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Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
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.
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C. Berge, Topological Spaces, Macmillan, New York, 1963. Google Scholar |
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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. 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. |
[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. 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. |
[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. 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. |
[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. Google Scholar |
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