-
Previous Article
Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering
- JIMO Home
- This Issue
-
Next Article
Quality choice and capacity rationing in advance selling
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.
|
| [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. |
| [1] |
Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control and Related Fields, 2021, 11 (3) : 653-679. doi: 10.3934/mcrf.2021017 |
| [2] |
Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010 |
| [3] |
Meng Xue, Yun Shi, Hailin Sun. Portfolio optimization with relaxation of stochastic second order dominance constraints via conditional value at risk. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2581-2602. doi: 10.3934/jimo.2019071 |
| [4] |
Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661 |
| [5] |
Kevin Schober, Jürgen Prestin. Analysis of directional higher order jump discontinuities with trigonometric shearlets. Mathematical Foundations of Computing, 2021 doi: 10.3934/mfc.2021038 |
| [6] |
Fausto Ferrari. Mean value properties of fractional second order operators. Communications on Pure and Applied Analysis, 2015, 14 (1) : 83-106. doi: 10.3934/cpaa.2015.14.83 |
| [7] |
Farah Abdallah, Mouhammad Ghader, Ali Wehbe, Yacine Chitour. Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2789-2818. doi: 10.3934/cpaa.2019125 |
| [8] |
Nguyen Huy Chieu, Jen-Chih Yao. Subgradients of the optimal value function in a parametric discrete optimal control problem. Journal of Industrial and Management Optimization, 2010, 6 (2) : 401-410. doi: 10.3934/jimo.2010.6.401 |
| [9] |
Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 |
| [10] |
Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061 |
| [11] |
Michel C. Delfour. Hadamard Semidifferential, Oriented Distance Function, and some Applications. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1917-1951. doi: 10.3934/cpaa.2021076 |
| [12] |
Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335 |
| [13] |
Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure and Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164 |
| [14] |
Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817 |
| [15] |
Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064 |
| [16] |
Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001 |
| [17] |
Pierre-Étienne Druet. Some mathematical problems related to the second order optimal shape of a crystallisation interface. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2443-2463. doi: 10.3934/dcds.2015.35.2443 |
| [18] |
Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581 |
| [19] |
Changbing Hu. Stability of under-compressive waves with second and fourth order diffusions. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 629-662. doi: 10.3934/dcds.2008.22.629 |
| [20] |
Fasma Diele, Angela Martiradonna, Catalin Trenchea. Stability and errors estimates of a second-order IMSP scheme. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022076 |
2021 Impact Factor: 1.411
Tools
Article outline
[Back to Top]