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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 2019 Early access  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 and Management Optimization, 2019, 15 (3) : 1387-1397. doi: 10.3934/jimo.2018100
References:
[1]

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

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

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