10,370 research outputs found
Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization
Primal-dual gradient dynamics that find saddle points of a Lagrangian have
been widely employed for handling constrained optimization problems. Building
on existing methods, we extend the augmented primal-dual gradient dynamics
(Aug-PDGD) to incorporate general convex and nonlinear inequality constraints,
and we establish its semi-global exponential stability when the objective
function is strongly convex. We also provide an example of a strongly convex
quadratic program of which the Aug-PDGD fails to achieve global exponential
stability. Numerical simulation also suggests that the exponential convergence
rate could depend on the initial distance to the KKT point
A Primal-Dual Augmented Lagrangian
Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both the primal and the dual variables simultaneously. A benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of conventional primal methods are proposed: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual 1 linearly constrained Lagrangian (pd1-LCL) method
An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow
A novel trust region method for solving linearly constrained nonlinear
programs is presented. The proposed technique is amenable to a distributed
implementation, as its salient ingredient is an alternating projected gradient
sweep in place of the Cauchy point computation. It is proven that the algorithm
yields a sequence that globally converges to a critical point. As a result of
some changes to the standard trust region method, namely a proximal
regularisation of the trust region subproblem, it is shown that the local
convergence rate is linear with an arbitrarily small ratio. Thus, convergence
is locally almost superlinear, under standard regularity assumptions. The
proposed method is successfully applied to compute local solutions to
alternating current optimal power flow problems in transmission and
distribution networks. Moreover, the new mechanism for computing a Cauchy point
compares favourably against the standard projected search as for its activity
detection properties
An Augmented Lagrangian Approach for Sparse Principal Component Analysis
Principal component analysis (PCA) is a widely used technique for data
analysis and dimension reduction with numerous applications in science and
engineering. However, the standard PCA suffers from the fact that the principal
components (PCs) are usually linear combinations of all the original variables,
and it is thus often difficult to interpret the PCs. To alleviate this
drawback, various sparse PCA approaches were proposed in literature [15, 6, 17,
28, 8, 25, 18, 7, 16]. Despite success in achieving sparsity, some important
properties enjoyed by the standard PCA are lost in these methods such as
uncorrelation of PCs and orthogonality of loading vectors. Also, the total
explained variance that they attempt to maximize can be too optimistic. In this
paper we propose a new formulation for sparse PCA, aiming at finding sparse and
nearly uncorrelated PCs with orthogonal loading vectors while explaining as
much of the total variance as possible. We also develop a novel augmented
Lagrangian method for solving a class of nonsmooth constrained optimization
problems, which is well suited for our formulation of sparse PCA. We show that
it converges to a feasible point, and moreover under some regularity
assumptions, it converges to a stationary point. Additionally, we propose two
nonmonotone gradient methods for solving the augmented Lagrangian subproblems,
and establish their global and local convergence. Finally, we compare our
sparse PCA approach with several existing methods on synthetic, random, and
real data, respectively. The computational results demonstrate that the sparse
PCs produced by our approach substantially outperform those by other methods in
terms of total explained variance, correlation of PCs, and orthogonality of
loading vectors.Comment: 42 page
- …