Convergence of a block coordinate descent method for nondifferentiable minimization. (English) Zbl 1006.65062

Tseng, P.

J. Optimization Theory Appl. 109, No. 3, 475-494 (2001).

Author’s abstract: We study the convergence properties of a (block) coordinate descent method applied to minimize a nondifferentiable (nonconvex) function \(f(x_1,\dots,x_N)\) with certain separability and regularity properties.
Assuming that \(f\) is continuous on a compact level set, the subsequence convergence of the iterates to a stationary point is shown when either \(f\) is pseudoconvex in every pair of coordinate blocks from among \(N-1\) coordinate blocks or \(f\) has at most one minimum in each of \(N-2\) coordinate blocks. If \(f\) is quasiconvex and hemivariate in every coordinate block, then the assumptions of continuity of \(f\) and compactness of the level set may be relaxed further.
These results are applied to derive new (and old) convergence results for the proximal minimization algorithm, an algorithm of S. Arimoto [IEEE Trans. Inf. Theory 18, 14-20 (1972; Zbl 0227.94011)] and of R. E. Blahut [ibid. 18, 460-473 (1972; Zbl 0247.94017)], and an algorithm of S.-P. Han [Math. Oper. Res. 14, No. 2, 237-248 (1989; Zbl 0671.90062)]. They are applied also to a problem of blind source separation.

Reviewer: Berwin A.Turlach (Crawley)

MSC:

65K05	Numerical mathematical programming methods
90C26	Nonconvex programming, global optimization
90C30	Nonlinear programming

Keywords:

block coordinate descent method; nondifferentiable minimization; stationary point; Gauss-Seidel method; convergence; quasiconvex functions; pseudoconvex functions; proximal minimization algorithm

Citations:

Zbl 0227.94011; Zbl 0247.94017; Zbl 0671.90062