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Date Published:
10/10/2020
Online Published:
10/10/2020
Abstract Views: 42
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Views PDF: 11
Section
Natural Sciences Issue
How to Cite
Huynh, T. K. L., & Vo, D. T. (2020). Cones generated by semi-infinite systems and their applications on optimization. Dong Thap University Journal of Science, 9(5), 17-25. https://doi.org/10.52714/dthu.9.5.2020.813
Cones generated by semi-infinite systems and their applications on optimization
Main Article Content
Abstract
In this paper, we first introduce the cones generated by semi-infinite systems. Then we use approaches of the semi-infinite programming to obtain formulas of normal cones and tangent cones to those cones. Thereby, we use obtained results in providing optimality conditions for conic optimization problems. The obtained results in the paper are new and they are generalized from some existing ones in the literature.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Keywords
Cone generated by semi-infinite system, normal cone, optimality condition, tangent cone
References
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Chang, Y.L., Yang, c. Y., & Chen, J.s. (2013). Smooth and nonsmooth analyses of vector-valued functions associated with circular cones. Nonlinear Analysis, (8), 160-173.
Chuông, T.D., & Jeyakumar, V. (2016). Characterizing robust local error bounds for linear inequality systems under data uncertain. Linear Algebra and its Application, (489), 199-216.
Ferreira, O.P., & Nemeth, S.Z. (2018). How to project onto extended second order cones. Journal of Global Optimization, (70), 707-718.
Glineur, F., & Terlaky, T. (2004). Conic formulation for Ip-norm optimization. Journal of Optimization Theory and Applications, (122), 285-307.
Gotoh, J., & Uryasev, s. (2015). Two pairs of families of polyhedral norms versus Ip- norms: proximity and applications in optimization. Mathematical Programming, Series A, (96), 1-41.
Hang, N.T.V., Mordukhovich, B.S., & Sarabi, M.E. (2020). Second-order variational analysis in second-order cone programming. Mathematical Programming, (180), 75-116.
Liu, Y. J., & Zhang, L. W . (2008). Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones. Journal of Optimization Theory and Applications, (139), 557-575.
Nemeth, S. Z., & Xie, J. (2018). Linear complementarity problems on extended second order cones. Journal of
Optimization Theory and Applications, 176), 269-288.
Nemeth, S.Z., Xie, J., & Zhang, G. (2020). Positive operators on extended second order cones. Acta Mathematica Hungarica, (160), 390-404.
Mordukhovich, B. S. (2005). Variational analysis and generalized differentiation I. Springer, Berlin.
Mohammadi, A., Mordukhovich, B.S., & Sarabi, M.E. (2020). Parabolic regularity in geometric variational analysis, to appear in Transactions of the American Mathematical Society, arxiv: 1909.00241.
Mordukhovich, B.S., & Nam, N.M. (2014). An easy path to convex analysis and applications. Morgan & Claypool Publishers, Willistion.
Mordukhovich, B. S., Outrata, J., & Sarabi, M.E. (2014). Full stability of locally optimal solution in second-order cone programming. SIAM Journal on Optimization, (24), 1581-1613.
Outrata, J. V., & Ramírez C, H. (2011). On the Aubin property of critical points to perturbed second-order cone programs. SIAM Journal on Optimization, 21(3), 798-823.
Sznajder, R. (2016). The Lyapunov rank of extended second order cones. Journal of Global Optimization, (66), 585-593.
Thinh, V.D., Chuông, T.D., & Anh, N.L.H. (2020). Optimality conditions for circular cone complementarity programs. Optimization, accepted.
Vinel, A., & Krokhmal, P. (2014a). On valid inequalities for mixed integer p-order cone programming. Journal of Optimization Theory and Applications, 160, 439-456.
Vinel, A., & Krokhmal, P. (2014b). Polyhedral approximations in p-order cone programming. Optimization Methods Software, (29), 1210-1237.
Xue, G., & Ye, Y. (2000). An efficient algorithm for minimizing a sum of p-norms. SIAM Journal on Optimization, (10), 551-579.
Zhou, J., Chang, J.L., & Chen, J.S. (2015). The H-differentiability and calmness of the circular cone functions. Journal of Global Optimization, (63), 811-833.
Zhou, J., & Chen, J. S. (2013). Properties of circular cone and spectral factorization associated with circular cone. Journal of Nonlinear and Convex Analysis, (14), 807-816.
