Averaged Mappings and the Gradient-Projection Algorithm

• Published in: Journal of optimization theory and applications Vol. 150; no. 2; pp. 360 - 378
• Main Author: Xu, Hong-Kun
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2011; Springer; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Constraints; Convergence; Engineering; Exact sciences and technology; Hilbert space; Mapping; Mathematical programming; Mathematics; Mathematics and Statistics; Minimization; Operational research and scientific management; Operational research. Management science; Operations Research/Decision Theory; Optimization; Regularization; Regularization methods; Studies; Theory of Computation; Relaxed gradient-projection algorithm; Averaged mapping; Constrained convex minimization; Minimum-norm; Maximal monotone operator; Regularization; Gradient-projection algorithm; Monotonic function; Strong convergence; Maximal operator; Monotone operator; Convex programming; Search algorithm; Constrained optimization; Weak convergence; Norm minimization; Hilbert space; Infinite dimension; Numerical convergence
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-011-9837-z

It is well known that the gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this article, we first provide an alternative averaged mapping approach to the GPA. This approach is operator-oriented in nature. Since, in general, in infinite-dimensional Hilbert spaces, GPA has only weak convergence, we provide two modifications of GPA so that strong convergence is guaranteed. Regularization is also applied to find the minimum-norm solution of the minimization problem under investigation.