向量多项式优化问题的数值方法

doi:10.13438/j.cnki.jdzk.2019.04.003

吉首大学学报（自然科学版）

向量多项式优化问题的数值方法

彭雪珂，周光明，赵文杰

（湘潭大学数学与计算科学学院，湖南湘潭 411105）

出版日期:2019-07-25 发布日期:2019-07-31
通讯作者: 周光明（1971—），男，湖南涟源人，湘潭大学数学与计算科学学院教授，博士，硕士生导师，主要从事最优化理论与方法研究.
基金资助:
国家自然科学基金资助项目(11671342)

Numerical Methods for Vector Polynomial Optimization Problem

PENG Xueke, ZHOU Guangming, ZHAO Wenjie

(School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan China)

Online:2019-07-25 Published:2019-07-31

摘要/Abstract

摘要：

向量多项式优化问题中的目标函数和约束条件都是由多项式描述的.先将多目标多项式函数分别通过主要目标法、线性加权和法和理想点法等转化为单目标多项式函数,再利用Lasserre松弛方法求解该多项式优化问题,从而得到原向量多项式优化问题的弱有效解或有效解.数值实验结果表明该数值方法是有效的.

关键词： 向量多项式优化, 多项式优化, 目标函数, 约束条件, 弱有效解

Abstract:

For vector polynomial optimization problem, the objective function and constraint condition are all described in polynomial. Firstly, the multi-objective polynomial function is transformed into the single-objective polynomial function by using the main objective method, linear weighted sum method and ideal point method respectively. Then Lasserre relaxation method is applied to solve the polynomial optimization problem, and its optimal solution is obtained, which is also the weak efficient solution or effective solution of the original vector polynomial optimization problem. Numerical experiments show that the proposed numerical methods are effective.

Key words: vector polynomial optimization, polynomial optimization, objective function, constraint condition, weak efficient solution

彭雪珂，周光明，赵文杰. 向量多项式优化问题的数值方法[J]. 吉首大学学报（自然科学版）, DOI: 10.13438/j.cnki.jdzk.2019.04.003.

PENG Xueke, ZHOU Guangming, ZHAO Wenjie. Numerical Methods for Vector Polynomial Optimization Problem[J]. Journal of Jishou University（Natural Sciences Edition), DOI: 10.13438/j.cnki.jdzk.2019.04.003.

参考文献

[1] PARETO VILFREDO. Coursd' conomie Politique[M]. Lausanne: Rouge, 1896: 50-97.
[2] PARETO VILFREDO. Manuale di Economia Politica[M]. Milano: Societ a Editrice Linraria, 1906: 1-27.
[3] KOOPMANS TJALLING C. Analysis of Production as an Efficient Combination of Activities[J]. Analysis of Production and Allocation,1951,158(1): 33-97.
[4] KUHN HAROLD W, TUCKER ALBERT WILLIAM. Nonliear Programming[C]. Proceeding of the Second Berkeley Symposium on Mathematical Statisitics and Probability. California: University of California Press, 1953, 35(2): 481-492.
[5] 林锉云,董加礼.多目标优化的方法与理论[M].吉林:吉林教育出版社,1992:55-72.
[6] GIANNESSI FRANCO, MASTROENI GIANDOMENICO, PELLEGRINI LETIZIA. On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation[J]. Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, 2000, 38: 153-215.
[7] 陈光亚.向量优化问题某些基础理论及其发展[J].重庆师范大学学报(自然科学版),2005,22(3):6-9.
[8] JAHN JOHANNES. Vector Optimization: Theory, Applications and Extensions[M]. New York: SpringerVerlag, 2011: 190-207.
[9] FLORES-BAZN FABIN, HERNNDEZ ELVIRA. A Unified Vector Optimization Problem: Complete Scalarizations and Applications[J]. Optimization, 2011, 60(12): 1 399-1 419.
[10] GUTIRREZ CSAR, HUERGA L, JIMNEZ B, et al. Approximate Solutions of Vector Optimization Problems via Improvement Sets in Real Linear Spaces[J]. Journal of Global Optimization, 2018, 70(4): 875-901.
[11] FLOUDAS CHRISTODOULOS A, PARDALOS PANOS M. Recent Advances in Global Optimization[M]. Princeton University Press, 1992: 165-199.
[12] HORST REINER, PARDALOS PANOS M. Handbook of Global Optimization[J]. Nonconvex Optimization and Its Applications, 2002, 2(5): 217-269.
[13] RUBINOV ALEXANDER. Abstract Convexity and Global Optimization[M]. Berlin: Kluwer Academic Publishers, 2000: 450-470.
[14] LASSERRE JEAN BERNARD. Global Optimization with Polynomials and the Problem of Moments[J]. SIAM J Optim, 2001, 11(3): 796-817.
[15] FLOUDAS CHRISTODOULOS A, PARDALOS PNOS M, ADJIMAN CLAIRE, et al. Handbook of Test Problems in Local and Global Optimization[M]∥Nonconvex Optimization and Its Applications. Berlin: Kluwer Academic Publishers, 1999: 1-31.
[16] HENDRIX ELIGIUS M T, G-TTH BOGLRKA. Introduction to Nonlinear and Global Optimization[M]. New York: Springer, 2010: 137-170.
[17] HENRION DIDIER, LASSERRE JEAN-BERBARD, LFBERG JOHAN. GloptiPoly 3: Moments, Optimization and Semidefinite Programming[J]. Optimization Methods and Software, 2009, 24(4-5): 761-779.
[18] LASSERRE JEAN BERNARD. Moments, Positive Polynomials and Their Applications[M]. Imperial College Press, 2010: 1-144.
[19] FLOUDAS CHRISTODOULOS A, PARDALOS PANOS M. A Collection of Test Problems for Constrained Global Optimization Algorithms[M]∥Lecture Notes in Computer Science. Berlin: Springer-Verlag, 1990: 1-35.
[20] NIE Jiawang. Linear Optimization with Cones of Moments and Nonnegative Polynomials[J]. Mathematical Programming, 2015, 153(1): 247-274.
[21] NIE Jiawang, WANG Li, YE JANE J. Bilevel Polynomial Programs and Semidefinite Relaxation Methods[J]. SIAM Journal on Optimization, 2017, 27(3): 1 728-1 757.
[22] WEISSER TILLMANN, LASSERRE JEAN BERNARD, TOH KIM-CHUAN. Sparse-BSOS: A Bounded Degree SOS Hierarchy for Large Scale Polynomial Optimization with Sparsity[J]. Mathematical Programming Computation, 2018, 10(1): 1-32.

向量多项式优化问题的数值方法

Numerical Methods for Vector Polynomial Optimization Problem

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 4

编辑推荐

Metrics

本文评价

[1]	田超松，王仙云. 分式优化问题的局部和全局最优性条件[J]. 吉首大学学报（自然科学版）, 2020, 41(1): 1-5.
[2]	叶冬平，方东辉. 鲁棒锥规划的Fenchel-Lagrange稳定零对偶[J]. 吉首大学学报（自然科学版）, 2019, 40(6): 1-5.
[3]	陈炳权, 李波勇, 曾庆立. 基于目标函数的FPGA宏模块布局算法设计[J]. journal6, 2008, 29(3): 72-75.
[4]	丁雷, 陈炳权, 陈丽萍, 孟凡斌. 基于共轭梯度法的线性规划优化设计[J]. journal6, 2007, 28(1): 85-87.