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.
ASJC Scopus subject areas
- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics