The kernel function plays a central role in kernel methods. In this paper, we consider the automated learning of the kernel matrix over a convex combination of pre-specified kernel matrices in Regularized Kernel Discriminant Analysis (RKDA), which performs linear discriminant analysis in the feature space via the kernel trick. Previous studies have shown that this kernel learning problem can be formulated as a semidefinite program (SDP), which is however computationally expensive, even with the recent advances in interior point methods. Based on the equivalence relationship between RKDA and least square problems in the binary-class case, we propose a Quadratically Constrained Quadratic Programming (QCQP) formulation for the kernel learning problem, which can be solved more efficiently than SDP. While most existing work on kernel learning deal with binary-class problems only, we show that our QCQP formulation can be extended naturally to the multi-class case. Experimental results on both binary-class and multiclass benchmark data sets show the efficacy of the proposed QCQP formulations.