  In this paper, we consider the nonconvex optimization problem in which the
objective is continuously differentiable and the feasible set is a nonconvex
$\ell_{p}$ ball. Studying such optimization offers the possibility to bridge
the gap between a range of intermediate values $p \in (0,1)$ and $p \in
\{0,1\}$ in the norm-constraint sparse optimization, both in theory and
practice. We propose a novel hybrid method within a first-order algorithmic
framework for solving such problems by combining the Frank-Wolfe method and the
gradient projection method. During iterations, it solves a Frank-Wolfe
subproblem if the current iterate is in the interior of the $\ell_p$ ball, and
it solves a gradient projection subproblem with a weighted $\ell_1$-ball
constraint if the current iterate is on the boundary of the $\ell_p$ ball.
Global convergence is proved, and a worst-case iteration complexity
$O(1/\epsilon^2)$ of the optimality error is also established. We believe our
proposed method is the first practical algorithm for solving $\ell_p$-ball
constrained nonlinear problems with theoretical guarantees. Numerical
experiments demonstrate the practicability and efficiency of the proposed
algorithm.
