Solving an Augmented Nonnegative Matrix Factorization Model by Modified Scaled Nonmonotone Memoryless BFGS Methods Devised Based on the Ellipsoid Vector Norm

ABSTRACT We suggest a modified version of the nonnegative matrix factorization problem, adding penalty terms to the model with the aim of taking control of the condition number of the decomposition elements. This measure is taken to reinforce computational stability in the solution path of the relevant optimization model. Then, using the ellipsoid vector norm in the Barzilai–Borwein least‐squares model, we gain scaling parameters for the memoryless BFGS updating formula. Therefore, two augmented versions of the Oren–Spedicato self‐scaling parameter are proposed when the BFGS and DFP formulas for the Hessian are employed as the matrix index of the ellipsoid norm. A similar analysis is conducted to obtain two augmented versions of the Oren–Luenberger self‐scaling parameter using the mentioned quasi‐Newton formulas for the inverse Hessian. We address the global convergence of the method under a modified version of a nonmonotone backtracking line search. Then, we discuss accelerating the method using a local memoryless quadratic model considering the given scaling parameters as scalar approximations of the Hessian. Finally, the computational merits of the given augmented nonnegative matrix factorization model as well as the effectiveness of the proposed scaling parameters are assessed on some standard and randomly generated test problems, and also, a classic database of facial images. The outputs generally support our analytical spectrum.