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Discriminant analysis is a useful classification method. Variable selectionfor discriminant analysis is becoming more and more im- portant in ahigh-dimensional setting. This paper is concerned with the binary-classproblems of main and interaction effects selection for the quadraticdiscriminant analysis. We propose a new penalized quadratic discriminantanalysis (QDA) for variable selection in binary classification. Under sparsityassumption on the relevant variables, we conduct a penalized liner regressionto derive sparse QDA by plug- ging the main and interaction effects in themodel. Then the QDA problem is converted to a penalized sparse ordinary leastsquares op- timization by using the composite absolute penalties (CAP). Coor-dinate descent algorithm is introduced to solve the convex penalized leastsquares. The penalized linear regression can simultaneously se- lect the mainand interaction effects, and also conduct classification. Compared with theexisting methods of variable selection in QDA, the extensive simulation studiesand two real data analyses demon- strate that our proposed method works welland is robust in the performance of variable selection and classification.

Main and Interaction Effects Selection for Quadratic Discriminant Analysis via Penalized Linear Regression