The Combined Homotopy Methods for Optimizition under the Quasi-Normal Cone Condition

2012 ◽  
Vol 459 ◽  
pp. 16-18
Author(s):  
Yun Feng Gao
Keyword(s):  
Normal Cone ◽  
Homotopy Methods ◽  
Cone Condition

Data Processing with Combined Homotopy Methods for a Class of Nonconvex Optimization Problems

2014 ◽  
Vol 1046 ◽  
pp. 403-406 ◽  
Author(s):  
Yun Feng Gao ◽  
Ning Xu
Keyword(s):  
Data Processing ◽  
Nonconvex Optimization ◽  
Normal Cone ◽  
Optimization Problems ◽  
Feasible Region ◽  
Specific Class ◽  
Homotopy Methods ◽  
Cone Condition ◽  
Nonconvex Optimization Problems ◽  
Theoretical Results

On the existing theoretical results, this paper studies the realization of combined homotopy methods on optimization problems in a specific class of nonconvex constrained region. Contraposing to this nonconvex constrained region, we give the structure method of the quasi-normal, prove that the chosen mappings on constrained grads are positive independent and the feasible region on SLM satisfies the quasi-normal cone condition. And we construct combined homotopy equation under the quasi-normal cone condition with numerical value and examples, and get a preferable result by data processing.

scholarly journals A Comparison of Normal Cone Conditions for Homotopy Methods for Solving Inequality Constrained Nonlinear Programming Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Zhengyong Zhou ◽  
Ting Zhang
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Constraint Qualification ◽  
Normal Cone ◽  
Homotopy Methods ◽  
Cone Condition ◽  
Feasible Set ◽  
Constrained Nonlinear Programming ◽  
Cone Conditions

Homotopy methods are powerful tools for solving nonlinear programming. Their global convergence can be generally established under conditions of the nonemptiness and boundness of the interior of the feasible set, the Positive Linear Independent Constraint Qualification (PLICQ), which is equivalent to the Mangasarian-Fromovitz Constraint Qualification (MFCQ), and the normal cone condition. This paper provides a comparison of the existing normal cone conditions used in homotopy methods for solving inequality constrained nonlinear programming.

Aggregate Homotopy Method for Min-Max-Min Programming Satisfying a Weak-Normal Cone Condition

2011 ◽  
Vol 50-51 ◽  
pp. 669-672
Author(s):  
Hui Juan Xiong ◽  
B. Yu
Keyword(s):  
Data Mining ◽  
Support Vector Machine ◽  
Normal Cone ◽  
Support Vector Machine Model ◽  
Homotopy Method ◽  
Support Vector ◽  
Cone Condition ◽  
Machine Model ◽  
Nonsmooth Programming

Min-max-min programming is an important but difficult nonsmooth programming. An aggregate homotopy method was given for solving min-max-min programming by Bo Yu el al. However, the method requires a difficult to verify weak-normal cone condition. Moreover, the method is only theoretically with no algorithmic implementation. In this paper, the weak normal cone condition is discussed first. A class of min-max-min programming satisfying the condition is introduced. A detailed algorithm to implement the method is presented. Models arising from some applications such as support vector machine for multiple-instance classification in data mining, can be included in the problem. In the end, the aggregate homotopy method is given to solve the multiple-instance support vector machine model.

Aggregate Constraint Homotopy Method for Nonlinear Programming on Unbounded Set

2011 ◽  
Vol 50-51 ◽  
pp. 283-287
Author(s):  
Yu Xiao ◽  
Hui Juan Xiong ◽  
Zhi Gang Yan
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Additional Assumption ◽  
Normal Cone ◽  
Constraint Qualifications ◽  
Convergence Result ◽  
Homotopy Method ◽  
Cone Condition ◽  
Feasible Set

In [1], an aggregate constraint aggregate (ACH) method for nonconvex nonlinear programming problems was presented and global convergence result was obtained when the feasible set is bounded and satisfies a weak normal cone condition with some standard constraint qualifications. In this paper, without assuming the boundedness of feasible set, the global convergence of ACH method is proven under a suitable additional assumption.

Proximal Normal Cone Analysis on Smooth Banach Spaces and Applications

2014 ◽  
Vol 24 (1) ◽  
pp. 363-384 ◽  
Author(s):  
Xi Yin Zheng ◽  
Kung Fu Ng
Keyword(s):  
Banach Spaces ◽  
Normal Cone ◽  
Proximal Normal Cone

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

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