# American Institute of Mathematical Sciences

• Previous Article
The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex
• JIMO Home
• This Issue
• Next Article
Tabu search guided by reinforcement learning for the max-mean dispersion problem
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  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 & Management Optimization, 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.  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

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
 [1] Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 [2] Tomasz Szostok. Inequalities of Hermite-Hadamard type for higher order convex functions, revisited. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020296 [3] Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381 [4] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377 [5] Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117 [6] Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119 [7] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [8] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [9] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [10] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [11] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [12] Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017 [13] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 [14] Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006 [15] Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002 [16] Xuemei Chen, Julia Dobrosotskaya. Inpainting via sparse recovery with directional constraints. Mathematical Foundations of Computing, 2020, 3 (4) : 229-247. doi: 10.3934/mfc.2020025 [17] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [18] Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053 [19] Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296 [20] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

2019 Impact Factor: 1.366

Article outline