scholarly journals A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jie Shen ◽  
Dan Li ◽  
Li-Ping Pang
Keyword(s):  
Objective Function ◽  
Cutting Plane ◽  
Bundle Method ◽  
Approximation Model ◽  
Nonconvex Functions ◽  
Proximal Bundle Methods ◽  
New Variant ◽  
Optimal Value ◽  
Inexact Data ◽  
Optimality Gap

Under the condition that the values of the objective function and its subgradient are computed approximately, we introduce a cutting plane and level bundle method for minimizing nonsmooth nonconvex functions by combining cutting plane method with the ideas of proximity control and level constraint. The proposed algorithm is based on the construction of both a lower and an upper polyhedral approximation model to the objective function and calculates new iteration points by solving a subproblem in which the model is employed not only in the objective function but also in the constraints. Compared with other proximal bundle methods, the new variant updates the lower bound of the optimal value, providing an additional useful stopping test based on the optimality gap. Another merit is that our algorithm makes a distinction between affine pieces that exhibit a convex or a concave behavior relative to the current iterate. Convergence to some kind of stationarity point is proved under some looser conditions.

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 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.

scholarly journals An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

2021 ◽  
Author(s):  
E. Alper Yıldırım
Keyword(s):  
Objective Function ◽  
Convex Relaxation ◽  
Large Family ◽  
Quadratic Program ◽  
Feasible Region ◽  
Recession Cone ◽  
Convex Relaxations ◽  
Quadratic Programs ◽  
Alternative Perspective ◽  
Optimal Value

AbstractWe study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

New Approach for the Continuing Dynamic Programming

2012 ◽  
Vol 459 ◽  
pp. 575-578
Author(s):  
Peng Zhang ◽  
Xiang Huan Meng
Keyword(s):  
Dynamic Programming ◽  
Objective Function ◽  
Iteration Method ◽  
Programming Model ◽  
State Equations ◽  
New Approach ◽  
Optimal Value ◽  
Convergent Method ◽  
Optimal Values ◽  
Dynamic Programming Model

The paper proposes the discrete approximate iteration method to solve single-dimensional continuing dynamic programming model. The paper also presents a comparison of the discrete approximate iteration method and bi- convergent method to solve multi-dimensional continuing dynamic programming model. The algorithm is the following: Firstly, let state value of one of state equations be unknown and the others be known. Secondly, use discrete approximate iteration method to find the optimal value of the unknown state values, continue iterating until all state equations have found optimal values. If the objective function is convex, the algorithm is proved linear convergent. If the objective function is non-concave and non-convex, the algorithm is proved convergent.

Statistical Inference of Second-Order Cone Programming

2018 ◽  
Vol 35 (06) ◽  
pp. 1850044
Author(s):  
Jiani Wang ◽  
Liwei Zhang
Keyword(s):  
Objective Function ◽  
Second Order ◽  
Optimal Solutions ◽  
Random Vectors ◽  
Cone Programming ◽  
Second Order Cone Programming ◽  
Second Order Cone ◽  
Analysis Theory ◽  
Optimal Value ◽  
Sum Of Norms

The randomness of the second-order cone programming problems is mainly reflected in the objective function and the constraints both having random vectors. In this paper, we discuss the statistical properties of estimates of the respective optimal value and optimal solutions when the random vectors are estimated by their sample both in the objective function and the constraints, which are based on perturbation analysis theory of second-order cone programming. As an example we consider the problem of minimizing a sum of norms with weights.

Image denoising algorithm of compressed sensing based on alternating direction method of multipliers

2021 ◽  
pp. 2250009
Author(s):  
Changjie Fang ◽  
Jingyu Chen ◽  
Shenglan Chen
Keyword(s):  
Compressed Sensing ◽  
Convergence Rate ◽  
Objective Function ◽  
Image Denoising ◽  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Alternating Direction ◽  
Time Simulation ◽  
Optimal Value ◽  
Simulation Results

In this paper, we propose an image denoising algorithm for compressed sensing based on alternating direction method of multipliers (ADMM). We prove that the objective function of the iterates approaches the optimal value. We also prove the [Formula: see text] convergence rate of our algorithm in the ergodic sense. At the same time, simulation results show that our algorithm is more efficient in image denoising compared with existing methods.

scholarly journals PENERAPAN BRANCH AND BOUND ALGORITHM DALAM OPTIMALISASI PRODUKSI ROTI

2016 ◽  
Vol 5 (4) ◽  
pp. 148
Author(s):  
GEDE SURYAWAN ◽  
NI KETUT TARI TASTRAWATI ◽  
KARTIKA SARI
Keyword(s):  
Branch And Bound ◽  
Cutting Plane ◽  
Optimal Combination ◽  
Branch And Bound Algorithm ◽  
Cutting Plane Method ◽  
Plane Method ◽  
Optimal Value ◽  
Bread Production

Companies which engaged in production activities such as Ramadhan Bakery would want optimal profit in their every production. The aim of this study was to find optimal profit and optimal combination of bread production (original chocolate bread, extra chocolate bread, rounding chocolate bread and mattress chocolate bread) that was produced by Ramadhan Bakery by applying Branch and Bound Algorithm method. Branch and Bound Algorithm is one method to solve Integer Programming’s problems other than Cutting Plane method. Compared with Cutting Plane method, Branch and Bound Algorithm method is more effective in determining the optimal value. As the result of this study showed that to get optimal profit, Ramadhan Bakery should produce 360 pcs of original chocolate bread, 300 pcs of extra chocolate bread, 306 pcs of rounding chocolate bread and 129 pcs of mattress chocolate bread with optimal profit amounts Rp. 1.195.624,00.. The profit will increase amounts 25,2 % than before.

Sign in / Sign up

Export Citation Format

Share Document