In this paper, a kernel-free soft quadratic surface support vector machine model is proposed for binary classification directly using a quadratic function for separation. Properties (including the solvability, uniqueness and support vector representation of the optimal solution) of the proposed model are derived. Results of computational experiments on some artificial and real-world classifying data sets indicate that the proposed soft quadratic surface support vector machine model may outperform Dagher’s quadratic model and other soft support vector machine models with a Quadratic or Gaussian kernel in terms of the classification accuracy and robustness.

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We propose a new augmented Lagrangian (AL) method for solving the mathematical program with complementarity constraints (MPCC), where the complementarity constraints are left out of the AL function and treated directly. Two observations motivate us to propose this method: The AL subproblems are closer to the original problem in terms of the constraint structure; and the AL subproblems can be solved efficiently by a nonmonotone projected gradient method, in which we have closed-form solutions at each iteration. The former property helps us show that the proposed method converges globally to an M-stationary (better than C-stationary) point under MPCC relaxed constant positive linear dependence condition. Theoretical comparison with existing AL methods demonstrates that the proposed method is superior in terms of the quality of accumulation points and the strength of assumptions. Numerical comparison, based on problems in MacMPEC, validates the theoretical results.

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