Quaternion Matrix Optimization: Motivation and Analysis

The class of quaternion matrix optimization (QMO) problems, with quaternion matrices as decision variables, has been widely used in color image processing and other engineering areas in recent years. However, optimization theory for QMO is far from adequate. The main objective of this paper is to provide necessary theoretical foundations on optimality analysis, in order to enrich the contents of optimization theory and to pave way for the design of efficient numerical algorithms as well. We achieve this goal by conducting a thorough study on the first-order and second-order (sub)differentiation of real-valued functions in quaternion matrices, with a newly introduced operation called R-product as the key tool for our calculus. Combining with the classical optimization theory, we establish the first-order and the second-order optimality analysis for QMO. Particular treatments on convex functions, the similar to 0-norm and the rank function in quaternion matrices are tailored for a sparse low rank QMO model, arising from color image denoising, to establish its optimality conditions via stationarity.