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A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting.

Francisco J Aragón-Artacho¹, Radu I Boţ², David Torregrosa-Belén¹

¹Department of Mathematics, University of Alicante, San Vicente del Raspeig, 03690 Alicante Spain.

Numerical Algorithms

|April 11, 2023

Summary

We introduce a novel primal-dual splitting algorithm for composite monotone inclusion problems. This method achieves minimal lifting, reducing computational complexity for applications like image deblurring.

Keywords:

Minimal lifting Monotone inclusion Monotone operator Primal-dual algorithm Splitting algorithm

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Area of Science:

Optimization Theory
Numerical Analysis
Applied Mathematics

Background:

Composite monotone inclusion problems are crucial in various fields, including optimization and machine learning.
Existing resolvent splitting algorithms often require high-dimensional product spaces and multiple resolvent evaluations.
The minimal lifting concept aims to simplify the underlying fixed-point operator's space.

Purpose of the Study:

To develop a novel primal-dual splitting algorithm for composite monotone inclusion problems.
To introduce a method with minimal lifting, reducing the dimension of the product space.
To extend the theory of resolvent splitting algorithms concerning minimal lifting.

Main Methods:

Development of a new primal-dual splitting algorithm.
Application of the minimal lifting technique to reduce the product space dimension.
Convergence analysis of the proposed algorithm.
Extension of the minimal lifting theorem to resolvent parameters.

Main Results:

The first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting is established.
The proposed algorithm reduces the dimension of the product space without extra resolvent operator evaluations.
Convergence of the new algorithm is proven.
The algorithm's efficacy is demonstrated on an image deblurring and denoising problem.

Conclusions:

The novel primal-dual splitting algorithm offers an efficient approach to solving composite monotone inclusion problems.
This work advances the theory of resolvent splitting algorithms, particularly regarding minimal lifting.
The algorithm shows practical utility in image processing applications.