×

The vector-valued functions associated with circular cones. (English) Zbl 1474.49035

Summary: The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. Let \(\mathcal{L}_\theta\) denote the circular cone in \(\mathbb{R}^n\). For a 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 \(\mathcal{L}_\theta\). In this paper, we study properties that this vector-valued function inherits from \(f\), including Hölder continuity, \(B\)-subdifferentiability, \(\rho\)-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

MSC:

49J52 Nonsmooth analysis
90C22 Semidefinite programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chang, Y.-L.; Yang, C.-Y.; Chen, J.-S., Smooth and nonsmooth analyses of vector-valued functions associated with circular cones, Nonlinear Analysis. Theory, Methods & Applications Series A: Theory and Methods, 85, 160-173 (2013) · Zbl 1282.49011 · doi:10.1016/j.na.2013.01.017
[2] Zhou, J.-C.; Chen, J.-S., Properties of circular cone and spectral factorization associated with circular cone, Journal of Nonlinear and Convex Analysis, 14, 4, 807-816 (2013) · Zbl 1294.49007
[3] Chen, J.-S., Alternative proofs for some results of vector-valued functions associated with second-order cone, Journal of Nonlinear and Convex Analysis, 6, 2, 297-325 (2005) · Zbl 1071.49014
[4] Chen, J.-S., The convex and monotone functions associated with second-order cone, Optimization. A Journal of Mathematical Programming and Operations Research, 55, 4, 363-385 (2006) · Zbl 1147.49014 · doi:10.1080/02331930600819514
[5] Chen, J.-S.; Chen, X.; Tseng, P., Analysis of nonsmooth vector-valued functions associated with second-order cones, Mathematical Programming. A Publication of the Mathematical Programming Society B, 101, 1, 95-117 (2004) · Zbl 1065.49013 · doi:10.1007/s10107-004-0538-3
[6] Chen, X.; Qi, H.-D.; Tseng, P., Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complementarity problems, SIAM Journal on Optimization, 13, 4, 960-985 (2003) · Zbl 1076.90042 · doi:10.1137/S1052623400380584
[7] Sun, D. F.; Sun, J., Semismooth matrix-valued functions, Mathematics of Operations Research, 27, 1, 150-169 (2002) · Zbl 1082.49501 · doi:10.1287/moor.27.1.150.342
[8] Chang, Y.-L.; Chen, J.-S., The Hölder continuity of vector-valued function associated with second-order cone, Pacific Journal of Optimization, 8, 1, 135-141 (2012) · Zbl 1258.26010
[9] Wihler, T. P., On the Hölder continuity of matrix functions for normal matrices, JIPAM. Journal of Inequalities in Pure and Applied Mathematics, 10, 4, 1-5 (2009) · Zbl 1180.15022
[10] Qi, L. Q., Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18, 1, 227-244 (1993) · Zbl 0776.65037 · doi:10.1287/moor.18.1.227
[11] Qi, H.-D., A semismooth Newton method for the nearest Euclidean distance matrix problem, SIAM Journal on Matrix Analysis and Applications, 34, 1, 67-93 (2013) · Zbl 1266.49052 · doi:10.1137/110849523
[12] Qi, L. Q.; Sun, J., A nonsmooth version of Newton’s method, Mathematical Programming A, 58, 3, 353-367 (1993) · Zbl 0780.90090 · doi:10.1007/BF01581275
[13] Sun, D. F.; Sun, J., Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems, SIAM Journal on Numerical Analysis, 40, 6, 2352-2367 (2002) · Zbl 1041.65037 · doi:10.1137/S0036142901393814
[14] Sun, D. F.; Sun, J., Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions, Mathematical Programming. A Publication of the Mathematical Programming Society A, 103, 3, 575-581 (2005) · Zbl 1099.90062 · doi:10.1007/s10107-005-0577-4
[15] Wang, Y. N.; Xiu, N. H., Strong semismoothness of projection onto slices of second-order cone, Journal of Optimization Theory and Applications, 150, 3, 599-614 (2011) · Zbl 1231.49012 · doi:10.1007/s10957-011-9834-2
[16] Facchinei, F.; Pang, J.-S., Finite Dimensional Variational Inequalities and Complementarity Problems (2003), New York, NY, USA: Springer, New York, NY, USA · Zbl 1062.90001
[17] Ding, C., An introduction to a class of matrix optimization problems [Ph.D. thesis] (2012), Singapore: National University of Singapore, Singapore
[18] Aubin, J.-P.; Frankowska, H., Set-Valued Analysis (2009), New York, NY, USA: Springer, New York, NY, USA · doi:10.1007/978-0-8176-4848-0
[19] Rockafellar, R. T.; Wets, R. J., Variational Analysis (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3
[20] Shapiro, A., On concepts of directional differentiability, Journal of Optimization Theory and Applications, 66, 3, 477-487 (1990) · Zbl 0682.49015 · doi:10.1007/BF00940933
[21] Zhou, J.-C.; Chen, J.-S.; Hung, H.-F., Circular cone convexity and some inequalities associated with circular cones, Journal of Inequalities and Applications, 2013, 571 (2013) · Zbl 1297.26025 · doi:10.1186/1029-242X-2013-571
[22] Ding, C.; Sun, D. F.; Toh, K.-C., An introduction to a class of matrix cone programming, Mathematical Programming. A Publication of the Mathematical Programming Society A, 144, 1-2, 141-179 (2014) · Zbl 1301.65043 · doi:10.1007/s10107-012-0619-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.