TY - JOUR

T1 - Interior proximal methods and central paths for convex second-order cone programming

AU - Pan, Shaohua

AU - Chen, Jein Shan

PY - 2010/11/1

Y1 - 2010/11/1

N2 - We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second-order cones.

AB - We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second-order cones.

KW - Central path

KW - Convergence

KW - Convex second-order cone optimization

KW - Interior proximal methods

KW - Proximal distances with respect to SOCs

UR - http://www.scopus.com/inward/record.url?scp=77955664250&partnerID=8YFLogxK

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U2 - 10.1016/j.na.2010.06.079

DO - 10.1016/j.na.2010.06.079

M3 - Article

AN - SCOPUS:77955664250

VL - 73

SP - 3083

EP - 3100

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 9

ER -