An Iteratively Reweighted Method for Sparse Optimization on Nonconvex ℓ_p Ball
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This paper is intended to solve the nonconvex ℓ_p-ball constrained nonlinear optimization problems. An iteratively reweighted method is proposed, which solves a sequence of weighted ℓ_1-ball projection subproblems. At each iteration, the next iterate is obtained by moving along the negative gradient with a stepsize and then projecting the resulted point onto the weighted ℓ_1 ball to approximate the ℓ_p ball. Specifically, if the current iterate is in the interior of the feasible set, then the weighted ℓ_1 ball is formed by linearizing the ℓ_p norm at the current iterate. If the current iterate is on the boundary of the feasible set, then the weighted ℓ_1 ball is formed differently by keeping those zero components in the current iterate still zero. In our analysis, we prove that the generated iterates converge to a first-order stationary point. Numerical experiments demonstrate the effectiveness of the proposed method.
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