Greedy Projected Gradient-Newton Method for Sparse Logistic Regression

Fast ramp fraction loss SVM classifier with low computational complexity for pattern classification

The support vector machine (SVM) is a powerful tool for pattern classification thanks to its outstanding efficiency. However, when encountering extensive classification tasks, the considerable computational complexity may present a substantial barrier. To reduce computational complexity, the novel ramp fraction loss SVM model called Lrf-SVM is introduced. The aim of this model is to simultaneously achieve sparsity and robustness. Utilizing our proposed proximal stationary point, we develop a novel optimality theory to the nonsmooth and nonconvex Lrf-SVM. Drawing upon this innovative theory, a novel efficient alternating direction method of multipliers (ADMM) incorporating a working set with low computational complexity will be introduced for handing Lrf-SVM. Moreover, our algorithm has shown that it can achieve global convergence. Our algorithm has been shown to be highly efficient through numerical experiments, surpassing nine other top solvers regarding number of support vectors, speed of computation, accuracy of classification and robust to outliers. For example, for addressing the real dataset over 107 samples, our algorithm can finish the classification in just 18.67 s, a notable enhancement in comparison to other algorithms that need at least 605.3 s.

Sparse Index Tracking With K-Sparsity or ϵ-Deviation Constraint via ℓ0-Norm Minimization

Sparse index tracking, as one of the passive investment strategies, is to track a benchmark financial index via constructing a portfolio with a few assets in a market index. It can be considered as parameter learning in an adaptive system, in which we periodically update the selected assets and their investment percentages based on the sliding window approach. However, many existing algorithms for sparse index tracking cannot explicitly and directly control the number of assets or the tracking error. This article formulates sparse index tracking as two constrained optimization problems and then proposes two algorithms, namely, nonnegative orthogonal matching pursuit with projected gradient descent (NNOMP-PGD) and alternating direction method of multipliers for l0 -norm (ADMM- l0 ). The NNOMP-PGD aims at minimizing the tracking error subject to the number of selected assets less than or equal to a predefined number. With the NNOMP-PGD, investors can directly and explicitly control the number of selected assets. The ADMM- l0 aims at minimizing the number of selected assets subject to the tracking error that is upper bounded by a preset threshold. It can directly and explicitly control the tracking error. The convergence of the two proposed algorithms is also presented. With our algorithms, investors can explicitly and directly control the number of selected assets or the tracking error of the resultant portfolio. In addition, numerical experiments demonstrate that the proposed algorithms outperform the existing approaches.