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 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 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.

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 On stability and stationary points in nonlinear optimization

1986 ◽  
Vol 28 (1) ◽  
pp. 36-56 ◽  
Author(s):  
J. Guddat ◽  
H. Th. Jongen ◽  
J. Rueckmann
Keyword(s):  
Stability Condition ◽  
Minimization Problem ◽  
Level Sets ◽  
Constraint Qualification ◽  
Stationary Points ◽  
Feasible Set ◽  
Manifold With Boundary ◽  
Feasible Sets ◽  
Constrained Minimization Problem ◽  
Image Position

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:In summary, we provethat, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible setM[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible setM[H, G] is stable (perturbations ofHandGproduce homeomorphic feasible sets) if and only if MFCQ holds;under a stability condition, two lower level sets offwith a Kuhn-Tucker point between them are homotopically related by attachment of ak-cell (kbeing the stationary index in the sense of Kojima).

scholarly journals The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Zhengyong Zhou ◽  
Bo Yu
Keyword(s):  
Nonlinear Programming ◽  
Interior Point Method ◽  
Large Scale ◽  
State Of The Art ◽  
Numerical Procedure ◽  
Homotopy Method ◽  
Nonlinear Constraints ◽  
Computational Results ◽  
Homotopy Methods ◽  
Constraint Function

The aggregate constraint homotopy method uses a single smoothing constraint instead ofm-constraints to reduce the dimension of its homotopy map, and hence it is expected to be more efficient than the combined homotopy interior point method when the number of constraints is very large. However, the gradient and Hessian of the aggregate constraint function are complicated combinations of gradients and Hessians of all constraint functions, and hence they are expensive to calculate when the number of constraint functions is very large. In order to improve the performance of the aggregate constraint homotopy method for solving nonlinear programming problems, with few variables and many nonlinear constraints, a flattened aggregate constraint homotopy method, that can save much computation of gradients and Hessians of constraint functions, is presented. Under some similar conditions for other homotopy methods, existence and convergence of a smooth homotopy path are proven. A numerical procedure is given to implement the proposed homotopy method, preliminary computational results show its performance, and it is also competitive with the state-of-the-art solver KNITRO for solving large-scale nonlinear optimization.

Sign in / Sign up

Export Citation Format

Share Document