  In this paper, we consider an $\ell_{0}$-norm penalized formulation of the
generalized eigenvalue problem (GEP), aimed at extracting the leading sparse
generalized eigenvector of a matrix pair. The formulation involves maximization
of a discontinuous nonconcave objective function over a nonconvex constraint
set, and is therefore computationally intractable. To tackle the problem, we
first approximate the $\ell_{0}$-norm by a continuous surrogate function. Then
an algorithm is developed via iteratively majorizing the surrogate function by
a quadratic separable function, which at each iteration reduces to a regular
generalized eigenvalue problem. A preconditioned steepest ascent algorithm for
finding the leading generalized eigenvector is provided. A systematic way based
on smoothing is proposed to deal with the "singularity issue" that arises when
a quadratic function is used to majorize the nondifferentiable surrogate
function. For sparse GEPs with special structure, algorithms that admit a
closed-form solution at every iteration are derived. Numerical experiments show
that the proposed algorithms match or outperform existing algorithms in terms
of computational complexity and support recovery.
