A piecewise-linear homotopy method for nonlinear programming

1991 ◽  
Vol 74 (10) ◽  
pp. 28-38 ◽  
Author(s):  
Kiyotaka Yamamura ◽  
Kaori Arai ◽  
Masahiro Kiyoi
Keyword(s):  
Nonlinear Programming ◽  
Piecewise Linear ◽  
Homotopy Method ◽  
Piecewise Linear Homotopy

scholarly journals Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems

2013 ◽  
Vol 22 (1) ◽  
pp. 41-46
Author(s):  
ANDREI BOZANTAN ◽  
◽  
VASILE BERINDE ◽  
Keyword(s):  
Fixed Points ◽  
Unconstrained Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Local Minima ◽  
Homotopy Algorithm ◽  
Unconstrained Optimization Problems ◽  
Piecewise Linear Homotopy ◽  
Computation Of Fixed Points ◽  
Modern Programming Language

This paper describes the main aspects of the ”piecewise-linear homotopy method” for fixed point approximation proposed by Eaves and Saigal [Eaves, C. B. and Saigal, R., Homotopies for computation of fixed points on unbounded regions, Mathematical Programming, 3 (1972), No. 1, 225–237]. The implementation of the method is developed using the modern programming language C# and then is used for solving some unconstrained optimization problems. The PL homotopy algorithm appears to be more reliable than the classical Newton method in the case of the problem of finding a local minima for Schwefel’s function and other optimization problems.

scholarly journals An interior-point piecewise linear penalty method for nonlinear programming

2009 ◽  
Vol 128 (1-2) ◽  
pp. 73-122 ◽  
Author(s):  
Lifeng Chen ◽  
Donald Goldfarb
Keyword(s):  
Nonlinear Programming ◽  
Interior Point ◽  
Penalty Method ◽  
Piecewise Linear

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.

The aggregate constraint homotopy method for nonconvex nonlinear programming

2001 ◽  
Vol 45 (7) ◽  
pp. 839-847 ◽  
Author(s):  
Bo Yu ◽  
Guo-chen Feng ◽  
Shao-Liang Zhang
Keyword(s):  
Nonlinear Programming ◽  
Homotopy Method

Optimal Fair Division for Measures with Piecewise Linear Density Functions

2017 ◽  
Vol 19 (02) ◽  
pp. 1750009 ◽  
Author(s):  
Jerzy Legut
Keyword(s):  
Nonlinear Programming ◽  
Linear Density ◽  
Piecewise Linear ◽  
Fair Division ◽  
Unit Interval ◽  
Programming Method ◽  
Probability Measures ◽  
Density Functions ◽  
Nonlinear Programming Method

A nonlinear programming method is used for finding an optimal fair division of the unit interval [Formula: see text] among [Formula: see text] players. Preferences of players are described by nonatomic probability measures [Formula: see text] with piecewise linear (PWL) density functions. The presented algorithm can be applied for obtaining “almost” optimal fair divisions for measures with arbitrary density functions approximable by PWL functions. The number of cuts needed for obtaining such divisions is given.

scholarly journals A new nonlinear model for the two-dimensional rectangle packing problem

2013 ◽  
Vol 93 (107) ◽  
pp. 95-107
Author(s):  
Aleksandar Savic ◽  
Jozef Kratica ◽  
Vladimir Filipovic
Keyword(s):  
Nonlinear Programming ◽  
Piecewise Linear ◽  
Packing Problem ◽  
Linear Formulation ◽  
Mixed Integer ◽  
Two Dimensional ◽  
Programming Formulation ◽  
Real Numbers ◽  
Rectangle Packing ◽  
Rectangle Packing Problem

This paper deals with the rectangle packing problem, of filling a big rectangle with smaller rectangles, while the rectangle dimensions are real numbers. A new nonlinear programming formulation is presented and the validity of the formulation is proved. In addition, two cases of the problem are presented, with and without rotation of smaller rectangles by 90?. The mixed integer piecewise linear formulation derived from the model is given, but with a simple form of the objective function.

Sign in / Sign up

Export Citation Format

Share Document