A Global Dual Error Bound and Its Application to the Analysis of Linearly Constrained Nonconvex Optimization

Error bound analysis, which estimates the distance of a point to solution set an optimization problem4 using optimality residual, is powerful tool for analysis first-order algorithms. In this paper, we use global error study iteration complexity algorithm linearly constrained nonconvex minimization problem. We develop dual regularized version problem by novel “decomposition” technique. Equipped with bound, prove that suitably designed primal-dual method can generate $\epsilon$-stationary within $\mathcal{O}(1/\epsilon^2)$ iterations, best known class problems. The our finding $\mathcal{O}(1/\epsilon^2)$, improves $\mathcal{O}(1/\epsilon^3)$ under consideration. Furthermore, when objective function quadratic, establish linear convergence algorithm. Our proof based on new potential and bounds.