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

Chunyang Zhang; Shugong Zhang; Qinghuai Liu

doi:10.3934/jimo.2016005

Previous Article
A smoothing iterative method for quantile regression with nonconvex $ \ell_p $ penalty
JIMO Home
This Issue
Next Article
The stable duality of DC programs for composite convex functions

January 2017, 13(1): 81-92. doi: 10.3934/jimo.2016005

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

Chunyang Zhang ^1,2, , Shugong Zhang ^1, and Qinghuai Liu ^2,,

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

Received December 2014 Revised December 2015 Published March 2016

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

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: Homotopy method, Normal cone condition, Multiobjective optimization, Equilibrium constraints.

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

Citation: Chunyang Zhang, Shugong Zhang, Qinghuai Liu. Homotopy method for a class of multiobjective optimization problems with equilibrium constraints. Journal of Industrial & Management Optimization, 2017, 13 (1) : 81-92. doi: 10.3934/jimo.2016005

References:

[1]	E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin-New York, 1990. doi: 10.1007/978-3-642-61257-2. Google Scholar
[2]	T. Q. Bao, P. Gupta and B. S. Mordukhovich, Necessary conditions in multiobjective optimization with equilibrium constraints, J. Optim. Theory Appl., 135 (2007), 179-203. doi: 10.1007/s10957-007-9209-x. Google Scholar
[3]	S. N. Chow, J. Mallet-Paret and J. A. Yorke, Finding zeros of maps: Homotopy methods that are constructive with probability one, Math. Compu., 13 (1978), 887-899. doi: 10.1090/S0025-5718-1978-0492046-9. Google Scholar
[4]	S. Dempe, Annotated bibliographa on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. Google Scholar
[5]	X. N. Fan and Q. L. Yan, Homotopy method for solving ball-constrained variational inequalities, Nonlinear Analysis, 74 (2011), 1539-1544. doi: 10.1016/j.na.2010.09.041. Google Scholar
[6]	M. Fukushima and P. Tseng, An Implementable Active-Set Algorithm for Computing a B-Stationary Point of the Mathematical Program with Linear Complementarity Constraints, Manuscript, Department of Applied Mathematics and Physics, Graduate School of Informations, Kyoto University, Japan, 1999. Google Scholar
[7]	M. Fukushima and P. Tseng, An Implementable active-set algorithm for computing a b-stationary point of mathematical program with linear complemetarity constraints, SIAM J. Optim., 12 (2002), 724-739. doi: 10.1137/S1052623499363232. Google Scholar
[8]	L. Guo and G. H. Lin, Globally convergent algorithm for solving stationary points for mathematical programs with complementarity constraints via nonsmooth reformulations, Journal of Industrial and Management Optimization, 9 (2013), 305-322. doi: 10.3934/jimo.2013.9.305. Google Scholar
[9]	L. Guo, G. H. Lin and J. J. Ye, Stability analysis for parametric mathematical programs with geometric constraints and its applications, SIAM Journal on Optimization, 22 (2012), 1151-1176. doi: 10.1137/120868657. Google Scholar
[10]	P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and non-linear complementarity problem: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161-220. doi: 10.1007/BF01582255. Google Scholar
[11]	R. B. Kellogg, T. Y. Li and J. Yorke, A constructive proof the Brouwer fixed-point theorem and computational results, SIAM J. Numer. Anal., 13 (1976), 473-483. doi: 10.1137/0713041. Google Scholar
