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On Solving of Constrained Convex Minimize Problem Using Gradient Projection Method

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  • Taksaporn Sirirut
  • Pattanapong Tianchai

Abstract

Let and be closed convex subsets of real Hilbert spaces and , respectively, and let be a strictly real-valued convex function such that the gradient is an -ism with a constant . In this paper, we introduce an iterative scheme using the gradient projection method, based on Mann’s type approximation scheme for solving the constrained convex minimization problem (CCMP), that is, to find a minimizer of the function over set . As an application, it has been shown that the problem (CCMP) reduces to the split feasibility problem (SFP) which is to find such that where is a linear bounded operator. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the CCMP and the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (2017) and many others. The data availability for the proposed SFP is shown and the example of this problem is also shown through numerical results.

Suggested Citation

  • Taksaporn Sirirut & Pattanapong Tianchai, 2018. "On Solving of Constrained Convex Minimize Problem Using Gradient Projection Method," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2018, pages 1-10, October.
    • Handle: RePEc:hin:jijmms:1580837
    DOI: 10.1155/2018/1580837
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    References listed on IDEAS

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    1. Hong-Kun Xu, 2011. "Averaged Mappings and the Gradient-Projection Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 360-378, August.
    2. Yonghong Yao & Shin Min Kang & Wu Jigang & Pei-Xia Yang, 2012. "A Regularized Gradient Projection Method for the Minimization Problem," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-9, February.
    3. Ming Tian & Min-Min Li, 2014. "A General Iterative Method for Solving Constrained Convex Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 202-207, July.
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