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Hadamard directional differentiability of the optimal value of a linear second-order conic programming problem
1. | China Bohai Bank and, School of Economics and Management, University of Chinese Academy of Sciences, Tianjin, MO 300012, China |
2. | Department of Applied Mathematics, Hebei University of Technology, Tianjin, MO 300401, China |
3. | School of Mathematical Sciences, Dalian University of Technology, Dalian, MO 116024, China |
4. | School of Mathematics, Dongbei University of Finance and Economics, Dalian, MO 116025, China |
In this paper, we consider perturbation properties of a linear second-order conic optimization problem and its Lagrange dual in which all parameters in the problem are perturbed. We prove the upper semi-continuity of solution mappings for the pertured problem and its Lagrange dual problem. We demonstrate that the optimal value function can be expressed as a min-max optimization problem over two compact convex sets, and it is proven as a Lipschitz continuous function and Hadamard directionally differentiable.
References:
[1] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[2] |
B. Bereanu,
The continuity of the optimum in parametric programming and applications to stochastic programming, Journal of Optimization Theory and Applications, 18 (1976), 319-333.
doi: 10.1007/BF00933815. |
[3] |
D. Bertsimas, Theory and applications of robust optimization, SIAM Review, 53 (2011), 464-501. Google Scholar |
[4] |
A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer, New York, 2009.
doi: 10.1007/978-0-387-87821-8. |
[5] |
M. S. Gowda and J. S. Pang,
On solution stability of the linear complementarity problem, Mathematics of Operation Reseach, 17 (1992), 77-83.
doi: 10.1287/moor.17.1.77. |
[6] |
M. S. Gowda and J.-S. Pang,
On the boundedness and stability of solutions to the affine variational inequality problem, SIAM J. Control Optim., 32 (1994), 421-441.
doi: 10.1137/S036301299222888X. |
[7] |
D. Goldfarb and G. Iyengar,
Robust portfolio selection problems, Mathematics of Operation Reseach, 28 (2003), 1-38.
doi: 10.1287/moor.28.1.1.14260. |
[8] |
Y. Han and Z. Chen,
Quantitative stability of full random two-stage stochastic programs with recourse, Optim. Lett., 9 (2015), 1075-1090.
doi: 10.1007/s11590-014-0827-6. |
[9] |
G. M. Lee, N. N. Tam and N. D. Yen, Quadratic Programming and Affine Variational Inequalities, A Qualitative Study, Springer, New York, 2005. |
[10] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret,
Applications of second-ordr cone programming, Linear Algebra and its Applications, 284 (1998), 193-228.
doi: 10.1016/S0024-3795(98)10032-0. |
[11] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, in Sobolev and BV Spaces, MPS-SIAM Series on Optimization, 30 (1998), 324-326. Google Scholar |
[12] |
W. Römisch and R. J.-B. Wets,
Stability of $\varepsilon$-approximate solutions to convex stochastic programs, SIAM J. Optim., 18 (2007), 961-979.
doi: 10.1137/060657716. |
[13] |
A. Shapiro, D. Dentcheva and A. Ruszcy$\acute{n}$ski, Lectures on Stochastic Programming Modeling and Theory, SIAM, Philadelphia, 2009. Google Scholar |
show all references
References:
[1] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[2] |
B. Bereanu,
The continuity of the optimum in parametric programming and applications to stochastic programming, Journal of Optimization Theory and Applications, 18 (1976), 319-333.
doi: 10.1007/BF00933815. |
[3] |
D. Bertsimas, Theory and applications of robust optimization, SIAM Review, 53 (2011), 464-501. Google Scholar |
[4] |
A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer, New York, 2009.
doi: 10.1007/978-0-387-87821-8. |
[5] |
M. S. Gowda and J. S. Pang,
On solution stability of the linear complementarity problem, Mathematics of Operation Reseach, 17 (1992), 77-83.
doi: 10.1287/moor.17.1.77. |
[6] |
M. S. Gowda and J.-S. Pang,
On the boundedness and stability of solutions to the affine variational inequality problem, SIAM J. Control Optim., 32 (1994), 421-441.
doi: 10.1137/S036301299222888X. |
[7] |
D. Goldfarb and G. Iyengar,
Robust portfolio selection problems, Mathematics of Operation Reseach, 28 (2003), 1-38.
doi: 10.1287/moor.28.1.1.14260. |
[8] |
Y. Han and Z. Chen,
Quantitative stability of full random two-stage stochastic programs with recourse, Optim. Lett., 9 (2015), 1075-1090.
doi: 10.1007/s11590-014-0827-6. |
[9] |
G. M. Lee, N. N. Tam and N. D. Yen, Quadratic Programming and Affine Variational Inequalities, A Qualitative Study, Springer, New York, 2005. |
[10] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret,
Applications of second-ordr cone programming, Linear Algebra and its Applications, 284 (1998), 193-228.
doi: 10.1016/S0024-3795(98)10032-0. |
[11] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, in Sobolev and BV Spaces, MPS-SIAM Series on Optimization, 30 (1998), 324-326. Google Scholar |
[12] |
W. Römisch and R. J.-B. Wets,
Stability of $\varepsilon$-approximate solutions to convex stochastic programs, SIAM J. Optim., 18 (2007), 961-979.
doi: 10.1137/060657716. |
[13] |
A. Shapiro, D. Dentcheva and A. Ruszcy$\acute{n}$ski, Lectures on Stochastic Programming Modeling and Theory, SIAM, Philadelphia, 2009. Google Scholar |
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