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.c.

#include <qpalm/nonconvex.h>
#include <qpalm/types.h>
#include <qpalm/constants.h>
#include <qpalm/global_opts.h>
#include <qpalm/lin_alg.h>
#include <qpalm/util.h>
#include <math.h>

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Macros
#define	LOBPCG_TOL 1e-5
	Tolerance on the infinity norm of the residual in lobpcg.

#define	M_PI 3.14159265358979323846

#define	RREF_TOL 1e-8

Functions
void	set_settings_nonconvex (QPALMWorkspace work, solver_common c)
	Set the proximal parameters for nonconvex QPs.

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.

Macro Definition Documentation

◆ LOBPCG_TOL

#define LOBPCG_TOL 1e-5

Tolerance on the infinity norm of the residual in lobpcg.

Definition at line 22 of file nonconvex.c.

◆ M_PI

#define M_PI 3.14159265358979323846

Definition at line 25 of file nonconvex.c.

◆ RREF_TOL

#define RREF_TOL 1e-8

Definition at line 28 of file nonconvex.c.

Function Documentation

◆ 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.

◆ 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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