A proximal linearized NEP method for composite optimization with conic and manifold constraints

This paper studies DC composite optimization problems with simultaneous conic and manifold constraints. We propose an inexact proximal linearized exact penalty algorithm based on a distance-based penalty function for the conic constraint. The algorithm successively computes approximate minimizers of proximal linearizations of the penalty function over the tangent spaces of the manifold under a readily verifiable inexactness criterion, with adaptive updates of the proximal and penalty parameters. Under the boundedness assumption on the sequences of iterates and penalty parameters, we establish an $O({\epsilon }^{-2})$ worst-case complexity bound for finding an $\epsilon$-stationary point. Moreover, when semi-proximal ADMM is employed to solve the subproblems, the resulting oracle complexity is $O({\epsilon }^{-2}\log {\epsilon }^{-1})$ provided that the distance function and the first DC component are piecewise linear-quadratic, while an accelerated variant achieves an $O({\epsilon }^{-4})$ oracle complexity. If, in addition, the associated potential function satisfies the KL property, the whole sequence of iterates converges to a stationary point. Extensive numerical experiments on composite optimization problems with orthogonal manifold, nonnegative cone, and second-order cone (SOC) constraints demonstrate the effectiveness of the proposed method.