Homotopy method for a class of multiobjective optimization problems with equilibrium constraints

Chunyang Zhang; Shugong Zhang; Qinghuai Liu

doi:10.3934/jimo.2016005

Article Contents

2017, Volume 13, Issue 1: 81-92. Doi: 10.3934/jimo.2016005

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Homotopy method for a class of multiobjective optimization problems with equilibrium constraints

1.
Department of Mathematics, Jilin University, Changchun 130012, China
2.
Institute of Applied Mathematics, Changchun, University of Technology, Changchun 130012, China

*Corresponding author: Qinghuai Liu
*Corresponding author: Qinghuai Liu

Received: November 30, 2014

Revised: November 30, 2015

Published: August 2017

The first author is supported by (Grant No.51278065) and the Jilin Province Natural Science Foundation (Grant No.20130101061 JC.).

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Abstract

In this paper, we present a combined homotopy interior point method for solving multiobjective programs with equilibrium constraints. Under suitable conditions, we prove the existence and convergence of a smooth homotopy path from almost any interior point to a solution of the K-K-T system. Numerical results are presented to show the effectiveness of this algorithm.

Keywords:

Mathematics Subject Classification: Primary: 90C30; Secondary: 65H10.

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Table 1. The numerical results of Example1.

$(x_1^{(0)}, x_2^{(0)}, y_1^{(0)}, y_2^{(0)}, t^{(0)})$	$(x_1^, x_2^, y_1^, y_2^, t^*)$	$(f_1^, f_2^)$
(3.3750, 9.2500, -6.2500, -0.3750, 1)	(0.2230, -0.0470, -1.8319, 0.3554, 0.0001)	(2.2732, 0.0373)
(2.0625, 7.3750, -5.3750, -0.0625, 1)	(0.2231, -0.0473, -1.8318, 0.3554, 0.0001)	(2.2735, 0.0372)
(4.8750, 13.4500, -9.6500, -1.0750, 1)	(-4.2060, 1.0421, 2.9579, -4.9790, 0.0001)	(61.9144, 2.2810)
(4.6875, 12.6250, -8.875, -0.9375, 1)	(-4.2059, 1.0423, 2.9577, -4.9788, 0.0001)	(61.9122, 2.2811)

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Table 2. The numerical results of Example2.

$(x_1^{(0)}, x_2^{(0)}, y_1^{(0)}, y_2^{(0)}, t^{(0)})$	$(x_1^, x_2^, y_1^, y_2^, t^*)$	$(f_1^, f_2^)$
(0, 3.1380, 0.9653, -2.0545, 1)	(-2.2955, -0.7998, 1.1667, -2.6666, 0.0001)	(9.4784, 0.9151)
(3.5326, 0, 0.8932, -1.8410, 1)	(-2.2954, -0.7999, 1.1666, -2.6666, 0.0001)	(9.4773, 0.9154)
(4.2426, 0, 0.7826, -1.5217, 0.5, 1)	(-2.9356, -1.1234, 1.2501, -2.9256, 0.0001)	(12.0080, 1.2622)
(7.4907, 0, 0.8333, -1.6667, 0.4, 1)	(-2.9356, -1.1234, 1.2501, -2.9254, 0.0001)	(12.0072, 1.2622)

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