Soft Computing (2020) 24:1539–1552 https://doi.org/10.1007/s00500-019-03984-7 METHODOLOGIES AND APPLICATION Application of a new accelerated algorithm to regression problems Avinash Dixit 1 · D. R. Sahu 2 · Amit Kumar Singh 2 · T. Som 1 Published online: 13 April 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Many iterative algorithms like Picard, Mann, Ishikawa are very useful to solve fixed point problems of nonlinear operators in real Hilbert spaces. The recent trend is to enhance their convergence rate abruptly by using inertial terms. The purpose of this paper is to investigate a new inertial iterative algorithm for finding the fixed points of nonexpansive operators in the framework of Hilbert spaces. We study the weak convergence of the proposed algorithm under mild assumptions. We apply our algorithm to design a new accelerated proximal gradient method. This new proximal gradient technique is applied to regression problems. Numerical experiments have been conducted for regression problems with several publicly available high-dimensional datasets and compare the proposed algorithm with already existing algorithms on the basis of their performance for accuracy and objective function values. Results show that the performance of our proposed algorithm overreaches the other algorithms, while keeping the iteration parameters unchanged. Keywords Nonexpansive mapping · S-iteration method · Regression · Composite minimization problems 1 Introduction Fixed point theory plays very crucial role in the fields of pure and applied mathematics as well as in many other branches of science (see Franklin 1980; Lions and Stam- pacchia 1967; Ege and Karaca 2015; Uko 1993, 1996 and references therein). One of the most fundamental prob- lems in the operator theory is to find fixed points of nonlinear operators. Many problems arising in different areas such as image reconstruction (Byrne 2004), signal processing (Byrne 2004), variational inequality (Mercier 1980), convex feasibility problems (Bauschke and Bor- Communicated by V. Loia. B Amit Kumar Singh amit.bsingh1992@gmail.com Avinash Dixit discover.avi92@gmail.com D. R. Sahu drsahudr@gmail.com T. Som tsom.apm@itbhu.ac.in 1 Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, India 2 Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India wein 1996) can be modeled in the form of fixed point problems: find x ∈ C such that Tx = x , (1) where C is a nonempty closed convex subset of a real Hilbert space H and T : C → C is a nonlinear operator. The solution set of the fixed point problem (1) is denoted by Fix (T ). As we know, many literatures have been published in both direct and iterative technique to find the fixed points of non- expansive mappings. The iterative technique is used to solve problems in information theory, game theory, optimization, etc., by formulating them into fixed point problems. One of the most used iterative techniques was introduced by Mann (1953), which is given as follows: for any initial point x 1 ∈ C , x n+1 = (1 − α n )x n + α n Tx n , for all n ∈ N, (2) where {α n } is real sequence in (0, 1). If T is nonexpansive mapping and iteration parameter {α n } satisfies the condition ∑ ∞ n=1 α n (1 − α n ) =∞. Then, sequence {x n } defined by (2) converges weakly to a fixed point of T. It is well known that the Mann iteration method for the approximation of fixed points of pseudocontractive map- pings may not well behave (see Chidume and Mutangadura 123