Regularized gradient-projection methods for finding the minimum-norm solution of the constrained convex minimization problem

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• 作者：Ming Tian ; Hui-Fang Zhang
• 关键词：regularized gradient ; projection method ; minimum ; norm ; the constrained convex minimization problem ; variational inequality
• 刊名：Journal of Inequalities and Applications
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1623KB
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X
• 卷排序：2017

文摘

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Assume that g is a real-valued convex function and the gradient ∇g is \(\frac{1}{L}\)-ism with \(L>0\). Let \(0<\lambda <\frac{2}{L+2}\), \(0<\beta_{n}<1\). We prove that the sequence \(\{x_{n}\} \) generated by the iterative algorithm \(x_{n+1}=P_{C}(I-\lambda(\nabla g+\beta_{n}I))x_{n}\), \(\forall n\geq0\) converges strongly to \(q\in U\), where \(q=P_{U}(0)\) is the minimum-norm solution of the constrained convex minimization problem, which also solves the variational inequality \(\langle-q, p-q\rangle\leq0\), \(\forall p\in U\). Under suitable conditions, we obtain some strong convergence theorems. As an application, we apply our algorithm to solving the split feasibility problem in Hilbert spaces.