For the solution of a type of discrete global optimization problems, we provide a unique non-parameter filled function. A relevant solution algorithm that combines the discrete steepest approach with numerical and theoretical characterization of the suggested filled function is created. The suggested approach is a global optimization technique that revises the objective function into a filled function through optimizing a filled function built on earlier found minimum points to find better local minimum points step by step. A global minimum point can be derived through iterating over these programs. The numerical outcomes generated demonstrate the effectiveness of the filled function method. Moreover, we verify that the presented method is promising through the application of the filled function to the assignment problem.

We consider a fundamental problem in the field of machine learning—structural risk minimization, which can be represented as the average of a large number of smooth component functions plus a simple and convex (but possibly non-smooth) function. In this paper, we propose a novel proximal variance reducing stochastic method building on the introduced Point-SAGA. Our method achieves two proximal operator calculations by combining the fast Douglas–Rachford splitting and refers to the scheme of the FISTA algorithm in the choice of momentum factors. We show that the objective function value converges to the iteration point at the rate of $𝒪\mathsf{}\mathrm{(}\mathrm{1}\mathrm{/}k\mathrm{)}$ when each loss function is convex and smooth. In addition, we prove that our method achieves a linear convergence rate for strongly convex and smooth loss functions. Experiments demonstrate the effectiveness of the proposed algorithm, especially when the loss function is ill-conditioned with good acceleration.