Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems
- PDF / 2,064,350 Bytes
- 25 Pages / 595.276 x 790.866 pts Page_size
- 43 Downloads / 181 Views
METHODOLOGIES AND APPLICATION
Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems D. R. Sahu1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper studies the convergence rates of two accelerated computational methods without assuming nonexpansivity of the underlying operators with convex and affine domains in infinite-dimensional Hilbert spaces. One method is a noninertial 1 method, and its convergence rate is estimated as RT ,{xn } (n) = o √n in worst case. The other is an inertial method, and its convergence rate is estimated as RT ,{yn } (n) = o √1n under practical conditions. Then, we apply our results to give new results on convergence rates for solving generalized split common fixed-point problems for the class of demimetric operators. We also apply our results to variational inclusion problems and convex optimization problems. Our results significantly improve and/or develop previously discussed fixed-point problems and splitting problems and related algorithms. To demonstrate the applicability of our methods, we provide numerical examples for comparisons and numerical experiments on regression problems for publicly available high-dimensional real datasets taken from different application domains. Keywords Convex optimization · Fixed-point set · Nonexpansive · Proximal gradient method · Fixed-point algorithm · S-iteration method · Maximal monotone operator · Lasso problem
1 Introduction 1.1 Multiple-set split feasibility problems Various problems in mathematics and science can be formulated as finding an element in the intersection of convex sets, which is called the convex feasibility problem (CFP). A number of inverse problems can be formulated as convex feasibility problem, for example, phase retrievals, medical image reconstructions and image processing. A special case of CFP is called the split feasibility problem (SFP). It was first introduced by Censor and Elfving (1994) as follows: find a point x ∈ C such that Ax ∈ Q,
(1)
where C is a closed convex subset of a Hilbert space X , Q is a closed convex subset of a Hilbert space Y , and A : X → Y Communicated by V. Loia.
B 1
D. R. Sahu [email protected] Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
is a bounded linear operator. When Q = {b} is a singleton set, SFP(1) reduces to the constrained linear inverse problem (CLIP). Let U : X → X and T : Y → Y be operators. Then, the mathematical formulation of two-set split common fixedpoint problem (SCFPP) is
to find a point x ∈ Fix(U ) such that Ax ∈ Fix(T ).
(2)
Clearly, the split common fixed-point problem (SCFP) is a natural generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP). The two-set SCFPP (2) was first introduced by Censor and Segal (2009) in finite-dimensional Hilbert spaces for directed operators U and T . In recent years, the split common fixed-point problems and the multiple-set split feasibility problems (MSSFP) and their
Data Loading...
