American Institute of Mathematical Sciences

Advanced
July  2019, 15(3): 1387-1397. doi: 10.3934/jimo.2018100

Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions

Ziran Yin , and  Liwei Zhang

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Z. R. Yin

Received  September 2017 Revised  March 2018 Published  July 2018

Fund Project: The second author is supported by the National Natural Science Foundation of China under projects No. 11571059, No. 11731013 and No. 91330206.

We consider the stability of a class of parameterized conic programming problems which are more general than $C^2$-smooth parameterization. We show that when the Karush-Kuhn-Tucker (KKT) condition, the constraint nondegeneracy condition, the strict complementary condition and the second order sufficient condition (named as Jacobian uniqueness conditions here) are satisfied at a feasible point of the original problem, the Jacobian uniqueness conditions of the perturbed problem also hold at some feasible point.

Keywords: Stability, $C^2$-cone reducible sets, Jacobian uniqueness conditions, KKT system, implicit function theorem.
Mathematics Subject Classification: 90C31, 49K40, 49J53.
Citation: Ziran Yin, Liwei Zhang. Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1387-1397. doi: 10.3934/jimo.2018100
References:
[1]

C. Berge, Topological Spaces, Macmillan, New York, 1963. Google Scholar

[2]

J. F. Bonnans, R. Cominetti and A. Shapiro, Sensitivity analysis of optimization problems under second order regular constraints, Mathematics of Operations Research, 23 (1998), 806-831.  doi: 10.1287/moor.23.4.806.  Google Scholar

[3]

J. F. Bonnans, R. Cominetti and A. Shapiro, Second order optimality conditions based on parabolic second order tangent sets, SIAM Journal on Optimization, 9 (1999), 466-492.  doi: 10.1137/S1052623496306760.  Google Scholar

[4]

J. F. Bonnans and H. Ramírez C., Perturbation analysis of second order cone programming problems, Mathematical Programming, 104 (2005), 205-227.  doi: 10.1007/s10107-005-0613-4.  Google Scholar

[5]

J. F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour, SIAM Review, 40 (1998), 228-264.  doi: 10.1137/S0036144596302644.  Google Scholar

[6]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[7]

J. F. Bonnans and A. Sulem, Pseudopower expansion of solutions of generalized equations and constrained optimization problems, Mathematical Programming, 70 (1995), 123-148.  doi: 10.1007/BF01585932.  Google Scholar

[8]

C. Ding, An Introduction to a Class of Matrix Optimization Problems, Ph. D thesis, National University of Singapore in Singapore, 2012. Google Scholar

[9]

C. Ding, D. F. Sun and L. W. Zhang, Characterization of the robust isolated calmness for a class of conic programming problems, SIAM Journal on Optimization, 27 (2017), 67-90.  doi: 10.1137/16M1058753.  Google Scholar

[10]

A. L. Dontchev and R. T. Rockafellar, Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM Journal on Optimization, 6 (1996), 1087-1105.  doi: 10.1137/S1052623495284029.  Google Scholar

[11]

H. T. Jongen, T. Mobert, J. Rückmann and K. Tammer, On inertia and schur complement in optimization, Linear Algebra and Its Applications, 95 (1987), 97-109.  doi: 10.1016/0024-3795(87)90028-0.  Google Scholar

[12]

L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Macmillan, New York, 1964.  Google Scholar

[13]

S. M. Robinson, Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms, Mathematical Programming, 7 (1974), 1-16.  doi: 10.1007/BF01585500.  Google Scholar

[14]

S. M. Robinson, Strongly regular generalized equations, Mathematics of Operations Research, 5 (1980), 43-62.  doi: 10.1287/moor.5.1.43.  Google Scholar

[15]

R. T. Rockafellar, Convex Analysis, Princeton, New Jersey, 1970.  Google Scholar

[16]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[17]

A. Shapiro, Sensitivity analysis of generalized equations, Journal of Mathematical Sciences, 115 (2003), 2554-2565.  doi: 10.1023/A:1022940300114.  Google Scholar

[18]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, Society for Industrial and Applied Mathematics, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718751.  Google Scholar

[19]

D. F. Sun, The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications, Mathematics of Operations Research, 31 (2006), 761-776.  doi: 10.1287/moor.1060.0195.  Google Scholar

[20]

Z. R. Yin and L. W. Zhang, Perturbation analysis of nonlinear semidefinite programming under Jacobian uniqueness conditions, 2017. Available from: http://www.optimization-online.org/DB_FILE/2017/09/6197.pdf. Google Scholar

show all references

References:
[1]

C. Berge, Topological Spaces, Macmillan, New York, 1963. Google Scholar

[2]

