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.

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.

scholarly journals Strong Convergence on the Aggregate Constraint-Shifting Homotopy Method for Solving General Nonconvex Programming

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Zhichuan Zhu ◽  
Yeong-Cheng Liou
Keyword(s):  
Nonlinear Programming ◽  
Strong Convergence ◽  
Nonconvex Programming ◽  
Inequality Constraint ◽  
Homotopy Method ◽  
Cone Condition ◽  
Globally Convergent ◽  
Weaker Conditions ◽  
Convergent Solution ◽  
T System

In the paper, the aggregate constraint-shifting homotopy method for solving general nonconvex nonlinear programming is considered. The aggregation is only about inequality constraint functions. Without any cone condition for the constraint functions, the existence and convergence of the globally convergent solution to the K-K-T system are obtained for both feasible and infeasible starting points under much weaker conditions.

On the Global Convergence of Nonlinear Programming Algorithms

1985 ◽  
Vol 107 (4) ◽  
pp. 454-458 ◽  
Author(s):  
K. Schittkowski
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Initial Point ◽  
Convergence Result ◽  
Gradient Algorithms ◽  
Reduced Gradient ◽  
Numerical Performance ◽  
Generalized Reduced Gradient ◽  
Nonlinear Programming Methods ◽  
Programming Algorithms

In a previous paper a unified outline of some of the most successful nonlinear programming methods was presented by the author, i.e. of penalty, multiplier, sequential quadratic programming, and generalized reduced gradient algorithms, to illustrate their common mathematical features and to explain the different numerical performance observed in practice. By defining a general algorithmic frame for all these approaches, a global convergence result can be achieved in the sense that starting from an arbitrary initial point, a stationary solution will be approximated.

scholarly journals Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1551
Author(s):  
Bothina El-Sobky ◽  
Yousria Abo-Elnaga ◽  
Abd Allah A. Mousa ◽  
Mohamed A. El-Shorbagy
Keyword(s):  
Nonlinear Programming ◽  
Global Convergence ◽  
Programming Problem ◽  
Trust Region ◽  
Convergence Theory ◽  
Benchmark Problems ◽  
Nonlinear Programming Problem ◽  
Barrier Method ◽  
Starting Point ◽  
Constrained Nonlinear Programming

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.

The stochastic equation Yt+1 = AtYt + Bt with non-stationary coefficients

2001 ◽  
Vol 38 (1) ◽  
pp. 80-94 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Global Convergence ◽  
Random Environment ◽  
Linear Relation ◽  
Stochastic Equation ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Stationary Case ◽  
Finite Dimensional ◽  
Long Run

In this paper, we consider the stochastic sequence {Yt}t∊ℕ defined recursively by the linear relation Yt+1 = AtYt + Bt in a random environment which is described by the non-stationary process {(At, Bt)}t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Yt}t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(At, Bt)}t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.

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