nonconvex.h File Reference

Detailed Description

Routines to deal with nonconvex QPs.

Author

Ben Hermans

The functions in this file serve to set up QPALM for a nonconvex QP. The main routine in this file computes the minimum eigenvalue of a square matrix, based on lobpcg [3]. Furthermore, some setting updates are performed. In addition, the spectrum of a matrix can be upper bounded using Gershgorin's circle theorem, which is used in the gamma_boost routine in iteration.c.

Definition in file nonconvex.h.

#include <qpalm/types.h>

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Functions
c_float	gershgorin_max (solver_sparse *M, c_float *center, c_float *radius)
	Calculate the Gershgorin upper bound for the eigenvalues of a symmetric matrix.
void	set_settings_nonconvex (QPALMWorkspace *work, solver_common *c)
	Set the proximal parameters for nonconvex QPs.

Function Documentation

gershgorin_max()

c_float gershgorin_max	(	solver_sparse *	M,
		c_float *	center,
		c_float *	radius
	)

Calculate the Gershgorin upper bound for the eigenvalues of a symmetric matrix.

This routine uses the Gershgorin circle theorem to compute an upper bound on the eigenvalues of a matrix.

Note

M is assumed to be symmetric

Parameters

M	Matrix
center	Vector of size M->ncol to hold the values of the centers of the discs
radius	Vector of size M->ncol to hold the values of the radii of the discs

Returns

Upper bound on the eigenvalues of M

Definition at line 353 of file nonconvex.c.

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set_settings_nonconvex()

void set_settings_nonconvex	(	QPALMWorkspace *	work,
		solver_common *	c
	)

Set the proximal parameters for nonconvex QPs.

QPALM can deal with nonconvex QPs, by setting the initial and maximal proximal penalty small enough (smaller than \( \frac{1}{|\lambda_\textrm{min}|} \)). This ensures positive definiteness of \( Q + \frac{1}{\gamma}I \) during the iterations. The minimum eigenvalue is computed using lobpcg.

Parameters

work	Workspace
c	Linear systems solver environment

Definition at line 331 of file nonconvex.c.