TY - JOUR

T1 - Entropy-like proximal algorithms based on a second-order homogeneous distance function for quasi-convex programming

AU - Pan, Shaohua

AU - Chen, Jein Shan

N1 - Funding Information:
Shaohua Pan 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. Jein-Shan Chen work is partially supported by National Science Council of Taiwan.

PY - 2007/12

Y1 - 2007/12

N2 - We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x) subject to non-negative 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 parameters approach 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 Rn to R and a continuously differentiable increasing function from R to R, and computational results indicate that these algorithms are very promising in finding a global optimal solution to these quasi-convex problems.

AB - We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x) subject to non-negative 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 parameters approach 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 Rn to R and a continuously differentiable increasing function from R to R, and computational results indicate that these algorithms are very promising in finding a global optimal solution to these quasi-convex problems.

KW - Entropy-like distance

KW - Proximal-like method

KW - Quasi-convex programming

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U2 - 10.1007/s10898-007-9156-y

DO - 10.1007/s10898-007-9156-y

M3 - Article

AN - SCOPUS:35748949858

VL - 39

SP - 555

EP - 575

JO - Journal of Global Optimization

JF - Journal of Global Optimization

SN - 0925-5001

IS - 4

ER -