J. F. Bonnans, R. Cominetti and A. Shapiro, Sensitivity analysis of optimization problems under second order regular constraints, Mathematics of Operations Research, 23 (1998), 806-831.  doi: 10.1287/moor.23.4.806.  Google Scholar

[3]

J. F. Bonnans, R. Cominetti and A. Shapiro, Second order optimality conditions based on parabolic second order tangent sets, SIAM Journal on Optimization, 9 (1999), 466-492.  doi: 10.1137/S1052623496306760.  Google Scholar

[4]

J. F. Bonnans and H. Ramírez C., Perturbation analysis of second order cone programming problems, Mathematical Programming, 104 (2005), 205-227.  doi: 10.1007/s10107-005-0613-4.  Google Scholar

[5]

J. F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour, SIAM Review, 40 (1998), 228-264.  doi: 10.1137/S0036144596302644.  Google Scholar

[6]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[7]

J. F. Bonnans and A. Sulem, Pseudopower expansion of solutions of generalized equations and constrained optimization problems, Mathematical Programming, 70 (1995), 123-148.  doi: 10.1007/BF01585932.  Google Scholar

[8]

C. Ding, An Introduction to a Class of Matrix Optimization Problems, Ph. D thesis, National University of Singapore in Singapore, 2012. Google Scholar

[9]

C. Ding, D. F. Sun and L. W. Zhang, Characterization of the robust isolated calmness for a class of conic programming problems, SIAM Journal on Optimization, 27 (2017), 67-90.  doi: 10.1137/16M1058753.  Google Scholar

[10]

A. L. Dontchev and R. T. Rockafellar, Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM Journal on Optimization, 6 (1996), 1087-1105.  doi: 10.1137/S1052623495284029.  Google Scholar

[11]

H. T. Jongen, T. Mobert, J. Rückmann and K. Tammer, On inertia and schur complement in optimization, Linear Algebra and Its Applications, 95 (1987), 97-109.  doi: 10.1016/0024-3795(87)90028-0.  Google Scholar

[12]

L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Macmillan, New York, 1964.  Google Scholar

[13]

S. M. Robinson, Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms, Mathematical Programming, 7 (1974), 1-16.  doi: 10.1007/BF01585500.  Google Scholar

[14]

S. M. Robinson, Strongly regular generalized equations, Mathematics of Operations Research, 5 (1980), 43-62.  doi: 10.1287/moor.5.1.43.  Google Scholar

[15]

R. T. Rockafellar, Convex Analysis, Princeton, New Jersey, 1970.  Google Scholar

[16]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[17]

A. Shapiro, Sensitivity analysis of generalized equations, Journal of Mathematical Sciences, 115 (2003), 2554-2565.  doi: 10.1023/A:1022940300114.  Google Scholar

[18]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, Society for Industrial and Applied Mathematics, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718751.  Google Scholar

[19]

D. F. Sun, The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications, Mathematics of Operations Research, 31 (2006), 761-776.  doi: 10.1287/moor.1060.0195.  Google Scholar

[20]

Z. R. Yin and L. W. Zhang, Perturbation analysis of nonlinear semidefinite programming under Jacobian uniqueness conditions, 2017. Available from: http://www.optimization-online.org/DB_FILE/2017/09/6197.pdf. Google Scholar

[1]

Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549
[2]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
[3]

Bassam Fayad, Zhiyuan Zhang. An effective version of Katok's horseshoe theorem for conservative C2 surface diffeomorphisms. Journal of Modern Dynamics, 2017, 11: 425-445. doi: 10.3934/jmd.2017017
[4]

Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919
[5]

Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[6]

Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173
[7]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659
[8]

Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617
[9]

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413
[10]

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641
[11]

Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems & Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631
[12]

Mark Pollicott. Closed orbits and homology for $C^2$-flows. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 529-534. doi: 10.3934/dcds.1999.5.529
[13]

Nanhee Kim. Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress. Evolution Equations & Control Theory, 2013, 2 (4) : 679-693. doi: 10.3934/eect.2013.2.679
[14]

Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218
[15]

Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure & Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014
[16]

Qigui Yang, Qiaomin Xiang. Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020335
[17]

Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055
[18]

Francisco Braun, José Ruidival dos Santos Filho. The real jacobian conjecture on $\R^2$ is true when one of the components has degree 3. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 75-87. doi: 10.3934/dcds.2010.26.75
[19]

S. Yu. Pilyugin, Kazuhiro Sakai, O. A. Tarakanov. Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 871-882. doi: 10.3934/dcds.2006.16.871
[20]

Juan Wang, Jing Wang, Yongluo Cao, Yun Zhao. Dimensions of $ C^1- $average conformal hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 883-905. doi: 10.3934/dcds.2020065

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (86)
  • HTML views (1145)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences