In this paper, we consider a family of linearly constrained convex minimization problems whose objective function is not necessarily smooth. A basic double-proximal augmented Lagrangian method (DP-ALM) is developed, which enjoys a flexible dual stepsize and a proximal subproblem with relatively smaller proximal parameter. By a novel prediction-correction reformulation for the proposed DP-ALM and by … Read more
We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point … Read more
Global convergence of augmented Lagrangian methods to a first-order stationary point is well-known to hold under considerably weak constraint qualifications. In particular, several constant rank-type conditions have been introduced for this purpose which turned out to be relevant also beyond this scope. In this paper we show that in fact under these conditions subsequences of … Read more
At each iteration of the Safeguarded Augmented Lagrangian algorithm Algencan, a bound-constrained subproblem consisting of the minimization of the Powell-Hestenes-Rockafellar augmented Lagrangian function is considered, for which a minimizer with tolerance tending to zero is sought. More precisely, a point that satisfies a subproblem first-order necessary optimality condition with tolerance tending to zero is required. … Read more
We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions. This approach incorporates the augmented Lagrangian method and the Burer-Monteiro factorization, and features the adaptive strategies for updating the factorization size and the penalty parameter. We prove that the present algorithm can solve SDPs to global optimality, despite of the … Read more
In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under … Read more
In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality constrained optimization when a nearly feasible point is known. In particular, we first propose a new Newton-CG method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method [56]. We … Read more
In this paper we study a class of constrained minimax problems. In particular, we propose a first-order augmented Lagrangian method for solving them, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method recently developed in [26] by the authors. Under some suitable assumptions, an … Read more
The Fritz-John (FJ) and KKT conditions are fundamental tools for characterizing minimizers and form the basis of almost all methods for constrained optimization. Since the seminal works of Fritz John, Karush, Kuhn and Tucker, FJ/KKT conditions have been enhanced by adding extra necessary conditions. Such an extension was initially proposed by Hestenes in the 1970s … Read more
This work presents an adaptive superfast proximal augmented Lagrangian (AS-PAL) method for solving linearly-constrained smooth nonconvex composite optimization problems. Each iteration of AS-PAL inexactly solves a possibly nonconvex proximal augmented Lagrangian (AL) subproblem obtained by an aggressive/adaptive choice of prox stepsize with the aim of substantially improving its computational performance followed by a full Lagrangian … Read more