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

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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 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:

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[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

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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

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