# American Institute of Mathematical Sciences

November  2021, 17(6): 3085-3098. doi: 10.3934/jimo.2020108

## 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

* Corresponding author: Mengwei Xu

Received  September 2019 Revised  March 2020 Published  November 2021 Early access  June 2020

Fund Project: The first author is supported by NSFC grant 11901556. The second author is supported by NSFC grant 11601376. The third author is supported by NSFC grant 11971089 and 11731013

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.

Citation: Qingsong Duan, Mengwei Xu, Liwei Zhang, Sainan Zhang. Hadamard directional differentiability of the optimal value of a linear second-order conic programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3085-3098. doi: 10.3934/jimo.2020108
##### 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. [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. [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.

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. [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. [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.
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