Reference for CVX built-in function convexity proof

nikku · February 26, 2024, 3:13am

According to the CVX documentation det_inv is a Convex function. Is there any reference for this claim?

I understand det(A) is NOT convex or concave, but is log-concave according to the Mosek documentation.

Michal_Adamaszek · February 26, 2024, 5:42am

The proof of convexity is always in the body of the cvx function where it constructs a conic model

cvxr/CVX/blob/master/functions/@cvx/det_inv.m

function cvx_optval = det_inv( X, p )

%DET_INV   Internal cvx version.

narginchk(1,2);
n = size( X, 1 );
if ndims( X ) > 2, %#ok
    error( 'N-D arrays are not supported.' );
elseif size( X, 2 ) ~= n,
    error( 'Matrix must be square.' );
elseif nargin < 2,
    p = 1;
elseif ~isnumeric( p ) || ~isreal( p ) || numel( p ) ~=  1 || p <= 0,
    error( 'Second argument must be a positive scalar.' );
end

w = [ ones(n,1) ; p ];
if cvx_isconstant( X ),
    
    cvx_optval = cvx( det_inv( cvx_constant( X ), p ) );

This file has been truncated. show original

This is the most relevant piece:

    cvx_begin
        epigraph variable y
        variable Z(n,n) lower_triangular complex
        D = diag( Z );
        [ diag( D ), Z' ; Z, X ] == hermitian_semidefinite(2*n); %#ok
        geo_mean( [ real( D ) ; y ], [], w ) >= 1; %#ok
    cvx_end