We propose a new first-order augmented Lagrangian algorithm ALCC for solving convex conic programs of the form min{rho(x)+gamma(x): Ax-b in K, x in chi}, where rho and gamma are closed convex functions, and gamma has a Lipschitz continuous gradient, A is mxn real matrix, K is a closed convex cone, and chi is a “simple” … Read more
We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschizt stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving … Read more
We consider the augmented Lagrangian dual for integer programming, and provide a primal characterization of the resulting bound. As a corollary, we obtain proof that the augmented Lagrangian is a strong dual for integer programming. We are able to show that the penalty parameter applied to the augmented Lagrangian term may be placed at a … Read more
We propose an augmented Lagrangian algorithm for solving large-scale constrained optimization problems. The novel feature of the algorithm is an adaptive update for the penalty parameter motivated by recently proposed techniques for exact penalty methods. This adaptive updating scheme greatly improves the overall performance of the algorithm without sacrificing the strengths of the core augmented … Read more
The alternating direction of multipliers (ADMM) is a form of augmented Lagrangian algorithm that has experienced a renaissance in recent years due to its applicability to optimization problems arising from “big data” and image processing applications, and the relative ease with which it may be implemented in parallel and distributed computational environments. This paper aims … Read more
We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution of the augmented … Read more
Based on the classic augmented Lagrangian multiplier method, we propose, analyze and test an algorithm for solving a class of equality-constrained non-smooth optimization problems (chiefly but not necessarily convex programs) with a particular structure. The algorithm effectively combines an alternating direction technique with a nonmonotone line search to minimize the augmented Lagrangian function at each … Read more
Sequential quadratic programming (SQP) methods are a popular class of methods for nonlinearly constrained optimization. They are particularly effective for solving a sequence of related problems, such as those arising in mixed-integer nonlinear programming and the optimization of functions subject to differential equation constraints. Recently, there has been considerable interest in the formulation of \emph{stabilized} … Read more
We present the formulation and analysis of a new sequential quadratic programming (\SQP) method for general nonlinearly constrained optimization. The method pairs a primal-dual generalized augmented Lagrangian merit function with a \emph{flexible} line search to obtain a sequence of improving estimates of the solution. This function is a primal-dual variant of the augmented Lagrangian proposed … Read more
We consider the image denoising problem using total variation (TV) regularization. This problem can be computationally challenging to solve due to the non-differentiability and non-linearity of the regularization term. We propose an alternating direction augmented Lagrangian (ADAL) method, based on a new variable splitting approach that results in subproblems that can be solved efficiently and … Read more