Entropy-like proximal algorithms based on a second-order homogeneous distance function for quasi-convex programming

We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x) subject to nonnegative constraints x ≥ 0. The algorithms are based on an entropy-like second-order homogeneous distance function. Under the assumption that the global minimizer set is nonempty and bounded, we prove the full convergence of the sequence generated by the algorithms, and furthermore, obtain two important convergence results through imposing certain conditions on the proximal parameters. One is that the sequence generated will converge to a stationary point if the proximal parameters are bounded and the problem is continuously differentiable, and the other is that the sequence generated will converge to a solution of the problem if the proximal parameter approaches to zero. Numerical experiments are done for a class of quasi-convex optimization problems where the function f(x) is a composition of a quadratic convex function from IR to IR and a continuously differentiable increasing function from IR to IR, and computational results indicate that these algorithms are very promising in finding a global optimal solution to these quasi-convex problems. E-mail: [email protected]. The author’s work was partially supported by the Doctoral Starting-up Foundation (05300161) of GuangDong Province. Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan. FAX: 886-2-29332342, E-mail: [email protected].