Zhou, J., & Chen, J. S. (2017). Monotonicity and circular cone monotonicity associated with circular cones. Set-Valued and Variational Analysis, (25), 211-232.
Zhou, J., Tang, J., & Chen, J. S. (2017). Parabolic second-order directional differentiability in the Hadamard sense of the vector-valued functions associated with circular cones. Journal of Optimization Theory and Applications, (172), 802-823.
Bonnans, J.F., & Ramirez H .c. (2005). Perturbation analysis of second-order cone programmming problems. Mathematical Programming, (104), 205-227.
Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
Chang, Y.L., Yang, c. Y., & Chen, J.s. (2013). Smooth and nonsmooth analyses of vector-valued functions associated with circular cones. Nonlinear Analysis, (8), 160-173.
Chuông, T.D., & Jeyakumar, V. (2016). Characterizing robust local error bounds for linear inequality systems under data uncertain. Linear Algebra and its Application, (489), 199-216.
Ferreira, O.P., & Nemeth, S.Z. (2018). How to project onto extended second order cones. Journal of Global Optimization, (70), 707-718.
Glineur, F., & Terlaky, T. (2004). Conic formulation for Ip-norm optimization. Journal of Optimization Theory and Applications, (122), 285-307.
Gotoh, J., & Uryasev, s. (2015). Two pairs of families of polyhedral norms versus Ip- norms: proximity and applications in optimization. Mathematical Programming, Series A, (96), 1-41.
Hang, N.T.V., Mordukhovich, B.S., & Sarabi, M.E. (2020). Second-order variational analysis in second-order cone programming. Mathematical Programming, (180), 75-116.
Liu, Y. J., & Zhang, L. W . (2008). Convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones. Journal of Optimization Theory and Applications, (139), 557-575.
Nemeth, S. Z., & Xie, J. (2018). Linear complementarity problems on extended second order cones. Journal of
Optimization Theory and Applications, 176), 269-288.
Nemeth, S.Z., Xie, J., & Zhang, G. (2020). Positive operators on extended second order cones. Acta Mathematica Hungarica, (160), 390-404.
Mordukhovich, B. S. (2005). Variational analysis and generalized differentiation I. Springer, Berlin.
Mohammadi, A., Mordukhovich, B.S., & Sarabi, M.E. (2020). Parabolic regularity in geometric variational analysis, to appear in Transactions of the American Mathematical Society, arxiv: 1909.00241.
Mordukhovich, B.S., & Nam, N.M. (2014). An easy path to convex analysis and applications. Morgan & Claypool Publishers, Willistion.
Mordukhovich, B. S., Outrata, J., & Sarabi, M.E. (2014). Full stability of locally optimal solution in second-order cone programming. SIAM Journal on Optimization, (24), 1581-1613.
Outrata, J. V., & Ramírez C, H. (2011). On the Aubin property of critical points to perturbed second-order cone programs. SIAM Journal on Optimization, 21(3), 798-823.
Sznajder, R. (2016). The Lyapunov rank of extended second order cones. Journal of Global Optimization, (66), 585-593.
Thinh, V.D., Chuông, T.D., & Anh, N.L.H. (2020). Optimality conditions for circular cone complementarity programs. Optimization, accepted.
Vinel, A., & Krokhmal, P. (2014a). On valid inequalities for mixed integer p-order cone programming. Journal of Optimization Theory and Applications, 160, 439-456.
Vinel, A., & Krokhmal, P. (2014b). Polyhedral approximations in p-order cone programming. Optimization Methods Software, (29), 1210-1237.
Xue, G., & Ye, Y. (2000). An efficient algorithm for minimizing a sum of p-norms. SIAM Journal on Optimization, (10), 551-579.
Zhou, J., Chang, J.L., & Chen, J.S. (2015). The H-differentiability and calmness of the circular cone functions. Journal of Global Optimization, (63), 811-833.
Zhou, J., & Chen, J. S. (2013). Properties of circular cone and spectral factorization associated with circular cone. Journal of Nonlinear and Convex Analysis, (14), 807-816.
Zhou, J., & Chen, J. S. (2017). Monotonicity and circular cone monotonicity associated with circular cones. Set-Valued and Variational Analysis, (25), 211-232.
Zhou, J., Tang, J., & Chen, J. S. (2017). Parabolic second-order directional differentiability in the Hadamard sense of the vector-valued functions associated with circular cones. Journal of Optimization Theory and Applications, (172), 802-823.
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