[12]	M. Ko$ \overset{''}{\mathop{\text{c}}}\, $vara and J. V. Outrata, Optimization problems with equilibrium constraints and their numerical solution, Math. Program. Ser. B, 101 (2004), 119-149. doi: 10.1007/s10107-004-0539-2. Google Scholar
[13]	J. M. Li, Combined Homotopy Interior-Point Method for Solving Mathematical Programs with Equilibrium Constraints, Ph. D. thesis, Ji Lin University, 2007. Google Scholar
[14]	Z. H. Lin, B. Yu and D. L. Zhu, A continuation method for solving fixed points of self-mappings in general nonconvex sets, Nonlinear Analysis, 52 (2003), 905-915. doi: 10.1016/S0362-546X(02)00140-2. Google Scholar
[15]	X. W. Liu and J. Sun, Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints, Math. Program. Ser. B, 101 (2004), 231-261. doi: 10.1007/s10107-004-0543-6. Google Scholar
[16]	Q. H. Liu, B. Yu and G. C. Feng, An interior point path-following method for convex programming with quasi normal cone condition, Advances in Mathematics, 29 (2000), 281-282. Google Scholar
[17]	Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511983658. Google Scholar
[18]	M. M. M$ \overset{''}{\mathop{\text{a}}}\, $kel$ \overset{''}{\mathop{\text{a}}}\, $ and P. Neittaanm$ \overset{''}{\mathop{\text{a}}}\, $ki, Nonsmooth Optimization: Analysis And Algorithms with Applications to Optimal Control, World Scientific Publishing Co. Pte. Ltd., 1992. Google Scholar
[19]	O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Book Company, New York, 1969; Japanese Edition, 1971; SIAM Classics in Applied Mathematics, 10, Philadelphia, 1994. doi: 10.1137/1.9781611971255. Google Scholar
[20]	B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Springer, Berlin, 2006. Google Scholar
[21]	B. S. Mordukhovich, Multiobjective optimization problems with equilibrium constraints, Math. Program. Ser. B, 117 (2009), 331-354. doi: 10.1007/s10107-007-0172-y. Google Scholar
[22]	G. L. Naber, Topological Methods in Euclidean Spaces, Cambridge University Press, Cambridge, UK, 1980. Google Scholar
[23]	S. Smale, A convergent process of price adjustment and global Newton methods, J. Math. Econom., 3 (1976), 107-120. doi: 10.1016/0304-4068(76)90019-7. Google Scholar
[24]	W. Song and G. M. Yao, Homotopy method for a general multiobjective programming problem, Journal of Optimization Theory and Applications, 138 (2008), 139-153. doi: 10.1007/s10957-008-9366-6. Google Scholar
[25]	Q. Xu, B. Yu and G. C. Feng, Homotopy method for solving variational inequalities in unbounded sets, Journal of Global Optimization, 31 (2005), 121-131. doi: 10.1007/s10898-004-4272-4. Google Scholar
[26]	Q. Xu and B. Yu, Solving the Karush-Kuhn-Tucker system of a nonconvex programming problem on an unbounded sets, Nonlinear Analysis, 70 (2009), 757-763. doi: 10.1016/j.na.2008.01.008. Google Scholar
[27]	J. J. Ye and Q. J. Zhu, Multiobjective optimization problem with variational inequality constraints, Math. Program. Ser. A, 96 (2003), 139-160. doi: 10.1007/s10107-002-0365-3. Google Scholar
[28]	Z. S. Zhang, Introduction to Differential Topology, Beijing University Press, Beijing, China, 2002. Google Scholar
[29]	X. Zhao, S. G. Zhang and Q. H. Liu, Homotopy interior-point method for a general multiobjective programming problem Journal of Applied Mathematics, (2012), Art. ID 497345, 12pp. doi: 10.1155/2012/497345. Google Scholar
[30]	Z. Y. Zhou and B. Yu, A smoothing homotopy method for variational inequality problems on polyhedral convex sets, J. Glob Optim., 58 (2013), 151-168. doi: 10.1007/s10898-013-0033-6. Google Scholar

