The Algebraic Structure of the Arbitrary-Order Cone

We study and analyze the algebraic structure of the arbitrary-order cones. We show that, unlike popularly perceived, the arbitrary-order cone is self-dual for any order greater than or equal to 1. We establish a spectral decomposition, consider the Jordan algebra associated with this cone, and prove that this algebra forms a Euclidean Jordan algebra with a certain inner product. We generalize some important notions and properties in the Euclidean Jordan algebra of the second-order cone to the Euclidean Jordan algebra of the arbitrary-order cone.

[1]  Alexander Vinel,et al.  Polyhedral approximations in p-order cone programming , 2014, Optim. Methods Softw..

[2]  P. Krokhmal.  Higher moment coherent risk measures , 2007 .

[3]  J. Faraut,et al.  Analysis on Symmetric Cones , 1995 .

[4]  Farid Alizadeh,et al.  Extension of primal-dual interior point algorithms to symmetric cones , 2003, Math. Program..

[5]  Elisabetta Allevi,et al.  Stochastic Second-Order Cone Programming in Mobile Ad Hoc Networks , 2009 .

[6]  Yurii Nesterov,et al.  Towards non-symmetric conic optimization , 2012, Optim. Methods Softw..

[7]  David S. Watkins,et al.  Fundamentals of Matrix Computations: Watkins/Fundamentals of Matrix Computations , 2005 .

[8]  David S. Watkins,et al.  Fundamentals of matrix computations , 1991 .

[9]  Alexander Vinel,et al.  On Valid Inequalities for Mixed Integer p-Order Cone Programming , 2014, J. Optim. Theory Appl..

[10]  Xinhe Miao,et al.  A Note on the Paper “The Algebraic Structure of the Arbitrary-Order Cone” , 2017, J. Optim. Theory Appl..

[11]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

[12]  Luca Bertazzi,et al.  Solution Approaches for the Stochastic Capacitated Traveling Salesmen Location Problem with Recourse , 2014, Journal of Optimization Theory and Applications.

[13]  Stan Uryasev,et al.  Two pairs of families of polyhedral norms versus $$\ell _p$$ℓp-norms: proximity and applications in optimization , 2016, Math. Program..

[14]  G. Lumer.  SEMI-INNER-PRODUCT SPACES , 1961 .

[15]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[16]  Yinyu Ye,et al.  An Efficient Algorithm for Minimizing a Sum of p-Norms , 1999, SIAM J. Optim..

[17]  Baha Alzalg,et al.  Stochastic second-order cone programming: Applications models , 2012 .

[18]  Pavlo A. Krokhmal,et al.  Risk optimization with p-order conic constraints: A linear programming approach , 2010, Eur. J. Oper. Res..

[19]  F. Glineur,et al.  Conic Formulation for lp-Norm Optimization , 2000 .

[20]  N. K. Sahu,et al.  Frames in Semi-inner Product Spaces , 2015 .

[21]  Vladimír Müller,et al.  Spectral Theory of Linear Operators: and Spectral Systems in Banach Algebras , 2003 .

[22]  M. Kloft,et al.  l p -Norm Multiple Kernel Learning , 2011 .