Lagrange multipliers and duality with applications to constrained support vector machine

In this paper, we employ the concept of quasi-relative interior to analyse the method of Lagrange multipliers and establish strong Lagrangian duality for nonsmooth convex optimization problems in Hilbert spaces. Then, we generalize the classical support vector machine (SVM) model by incorporating a new geometric constraint or a regularizer on the separating hyperplane, serving as a regularization mechanism for the SVM model. This new SVM model is examined using Lagrangian duality and other convex optimization techniques from both theoretical and numerical perspectives via a new subgradient algorithm as well as a primal-dual method.