It is proved that the Error Bound Constraint Qualification is the weakest constraint qualification that guarantees boundedness of the computed multiplier sequences generated by the augmented Lagrangian method, and the feasibility of accumulation points of primal sequences generated by the augmented Lagrangian method under a Polyak-Łojasiewicz inequality for the quadratic infeasibility measure.
This work considers convex optimization with nonlinear inequality constraints and develops a primal-dual multiplier framework that is consistent in continuous and discrete time and is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.
It is shown that strict complementarity, together with a quadratic facial-violation property of the associated complementary faces, implies uniform quadratic growth of both the primal and dual augmented Lagrangians near a strictly complementary solution, and the local equivalence of three regularity conditions is prove...
It is shown that any first-order method guaranteeing a bound on the primal objective gap f(x_N)-f(x_\star) assuming only a bound on $\|x_0-x_\star\|$ actually has a stronger guarantee on an explicit, computable primal-dual gap at the same rate.
The strong convergence of the primal trajectory is proved and fast convergence rates under the same parameter assumptions are derived under the same parameter assumptions, without imposing any eventual inside/outside-ball condition.
This work proposes a nonlinear-residual linearized augmented Lagrangian method (NR-LALM) that replaces this subproblem by a regularized Gauss-Newton-type step while retaining the classical multiplier update based on the nonlinear constraint residual.
Ben-Qi Liu, Kangkang Deng, Zichen Wang et al.· 0 citations
In this paper, we propose a balanced augmented Lagrangian method based on accelerated stochastic ADMM (b-ASADMM) to efficiently solve structured separable nonconvex optimization problems subject to linear constraints. The objective function in this problem comprises potentially nonsmooth and smooth functions, where the...
