Conic Relaxations for Semi-supervised Support Vector Machines

• Published in: Journal of optimization theory and applications Vol. 169; no. 1; pp. 299 - 313
• Main Authors: Bai, Yanqin, Yan, Xin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2016; Springer Nature B.V
• Subjects: Accuracy; Applications of Mathematics; Approximation; Artificial intelligence; Calculus of Variations and Optimal Control; Optimization; Classification; Conics; Convex analysis; Engineering; Equivalence; Integer programming; Linear programming; Machine learning; Mathematical analysis; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Programming; Quadratic programming; Studies; Support vector machines; Theory of Computation; Convex conic relaxation; Semi-supervised support vector machines; 90C11; 90C22; Semi-definite relaxation; 62H30; Completely positive programming; Doubly nonnegative relaxation
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-015-0843-4

Semi-supervised support vector machines arise in machine learning as a model of mixed integer programming problem for classification. In this paper, we propose two convex conic relaxations for the original mixed integer programming problem. The first one is a new semi-definite relaxation, and its possibly maximal ratio of the optimal value is estimated approximately. The second one is a doubly nonnegative relaxation, which is relaxed from a well-known conic programming problem called completely positive programming problem that is equivalent to the original problem. Furthermore, we prove that the doubly nonnegative relaxation is tighter than the semi-definite relaxation. Finally, the numerical results show that two proposed relaxations not only generate proper classifiers but also outperform some existing methods in classification accuracy.