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Differential properties of Euclidean projection onto power cone

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Abstract

In this paper, we study differential properties of Euclidean projection onto the high dimensional power cone

$$\begin{aligned} K^{\alpha }_{m,n}=\left\{ (x,z)\in \mathbb {R}^m_+ \times \mathbb {R}^n, \left| \left| {z} \right| \right| \le \prod \nolimits _{i=1}^m x_i^{\alpha _i}\right\} , \end{aligned}$$

where \(0<\alpha _i, \sum \nolimits _{i=1}^m \alpha _i=1\). Projector’s formulas, its directional derivative formulas, its first order Fréchet derivative formulas are considered.

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References

  • Alizadeh F, Goldfarb D (2003) Second-order cone programming. Math Program 95(1):3–51

    Article  MATH  MathSciNet  Google Scholar 

  • Bertsekas DP (1999) Nonlinear programming. Athena Scientific, Belmont

    MATH  Google Scholar 

  • Chares PR (2009) Cones and Interior-point algorithms for structured convex optimization involving powers and exponentials. PhD thesis, Uni. Catholique de Louvain

  • Chares R, Glineur F (2008) An interior-point method for the single-facility location problem with mixed norms using a conic formulation. Math Methods Oper Res 68(3):383–405

    Article  MATH  MathSciNet  Google Scholar 

  • Chen B, Xiu N (1999) A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen–Mangasarian smoothing functions. SIAM J Optim 9:605–623

    Article  MATH  MathSciNet  Google Scholar 

  • Chen J-S, Tseng P (2005) An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math Program 104(2–3):293–327

    Article  MATH  MathSciNet  Google Scholar 

  • Chua CB (2009) A t-algebraic approach to primal-dual interior-point algorithms. SIAM J Optim 20(1):503–523

    Article  MATH  MathSciNet  Google Scholar 

  • Chua CB, Hien LTK (2014) Superlinearly convergent smoothing Newton continuation algorithms for variational inequalities over definable sets. Technical report, Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

  • Chua CB, Li Z (2013) A barrier-based smoothing proximal point algorithm for NCPs over closed convex cones. SIAM J Optim 23:745–769

    Article  MATH  MathSciNet  Google Scholar 

  • Ding C, Sun D, Toh K-C (2014) An introduction to a class of matrix cone programming. Math Program 144(1–2):141–179

    Article  MATH  MathSciNet  Google Scholar 

  • Facchinei F, Pang J-S (2003) Finite-dimensional variational inequalities and complementarity problems, vol I. Springer, New York

    Google Scholar 

  • Facchinei F, Pang J-S (2003) Finite-dimensional variational inequalities and complementarity problems, vol II. Springer, New York

    Google Scholar 

  • Hayashi S, Yamashita N, Fukushima M (2005) A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J Optim 15(2):593–615

    Article  MATH  MathSciNet  Google Scholar 

  • Kanzow C, Ferenczi I, Fukushima M (2009) On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J Optim 20(1):297–320

    Article  MATH  MathSciNet  Google Scholar 

  • Kanzow C, Pieper H (1999) Jacobian smoothing methods for nonlinear complementarity problems. SIAM J Optim 9(2):342–373

    Article  MATH  MathSciNet  Google Scholar 

  • Kong L, Tunçel L, Xiu N (2009) Clarke generalized Jacobian of the projection onto symmetric cones. Set-Valued Var Anal 17(2):135–151

    Article  MATH  MathSciNet  Google Scholar 

  • Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applications of second-order cone programming. Linear Algebra Appl 284(1–3):193–228

    Article  MATH  MathSciNet  Google Scholar 

  • Malick J, Sendov H (2006) Clarke generalized Jacobian of the projection onto the cone of positive semidefinite matrices. Set-Valued Anal 14(3):273–293

    Article  MATH  MathSciNet  Google Scholar 

  • Matsukawa Y, Yoshise A (2012) A primal barrier function phase I algorithm for nonsymmetric conic optimization problems. Jpn J Ind Appl Math 29:499–517

    Article  MATH  MathSciNet  Google Scholar 

  • Nesterov YuE, Nemirovskii AS (1994) Interior point polynomial algorithms in convex programming. SIAM Publication, Philadelphia

    Book  MATH  Google Scholar 

  • Qi L, Sun J (1993) A nonsmooth version of Newton’s method. Math Program 58(1–3):353–367

    Article  MATH  MathSciNet  Google Scholar 

  • Shapiro A (1994) Existence and differentiability of metric projection in Hilbert spaces. SIAM J Optim 4(1):130–141

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Le Thi Khanh Hien

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  1. Le Thi Khanh Hien
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Correspondence to Le Thi Khanh Hien.

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Le Thi Khanh Hien was supported by a SINGA scholarship.

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Hien, L.T.K. Differential properties of Euclidean projection onto power cone. Math Meth Oper Res 82, 265–284 (2015). https://doi.org/10.1007/s00186-015-0514-0

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  • DOI: https://doi.org/10.1007/s00186-015-0514-0

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