Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming
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In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ℝ++. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 ), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 ; Chen and Teboulle in SIAM J. Optim. 3:538–543 ), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the proposed algorithm is bounded and that every accumulation point is a solution to the considered problem.
KeywordsBregman functions Quasi D-functions Proximal-like methods Convex second-order cone programming
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