Primal-dual Optimization Research Articles

Subspace regression in reproducing kernel hilbert space

We focus on three methods for finding a suitable subspace for regression in a reproducing kernel Hilbert space: kernel principal component analysis, kernel partial least squares and kernel canonical correlation analysis and we demonstrate how this fits within a more general context of subspace regression. For the kernel partial least squares case a least squares support vector machine style derivation is given with a primal-dual optimization problem formulation. The methods are illustrated and compared on a number of examples.

A support vector machine formulation to pca analysis and its kernel version

In this paper, we present a simple and straightforward primal-dual support vector machine formulation to the problem of principal component analysis (PCA) in dual variables. By considering a mapping to a high-dimensional feature space and application of the kernel trick (Mercer theorem), kernel PCA is obtained as introduced by Scholkopf et al. (2002). While least squares support vector machine classifiers have a natural link with the kernel Fisher discriminant analysis (minimizing the within class scatter around targets +1 and -1), for PCA analysis one can take the interpretation of a one-class modeling problem with zero target value around which one maximizes the variance. The score variables are interpreted as error variables within the problem formulation. In this way primal-dual constrained optimization problem interpretations to the linear and kernel PCA analysis are obtained in a similar style as for least square-support vector machine classifiers.

Optimal power flow solutions under variable load conditions

This work presents a methodology to calculate a sequence of optimal power flow (OPF) solutions under variable load conditions. The aim is to obtain a set of optimal operating points in the neighborhood of the bounds of the region defined by the load flow equations and a set of operational limits. For this, an algorithm based on the continuation method and on a primal-dual interior point optimization method is proposed. Such an algorithm consists of two main steps: the predictor step, which uses a linear approximation of the Karush-Kuhn-Tucker (KKT) conditions to estimate a new operating point for an increment in the system load; and the corrector step, which calculates the optimum corresponding to the new load level via a nonlinear primal-dual interior point method. Indices for critical buses and inequality constraints are a byproduct of the methodology. In addition, sensitivity analysis is performed to calculate the amount of reactive compensation which allows for a pre-specified increase in the system load. Results for realistic test systems are presented.

New decomposition and convexification algorithm for nonconvex large-scale primal-dual optimization

A new algorithm for solving nonconvex, equality-constrained optimization problems with separable structures is proposed in the present paper. A new augmented Lagrangian function is derived, and an iterative method is presented. The new proposed Lagrangian function preserves separability when the original problem is separable, and the property of linear convergence of the new algorithm is also presented. Unlike earlier algorithms for nonconvex decomposition, the convergence ratio for this method can be made arbitrarily small. Furthermore, it is feasible to extend this method to algorithms suited for inequality-constrained optimization problems. An example is included to illustrate the method.

A geometric projection-space reconstruction algorithm

We present a method to reconstruct images from finite sets of noisy projections that may be available only over limited or sparse angles. The algorithm calculates the maximum a posteriori (MAP) estimate of the full sinogram (which is an image of the 2-D Radon transform of the object) from the available data. It is implemented using a primal-dual constrained optimization procedure that solves a partial differential equation in the primal phase with an efficient local relaxation algorithm and uses a simple Lagrange-multiplier update in the dual phase. The sinogram prior probability is given by a Markov random field (MRF) that includes information about the mass, center of mass, and convex hull of the object, and about the smoothness, fundamental constraints, and periodicity of the 2-D Radon transform. The object is reconstructed using convolution backprojection applied to the estimated sinogram. We show several reconstructed objects which are obtained from simulated limited- and sparse-angle data using the described algorithm, and compare these results with images obtained using convolution backprojection directly.

Constrained sinogram restoration for limited-angle tomography

Tomographic reconstruction from incomplete data is required in many fields, including medical imaging, sonar, and radar. In this paper, we present a new reconstruction algorithm for limited-angle tomography, a problem that occurs when projections are missing over a range of angles. The approach uses a variational formulation that incorporates the Ludwig- Helgason consistency conditions, measurement noise statistics, and a sinogram smoothness condition. Optimal restored sinograms, therefore, satisfy an associated Euler-Lagrange partial differential equation, which we solve on a lattice using a primal-dual optimization procedure. Object estimates are then reconstructed using convolution back-projection applied to the restored sinogram. We present results of simulations that illustrate the performance of the algorithm and discuss directions for further research.