A Family of Optimal Settings for the Dai-Liao Parameter Based on An Ellipsoid Norm Least-Squares Problem with Application to A Revised Model of the Nonnegative Matrix Factorization

(Communicated by Xinwei Liu) Due to the significant need for efficiently handling the huge data sets which mostly emerge in the contemporary world models, here we focus on modifying a well-known memoryless continuous optimization algorithm as well as improving a classic data compression model. With these issues at the forefront, firstly we suggest two optimal settings for the parameter of the Dai--Liao conjugate gradient method by approaching the search direction of the method to that of the efficient memoryless BFGS quasi--Newton method in an ellipsoid norm least-squares framework. Then, we deal with modifying the optimization model of the nonnegative matrix factorization problem. More precisely, we add penalty terms to the classic least-squares models of the nonnegative matrix factorization subproblems in the popular alternative solution process, as a plan to control collinearity between the columns/rows of the factorization elements. We put our theoretical assertions to the test on the CUTEr problems as well as several random nonnegative matrix factorization cases, and assess the outputs in various aspects. The outputs generally show the acceptable impact of our modifications.