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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

Qingsong Duan 1,, , Mengwei Xu 2, , Liwei Zhang 3, and  Sainan Zhang 4,

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.

Keywords: Second order conic optimization, Optimal value function, Solution mapping, Hadamard directional differentiability, Stability analysis.
Mathematics Subject Classification: Primary: 90C15; Secondary: 90C31.
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 & Management Optimization, 2021, 17 (6) : 3085-3098. doi: 10.3934/jimo.2020108
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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. M. Lee, N. N. Tam and N. D. Yen, Quadratic Programming and Affine Variational Inequalities, A Qualitative Study, Springer, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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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