show all references

References:

[1]	E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin-New York, 1990. doi: 10.1007/978-3-642-61257-2. Google Scholar
[2]	T. Q. Bao, P. Gupta and B. S. Mordukhovich, Necessary conditions in multiobjective optimization with equilibrium constraints, J. Optim. Theory Appl., 135 (2007), 179-203. doi: 10.1007/s10957-007-9209-x. Google Scholar
[3]	S. N. Chow, J. Mallet-Paret and J. A. Yorke, Finding zeros of maps: Homotopy methods that are constructive with probability one, Math. Compu., 13 (1978), 887-899. doi: 10.1090/S0025-5718-1978-0492046-9. Google Scholar
[4]	S. Dempe, Annotated bibliographa on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. Google Scholar
[5]	X. N. Fan and Q. L. Yan, Homotopy method for solving ball-constrained variational inequalities, Nonlinear Analysis, 74 (2011), 1539-1544. doi: 10.1016/j.na.2010.09.041. Google Scholar
[6]	M. Fukushima and P. Tseng, An Implementable Active-Set Algorithm for Computing a B-Stationary Point of the Mathematical Program with Linear Complementarity Constraints, Manuscript, Department of Applied Mathematics and Physics, Graduate School of Informations, Kyoto University, Japan, 1999. Google Scholar
[7]	M. Fukushima and P. Tseng, An Implementable active-set algorithm for computing a b-stationary point of mathematical program with linear complemetarity constraints, SIAM J. Optim., 12 (2002), 724-739. doi: 10.1137/S1052623499363232. Google Scholar
[8]	L. Guo and G. H. Lin, Globally convergent algorithm for solving stationary points for mathematical programs with complementarity constraints via nonsmooth reformulations, Journal of Industrial and Management Optimization, 9 (2013), 305-322. doi: 10.3934/jimo.2013.9.305. Google Scholar
[9]	L. Guo, G. H. Lin and J. J. Ye, Stability analysis for parametric mathematical programs with geometric constraints and its applications, SIAM Journal on Optimization, 22 (2012), 1151-1176. doi: 10.1137/120868657. Google Scholar
[10]	P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and non-linear complementarity problem: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161-220. doi: 10.1007/BF01582255. Google Scholar
[11]	R. B. Kellogg, T. Y. Li and J. Yorke, A constructive proof the Brouwer fixed-point theorem and computational results, SIAM J. Numer. Anal., 13 (1976), 473-483. doi: 10.1137/0713041. Google Scholar
[12]	M. Ko$ \overset{''}{\mathop{\text{c}}}\, $vara and J. V. Outrata, Optimization problems with equilibrium constraints and their numerical solution, Math. Program. Ser. B, 101 (2004), 119-149. doi: 10.1007/s10107-004-0539-2. Google Scholar
[13]	J. M. Li, Combined Homotopy Interior-Point Method for Solving Mathematical Programs with Equilibrium Constraints, Ph. D. thesis, Ji Lin University, 2007. Google Scholar
[14]	Z. H. Lin, B. Yu and D. L. Zhu, A continuation method for solving fixed points of self-mappings in general nonconvex sets, Nonlinear Analysis, 52 (2003), 905-915. doi: 10.1016/S0362-546X(02)00140-2. Google Scholar
[15]	X. W. Liu and J. Sun, Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints, Math. Program. Ser. B, 101 (2004), 231-261. doi: 10.1007/s10107-004-0543-6. Google Scholar
[16]	Q. H. Liu, B. Yu and G. C. Feng, An interior point path-following method for convex programming with quasi normal cone condition, Advances in Mathematics, 29 (2000), 281-282. Google Scholar
[17]	Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511983658. Google Scholar
[18]	M. M. M$ \overset{''}{\mathop{\text{a}}}\, $kel$ \overset{''}{\mathop{\text{a}}}\, $ and P. Neittaanm$ \overset{''}{\mathop{\text{a}}}\, $ki, Nonsmooth Optimization: Analysis And Algorithms with Applications to Optimal Control, World Scientific Publishing Co. Pte. Ltd., 1992. Google Scholar
[19]	O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Book Company, New York, 1969; Japanese Edition, 1971; SIAM Classics in Applied Mathematics, 10, Philadelphia, 1994. doi: 10.1137/1.9781611971255. Google Scholar
[20]	B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Springer, Berlin, 2006. Google Scholar
[21]	B. S. Mordukhovich, Multiobjective optimization problems with equilibrium constraints, Math. Program. Ser. B, 117 (2009), 331-354. doi: 10.1007/s10107-007-0172-y. Google Scholar
[22]	G. L. Naber, Topological Methods in Euclidean Spaces, Cambridge University Press, Cambridge, UK, 1980. Google Scholar
[23]	S. Smale, A convergent process of price adjustment and global Newton methods, J. Math. Econom., 3 (1976), 107-120. doi: 10.1016/0304-4068(76)90019-7. Google Scholar
[24]	W. Song and G. M. Yao, Homotopy method for a general multiobjective programming problem, Journal of Optimization Theory and Applications, 138 (2008), 139-153. doi: 10.1007/s10957-008-9366-6. Google Scholar
[25]	Q. Xu, B. Yu and G. C. Feng, Homotopy method for solving variational inequalities in unbounded sets, Journal of Global Optimization, 31 (2005), 121-131. doi: 10.1007/s10898-004-4272-4. Google Scholar
[26]	Q. Xu and B. Yu, Solving the Karush-Kuhn-Tucker system of a nonconvex programming problem on an unbounded sets, Nonlinear Analysis, 70 (2009), 757-763. doi: 10.1016/j.na.2008.01.008. Google Scholar
[27]	J. J. Ye and Q. J. Zhu, Multiobjective optimization problem with variational inequality constraints, Math. Program. Ser. A, 96 (2003), 139-160. doi: 10.1007/s10107-002-0365-3. Google Scholar
[28]	Z. S. Zhang, Introduction to Differential Topology, Beijing University Press, Beijing, China, 2002. Google Scholar
[29]	X. Zhao, S. G. Zhang and Q. H. Liu, Homotopy interior-point method for a general multiobjective programming problem Journal of Applied Mathematics, (2012), Art. ID 497345, 12pp. doi: 10.1155/2012/497345. Google Scholar
[30]	Z. Y. Zhou and B. Yu, A smoothing homotopy method for variational inequality problems on polyhedral convex sets, J. Glob Optim., 58 (2013), 151-168. doi: 10.1007/s10898-013-0033-6. Google Scholar

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)

Table Options

Download as excel

$(x_1^{(0)}, x_2^{(0)}, y_1^{(0)}, y_2^{(0)}, t^{(0)})$	$(x_1^, x_2^, y_1^, y_2^, t^*)$	$(f_1^, f_2^)$
(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)

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)

Table Options

Download as excel

$(x_1^{(0)}, x_2^{(0)}, y_1^{(0)}, y_2^{(0)}, t^{(0)})$	$(x_1^, x_2^, y_1^, y_2^, t^*)$	$(f_1^, f_2^)$
(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)

[1]	Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[2]	Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[3]	Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[4]	David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics & Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002
[5]	Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[6]	Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[7]	Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[8]	Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[9]	Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[10]	Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[11]	Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051
[12]	Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[13]	Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
[14]	Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[15]	Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[16]	Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[17]	Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263
[18]	Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[19]	Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[20]	Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

2019 Impact Factor: 1.366

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.366

Tools

Figures and Tables

2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors