Canonical duality for solving nonconvex and nonsmooth optimization problem

2008 ◽  
Vol 10 (2) ◽  
pp. 153-165 ◽  
Author(s):  
Jing Liu ◽  
David Y. Gao ◽  
Yan Gao
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problem ◽  
Canonical Duality ◽  
Nonconvex And Nonsmooth Optimization ◽  
Nonsmooth Optimization Problem

scholarly journals An Infeasible Incremental Bundle Method for Nonsmooth Optimization Problem Based on CVaR Portfolio

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jia-Tong Li ◽  
Jie Shen ◽  
Na Xu
Keyword(s):  
Objective Function ◽  
Nonsmooth Optimization ◽  
Value At Risk ◽  
Optimization Problem ◽  
Main Idea ◽  
Approximate Model ◽  
Bundle Method ◽  
Conditional Value At Risk ◽  
Incremental Method ◽  
Nonsmooth Optimization Problem

For CVaR (conditional value-at-risk) portfolio nonsmooth optimization problem, we propose an infeasible incremental bundle method on the basis of the improvement function and the main idea of incremental method for solving convex finite min-max problems. The presented algorithm only employs the information of the objective function and one component function of constraint functions to form the approximate model for improvement function. By introducing the aggregate technique, we keep the information of previous iterate points that may be deleted from bundle to overcome the difficulty of numerical computation and storage. Our algorithm does not enforce the feasibility of iterate points and the monotonicity of objective function, and the global convergence of the algorithm is established under mild conditions. Compared with the available results, our method loosens the requirements of computing the whole constraint function, which makes the algorithm easier to implement.

scholarly journals The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Miao Chen ◽  
Shou-qiang Du
Keyword(s):  
Nonsmooth Optimization ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Optimization Problem ◽  
Optimization Problems ◽  
L1 Norm ◽  
Engineering Technology ◽  
The Given ◽  
Nonsmooth Optimization Problem

We study the method for solving a kind of nonsmooth optimization problems with l1-norm, which is widely used in the problem of compressed sensing, image processing, and some related optimization problems with wide application background in engineering technology. Transformated by the absolute value equations, this kind of nonsmooth optimization problem is rewritten as a general unconstrained optimization problem, and the transformed problem is solved by a smoothing FR conjugate gradient method. Finally, the numerical experiments show the effectiveness of the given smoothing FR conjugate gradient method.

scholarly journals Pessimistic Bilevel Linear Optimization

2018 ◽  
Vol 1 (1) ◽  
pp. 1-10
Author(s):  
S. Dempe ◽  
G. Luo ◽  
S. Franke
Keyword(s):  
Optimization Problem ◽  
Value Function ◽  
Linear Optimization ◽  
Equivalent Transformation ◽  
Optimal Solutions ◽  
Linear Optimization Problem ◽  
Optimal Value ◽  
Nonconvex And Nonsmooth Optimization ◽  
Bilevel Linear Optimization ◽  
Nonsmooth Optimization Problem

In this paper, we investigate the pessimistic bilevel linear optimization problem (PBLOP). Based on the lower level optimal value function and duality, the PBLOP can be transformed to a single-level while nonconvex and nonsmooth optimization problem. By use of linear optimization duality, we obtain a tractable and equivalent transformation and propose algorithms for computing global or local optimal solutions. One small example is presented to illustrate the feasibility of the method.  

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 Novel Global Harmony Search Algorithm for Least Absolute Deviation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Longquan Yong
Keyword(s):  
Optimization Problem ◽  
Search Algorithm ◽  
Convergence Property ◽  
Harmony Search ◽  
Harmony Search Algorithm ◽  
Least Squares Regression ◽  
Least Absolute Deviation ◽  
Absolute Deviation ◽  
Good Convergence ◽  
Nonsmooth Optimization Problem

The method of least absolute deviation (LAD) finds applications in many areas, due to its robustness compared to the least squares regression (LSR) method. LAD is robust in that it is resistant to outliers in the data. This may be helpful in studies where outliers may be ignored. Since LAD is nonsmooth optimization problem, this paper proposed a metaheuristics algorithm named novel global harmony search (NGHS) for solving. Numerical results show that the NGHS method has good convergence property and effective in solving LAD.

scholarly journals Global Optimal Trajectory in Chaos and NP-Hardness

2016 ◽  
Vol 26 (08) ◽  
pp. 1650142 ◽  
Author(s):  
Vittorio Latorre ◽  
David Yang Gao
Keyword(s):  
Dynamical Systems ◽  
Duality Theory ◽  
Optimization Problem ◽  
Nonlinear Dynamical Systems ◽  
Global Optimization Problem ◽  
Canonical Duality Theory ◽  
Nonlinear Dynamical ◽  
Canonical Duality ◽  
Global Optimal ◽  
Np Hardness

This paper presents an unconventional theory and method for solving general nonlinear dynamical systems. Instead of the direct iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. A newly developed canonical duality theory shows that this nonconvex minimization problem can be solved deterministically in polynomial time if a global optimality condition is satisfied. The so-called pseudo-chaos produced by linear iterative methods are mainly due to the intrinsic numerical error accumulations. Otherwise, the global optimization problem could be NP-hard and the nonlinear system can be really chaotic. A conjecture is proposed, which reveals the connection between chaos in nonlinear dynamics and NP-hardness in computer science. The methodology and the conjecture are verified by applications to the well-known logistic equation, a forced memristive circuit and the Lorenz system. Computational results show that the canonical duality theory can be used to identify chaotic systems and to obtain realistic global optimal solutions in nonlinear dynamical systems. The method and results presented in this paper should bring some new insights into nonlinear dynamical systems and NP-hardness in computational complexity theory.

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