Together with a weight (see BundleWeight), the proximal term $\|y-\hat{y}\|_H^2$ ( $H$ positive definite) of the augmented cutting model plays the role of a step length control and offers a possibility to bring in some metric information via the choice of $H$ (see Variable Metric). Two aspects are of importance in choosing this matrix. More...

Classes
class	ConicBundle::BundleProxObject
	abstract interface that allows to use different -norms $\\|y-\hat{y}\\|_H^2$ with a positive definite matrix in the proximal term of the augmented model of ConicBundle::BundleSolver. More...

class	ConicBundle::BundleIdProx
	implements the abstract interface ConicBundle::BundleProxObject for $\\|y-\hat{y}\\|_H^2$ with H=weight*I, giving rise to a pure augmented model without scaling More...

class	ConicBundle::BundleDiagonalTrustRegionProx
	implements the abstract interface ConicBundle::BundleProxObject for $\\|y-\hat{y}\\|_H^2$ with H=D+weight*I, where D is a diagonal matrix, giving rise to an augmented model with diagonal scaling More...

class	ConicBundle::BundleDLRTrustRegionProx
	implements the abstract interface ConicBundle::BundleProxObject for $\\|y-\hat{y}\\|_H^2$ with $H=VV^\top+D+u I$ , i.e., a low rank representation of a symmetric positive definite matrix with a diagonal and the weight times identity added as regularization More...

class	ConicBundle::BundleLowRankTrustRegionProx
	implements the abstract interface BundleProxObject for $\\|y-\hat{y}\\|_H^2$ with $H=V\Lambda V^\top+u I$ , i.e., a low rank representation of a symmetric positive definite matrix with the weight times identity added as regularization More...

class	ConicBundle::BundleDenseTrustRegionProx
	implements the abstract interface ConicBundle::BundleProxObject for $\\|y-\hat{y}\\|_H^2$ for general symmetric H+weight*I (H is assumed to be positive semidefinite without checking) giving rise to an augmented model with dense variable metric More...

Detailed Description

Together with a weight (see BundleWeight), the proximal term $\|y-\hat{y}\|_H^2$ ( $H$ positive definite) of the augmented cutting model plays the role of a step length control and offers a possibility to bring in some metric information via the choice of $H$ (see Variable Metric). Two aspects are of importance in choosing this matrix.

First order methods are, unfortunately, extremely sensitive to scaling. Depending on the nature of the problem it may be possible to choose so as to add some information about the curvature (e.g. the positive semidefinite part of the Hessian) or estimates and reasonable relative step sizes into the model. This step size also plays a role in the selection of subgradient information wihtin the cutting model.
Depending on the structural properties of (diagonal, low rank, full, etc.) it may be computationally very costly or rather easy to solve the various QP suproblems arising in the bundle method

The classes of this section help to set up appropriate specializations that allow to exploit structural properties of $H$ and to collect/compute the cost-coefficients for the QPs independent of the actual cutting models and ground sets in use. They may be brought to use via MatrixCBSolver::set_prox().

The abstract base class for providing quadratic prox terms is BundleProxObject . ConicBundle offers several implementations of this, some of them support VariableMetric, some do not. Currently all use the BundleWeight as an additive term on the diagonal in a trust region fashion:

BundleIdProx (Identity, no dynamic scaling, weight as a factor)
BundleDiagonalTrustRegionProx (diagonal matrix, offers dynamic diagonal/bounds scaling, additive trust region weight)
BundleLowRankTrustRegionProx (low rank Gram matrix, offers low rank variable metric, additive trust region weight)
BundleDLRTrustRegionProx (not properly tested yet) (low rank + diagonal, offers low rank/diagonal variable metric, additive trust region weight)
BundleDenseTrustRegionProx (dense symmetric, offers dense/low rank/diagonal variable metric, additive trust region weight; mainly for comparative testing)

Classes

Detailed Description