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Monotonicity and Circular Cone Monotonicity Associated with Circular Cones

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Abstract

The circular cone \(\mathcal {L}_{\theta }\) is not self-dual under the standard inner product and includes second-order cone as a special case. In this paper, we focus on the monotonicity of \(f^{\mathcal {L}_{\theta }}\) and circular cone monotonicity of f. Their relationship is discussed as well. Our results show that the angle πœƒ plays a different role in these two concepts.

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References

  1. 1.

    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)

  2. 2.

    Bonnans, J.F., Ramirez, H.: Perturbation analysis of second-order cone programming problems. Math. Program. 104, 205–227 (2005)

  3. 3.

    Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104, 293–327 (2005)

  4. 4.

    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)

  5. 5.

    Chen, J.-S.: The convex and monotone functions associated with second-order cone. Optimization 55(4), 363–385 (2006)

  6. 6.

    Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cone. Math. Program. 101, 95–117 (2004)

  7. 7.

    Chen, J.-S., Chen, X., Pan, S.-H., Zhang, J.: Some characterizations for SOC-monotone and SOC-convex functions. J. Glob. Optim. 45, 259–279 (2009)

  8. 8.

    Chen, J.-S., Liao, T.-K., Pan, S.-H.: Using Schur complement theorem to prove convexity of some SOC-functions. J. Nonlinear Convex Anal. 13, 421–431 (2012)

  9. 9.

    Chen, J.-S., Pan, S.-H.: A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs. Pac. J. Optim. 8, 33–74 (2012)

  10. 10.

    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

  11. 11.

    Dattorro, J.: Convex Optimization and Euclidean Distance Geometry. Meboo Publishing, California (2005)

  12. 12.

    Ding, C., Sun, D., Toh, K.C.: An introduction to a class of matrix cone programming. Math. Program. 144, 141–179 (2014)

  13. 13.

    Faraut, U., KorΓ‘nyi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs. Oxford University Press, New York (1994)

  14. 14.

    Fukushima, M., Luo, Z.Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002)

  15. 15.

    Guler, O., Tuncel, L.: Characterization of the barrier parameter of homogeneous convex cones. Math. Program. 81, 55–76 (1998)

  16. 16.

    Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15, 593–615 (2005)

  17. 17.

    Kanzow, C., Ferenczi, I., Fukushima, M.: On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J. Optim. 20, 297–320 (2009)

  18. 18.

    Kong, L.C., Tuncel, L., Xiu, N.H.: Monotonicity of Lowner operators and its applications to symmetric cone complementarity problems. Math. Program. 133, 327–336 (2012)

  19. 19.

    Liu, Y.J., Zhang, L.W.: Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Nonlinear Anal. Theory Methods Appl. 67, 1359–1373 (2007)

  20. 20.

    Miao, X.-H., Guo, S.-J., Qi, N., Chen, J.-S.: Constructions of complementarity functions and merit functions for circular cone complementarity problem. Comput. Optim. Appl. 63, 495–522 (2016)

  21. 21.

    Pan, S.-H., Chen, J.-S.: A class of interior proximal-like algorithms for convex second-order cone programming. SIAM J. Optim. 19, 883–910 (2008)

  22. 22.

    Pan, S.-H., Chen, J.-S.: An R-linearly convergent nonmonotone derivative-free method for symmetric cone complementarity problems. Adv. Model. Optim. 13, 185–211 (2011)

  23. 23.

    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, 372–377 (2010)

  24. 24.

    Pan, S.-H., Chiang, Y., Chen, J.-S.: SOC-Monotone and SOC-convex functions v.s. matrix-monotone and matrix-convex functions. Linear Algebra Appl. 437 (5), 1264–1284 (2012)

  25. 25.

    Roos, K. http://www.isa.ewi.tudelft.nl/~roos/course/WI4218/SLemma.pdf

  26. 26.

    Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, New York (1998)

  27. 27.

    Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior point algorithms to symmetric cones. Math. Program. 96, 409–438 (2003)

  28. 28.

    Sun, D., Sun, J.: Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Math. Program. 103, 575–581 (2005)

  29. 29.

    Truong, V.A., Tuncel, L.: Geometry of homogeneous cones: duality mapping and optimal selfconcordant barriers. Math. Program. 100, 295–316 (2004)

  30. 30.

    Xue, G., Ye, Y.: An efficient algorithm for minimizing a sum of p-norms. SIAM J. Optim. 10, 551–579 (2000)

  31. 31.

    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)

  32. 32.

    Zhou, J.-C., Chen, J.-S.: On the vector-valued functions associated with circular cones. Abstr. Appl. Anal. 2014(603542), 21 (2014)

  33. 33.

    Zhou, J.-C., Chen, J.-S., Hung, H.-F.: Circular cone convexity and some inequalities associated with circular cones. J. Inequalities Appl. 2013(571), 17 (2013)

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Author information

Correspondence to Jein-Shan Chen.

Additional information

The author’s work is supported by National Natural Science Foundation of China (11101248, 11271233) and Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2012AM016).

The author’s work is supported by Ministry of Science and Technology, Taiwan.

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Zhou, J., Chen, J. Monotonicity and Circular Cone Monotonicity Associated with Circular Cones. Set-Valued Var. Anal 25, 211–232 (2017). https://doi.org/10.1007/s11228-016-0374-7

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Keywords

  • Circular cone
  • Monotonicity
  • Circular cone monotonicity

Mathematics Subject Classification (2010)

  • 26A27
  • 26B35
  • 49J52
  • 65K10