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Zbl 0963.90058
Wang, Changyu; Xiu, Naihua
Convergence of the gradient projection method for generalized convex minimization.
(English)
[J] Comput. Optim. Appl. 16, No.2, 111-120 (2000). ISSN 0926-6003; ISSN 1573-2894/e

Summary: This paper develops convergence theory of the gradient projection method by {\it P. H. Calamai} and {\it J. J. Moré} [Math. Program. 39, 93-116 (1987; Zbl 0634.90064)] which, for minimizing a continuously differentiable optimization problem $\min \{f(x): x\in \Omega\}$ where $\Omega$ is a nonempty closed convex set, generates a sequence $x_{k+1}= P(x_k- \alpha_k \nabla f(x_k))$ where the stepsize $\alpha_k> 0$ is chosen suitably. It is shown that, when $f(x)$ is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either $x_k\to x^*$ and $x^*$ is a minimizer (stationary point); or $\|x_k\|\to \infty$, $\arg\min \{f(x): x\in \Omega\}= \emptyset$, and $f(x_k) \downarrow \inf\{f(x): x\in \Omega\}$.
MSC 2000:
*90C30 Nonlinear programming
90C52 Methods of reduced gradient type

Keywords: generalized convex minimization; gradient projection method; global convergence

Citations: Zbl 0634.90064

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