Abstract
Let \(\mathcal{L}_{\theta }\) be the circular cone in \({\mathbb {R}}^n\) which includes second-order cone as a special case. For any function f from \({\mathbb {R}}\) to \({\mathbb {R}}\), one can define a corresponding vector-valued function \(f^{\mathcal{L}_{\theta }}\) on \({\mathbb {R}}^n\) by applying f to the spectral values of the spectral decomposition of \(x \in {\mathbb {R}}^n\) with respect to \(\mathcal{L}_{\theta }\). The main results of this paper are regarding the H-differentiability and calmness of circular cone function \(f^{\mathcal{L}_{\theta }}\). Specifically, we investigate the relations of H-differentiability and calmness between f and \(f^{\mathcal{L}_{\theta }}\). In addition, we propose a merit function approach for solving the circular cone complementarity problems under H-differentiability. These results are crucial to subsequent study regarding various analysis towards optimizations associated with circular cone.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. program. 95, 3–51 (2003)
Andersen, E.D., Roos, C., Terlaky, T.: Notes on duality in second order and \(p\)-order cone optimization. Optimization 51, 627–643 (2002)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Chang, Y.-L., Chen, J.-S., Gu, W.-Z.: On the \(H\)-differentiability of Löwner function with application in symmetric cone complementarity problem. J. Nonlinear and Convex Anal. 14(2), 231–243 (2013)
Chang, Y.-L., Yang, C.-Y., Chen, J.-S.: Smooth and nonsmooth analysis of vector-valued functions associated with circular cones. Nonlinear Anal: Theory Methods Appl. 85, 160–173 (2013)
Chen, J.-S.: The convex and monotone functions associated with second-order cone. Optimization 55(4), 363–385 (2006)
Chen, J.-S.: The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem. J. Glob. Optim. 36, 565–580 (2006)
Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of second-order cone complementarity problem. Math. Program. 104(2–3), 293–327 (2005)
Chen, J.-S., Chen, X., Pan, S.-H., Zhang, J.: Some characterizations for SOC-monotone and SOC-convex functions. J. Glob. Optim. 45(2), 259–279 (2009)
Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cone. Math. Program. 101(1), 95–117 (2004)
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM’s Classics Appl. Math. 5, (1990)
Dür, M.: Copositive Programming-A Survey, Recent Advances in Optimization and Its Applications in Engineering. Springer, Berlin Heidelberg (2010)
Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Clarendon Press, Oxford (1994)
Gale, D., Nikaido, H.: The Jacobian matrix and global univalence of mappings. Math. Annal. 159, 81–93 (1965)
Glineur, F., Terlaky, T.: Conic formulation for \(l_p\)-norm optimization. J. Optim. Theory Appl. 122, 285–307 (2004)
Gowda, M.S.: Inverse and implicit function theorems for \(H\)-differentiable and semismooth functions. Optim. Methods Softw. 19, 443–461 (2004)
Gowda, M.S., Ravindran, G.: Algebraic univalence theorems for nonsmooth functions. J. Math. Anal. Appl. 252, 917–935 (2000)
Jiang, H.: Unconstrained minimization approaches to nonlinear complementarity problems. J. Glob. Optim. 9, 169–181 (1996)
Miao, X.-H., Guo, S.-J., Qi, N., Chen, J.-S.: Constructions of complementarity functions and merit functions for circular cone complementarity problem, submitted manuscript, (2014)
Pan, S.-H., Chang, Y.-L., Chen, J.-S.: Stationary point conditions for the FB merit function associated with symmetric cones. Oper. Res. Lett. 38(5), 372–377 (2010)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer-Verlag, Berlin (1998)
Tawhid, M.A.: On the local uniqueness of solutions of variational inequalities under \(H\)-differentiability. J. Optim. Theory Appl. 113, 149–164 (2002)
Tawhid, M.A.: On strictly semi-monotone (semi-monotone) properties in nonsmooth functions. Appl. Math. Comput. 175, 1609–1618 (2006)
Tawhid, M.A.: An application of \(H\)-differentiability to nonnegative and unrestricted generalized complementarity problems. Comput. Optim. Appl. 39, 51–74 (2008)
Tawhid, M.A., Gowda, M.S.: On two applications of H-differentiability to optimization and complementarity problems. Comput. Optim. Appl. 17, 279–299 (2000)
Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)
Zhou, J.-C., Chen, J.-S.: Properties of circular cone and spectral factorization associated with circular cone. J. Nonlinear Convex Anal. 14, 807–816 (2013)
Zhou, J.-C., Chen, J.-S.: The vector-valued functions associated with circular cones. Abstr. Appl. Anal. 2014, Article ID 603542, 21, (2014)
Zhou, J.-C., Chen, J.-S., Hung, H.-F.: Circular cone convexity and some inequalities associated with circular cones. J. Inequalities Appl. 2013, Article ID 571, 17, (2013)
Zhou, J.-C., Chen, J.-S., Mordukhovich, B.S.: Variational Analysis of circular cone programs. Optimization 64, 113–147 (2015)
Acknowledgments
We are gratefully indebted to anonymous referees and the editor for their valuable suggestions that help us to essentially improve the paper a lot. In particular, the content of Sect. 5 is inspired by one referee’s suggestion.
Author information
Authors and Affiliations
Corresponding author
Additional information
Jinchuan Zhou’s work is supported by National Natural Science Foundation of China (11101248, 11271233, 11171247) and Shandong Province Natural Science Foundation (ZR2012AM016).
Jein-Shan Chen’s work is supported by Ministry of Science and Technology, Taiwan.
Rights and permissions
About this article
Cite this article
Zhou, J., Chang, YL. & Chen, JS. The H-differentiability and calmness of circular cone functions. J Glob Optim 63, 811–833 (2015). https://doi.org/10.1007/s10898-015-0312-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-015-0312-5