Detailed Description

Problem data scaling during setup.

Author: Ben Hermans

This file includes the routine that is called during setup to scale the problem data, and initial guesses if the problem is warm-started. Scaling the problem is useful to prevent large changes in the active set and to guard against ill-conditioning in the objective function.

Definition in file scaling.h.

#include "types.h"
#include "lin_alg.h"
#include "constants.h"

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Functions
void	scale_data (QPALMWorkspace *work)
	Scale problem matrices.

void	unscale_data (QPALMWorkspace *work)
	Unscale the problem data.

Function Documentation

◆ scale_data()

void scale_data ( QPALMWorkspace * work )

Scale problem matrices.

Ruiz scaling [4] is applied to the constraint matrix A. This means that the rows and columns of A are scaled elementwise by the square root of their infinity norm, and this for a number of work->settings->scaling iterations. The resulting scaling can be written as \(\bar{A}\leftarrow EAD\), where \(E\) and \(D\) are the row and column scaling diagonal matrices respectively. The upper and lower bounds are also scaled with \(E\), thus \(\bar{b}_\textrm{min}, \bar{b}_\textrm{max} \leftarrow E b_\textrm{min}, E b_\textrm{max}\). The primal variables are transformed using \(D^{-1}\), resulting in \(\bar{x}\leftarrow D^{-1}x\). Therefore, also the cost matrix Q and vector q have to be scaled with \(D\), \(\bar{Q}\leftarrow DQD\) and \(\bar{q}\leftarrow Dq\). Finally the objective function is scaled with a scalar \(c\), thus \(\bar{Q}, \bar{q} \leftarrow c\bar{Q}, c\bar{q} \), where \(c=1/\textrm{max}(1, \nabla f(x_0))\). The dual variables in the scaled problem become \(\bar{y} \leftarrow c E^{-1} y\). The diagonals of \(D\) and \(E\) are stored and used in the remainder of the problem.

Parameters

work Workspace

Definition at line 31 of file scaling.c.

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◆ unscale_data()

void unscale_data ( QPALMWorkspace * work )

Unscale the problem data.

See scale_data.