvex functions. The cutting-planes model we construct is not the approximation to the whole nonconvex function, but to the local convexification of the approximate objective function, and this kind of local convexification is modified dynamically in order to always yield nonnegative linearization errors. Since we only employ the approximate function values and approximate subgradients, theoretical convergence analysis shows that an approximate stationary point or some double approximate stationary point can be obtained under some mild conditions.

A Redistributed Bundle Algorithm Based on Local Convexification Models for Nonlinear Nonsmooth DC Programming

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jie Shen ◽  
Na Xu ◽  
Fang-Fang Guo ◽  
Han-Yang Li ◽  
Pan Hu
Keyword(s):  
Cutting Plane ◽  
Dc Programming ◽  
Bundle Method ◽  
Difference Of Convex Functions ◽  
Bundle Algorithm ◽  
Difference Of Convex ◽  
Proximal Bundle Method ◽  
Local Convexification ◽  
Original Objective ◽  
Linearization Errors

Abstract For nonlinear nonsmooth DC programming (difference of convex functions), we introduce a new redistributed proximal bundle method. The subgradient information of both the DC components is gathered from some neighbourhood of the current stability center and it is used to build separately an approximation for each component in the DC representation. Especially we employ the nonlinear redistributed technique to model the second component of DC function by constructing a local convexification cutting plane. The corresponding convexification parameter is adjusted dynamically and is taken sufficiently large to make the ”augmented” linearization errors nonnegative. Based on above techniques we obtain a new convex cutting plane model of the original objective function. Based on this new approximation the redistributed proximal bundle method is designed and the convergence of the proposed algorithm to a Clarke stationary point is proved. A simple numerical experiment is given to show the validity of the presented algorithm.

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.

A Redistributed Bundle Algorithm for Generalized Variational Inequality Problems in Hilbert Spaces

2018 ◽  
Vol 35 (04) ◽  
pp. 1850019
Author(s):  
Jie Shen ◽  
Ya-Li Gao ◽  
Fang-Fang Guo ◽  
Rui Zhao
Keyword(s):  
Variational Inequality ◽  
Auxiliary Problem ◽  
Zero Point ◽  
Weak Limit ◽  
Bundle Method ◽  
Generalized Variational Inequality ◽  
Nonconvex Function ◽  
Bundle Algorithm ◽  
Local Convexification ◽  
Linearization Errors

Based on the redistributed technique of bundle methods and the auxiliary problem principle, we present a redistributed bundle method for solving a generalized variational inequality problem which consists of finding a zero point of the sum of two multivalued operators. The considered problem involves a nonsmooth nonconvex function which is difficult to approximate by workable functions. By imitating the properties of lower-[Formula: see text] functions, we consider approximating the local convexification of the nonconvex function, and the local convexification parameter is modified dynamically in order to make the augmented function produce nonnegative linearization errors. The convergence of the proposed algorithm is discussed when the sequence of stepsizes converges to zero, any weak limit point of the sequence of serious steps [Formula: see text] is a solution of problem (P) under some conditions. The presented method is the generalization of the convex bundle method [Salmon, G, JJ Strodiot and VH Nguyen (2004). A bundle method for solving variational inequalities. SIAM Journal on Optimization, 14(3), 869–893].

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 Benders decomposition with adaptive oracles for large scale optimization

2020 ◽  
Author(s):  
Nicolò Mazzi ◽  
Andreas Grothey ◽  
Ken McKinnon ◽  
Nagisa Sugishita
Keyword(s):  
Large Scale ◽  
Benders Decomposition ◽  
Optimization Problems ◽  
Cutting Plane ◽  
Computational Effort ◽  
Planning Problem ◽  
Illustrative Case ◽  
Investment Planning ◽  
Cutting Plane Algorithms ◽  
Inexact Information

AbstractThis paper proposes an algorithm to efficiently solve large optimization problems which exhibit a column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. Similar problems are often tackled using cutting-plane algorithms, which allow for an iterative and decomposed solution of the problem. When solving subproblems is computationally expensive and the set of subproblems is large, cutting-plane algorithms may slow down severely. In this context we propose two novel adaptive oracles that yield inexact information, potentially much faster than solving the subproblem. The first adaptive oracle is used to generate inexact but valid cutting planes, and the second adaptive oracle gives a valid upper bound of the true optimal objective. These two oracles progressively “adapt” towards the true exact oracle if provided with an increasing number of exact solutions, stored throughout the iterations. These adaptive oracles are embedded within a Benders-type algorithm able to handle inexact information. We compare the Benders with adaptive oracles against a standard Benders algorithm on a stochastic investment planning problem. The proposed algorithm shows the capability to substantially reduce the computational effort to obtain an $$\epsilon $$ ϵ -optimal solution: an illustrative case is 31.9 times faster for a $$1.00\%$$ 1.00 % convergence tolerance and 15.4 times faster for a $$0.01\%$$ 0.01 % tolerance.

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