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