function op = prox_hinge( q , r, y)

%PROX_HINGE Hinge-loss function.

% OP = PROX_HINGE( q , r, y ) implements the nonsmooth function

% OP(X) = q * sum( max( r - y.*x, 0 ) ).

% Q is optional; if omitted, Q=1 is assumed. But if Q is supplied,

% then it must be a positive real scalar.

% R is also optional; if omitted, R = 1 is assumed. R may be any real number.

% Y is also optional; if omitted, Y = 1 is assumed. Y may be any scalar

% or vector of the same size as X

% Dual: prox_hingeDual.m

%

% See also PROX_HINGEDUAL

if nargin < 3

y = [];

elseif ~isempty(y) && ( ~isnumeric(y) || ~isreal(y) )

error( 'Argument 3 must be a real vector');

end

if nargin < 2

r = 1;

elseif ~isnumeric( r ) || ~isreal( r ) %|| numel( r ) ~= 1

error( 'Argument 2 must be real.' );

end

if nargin < 1

q = 1;

elseif ~isnumeric( q ) || ~isreal( q ) || numel( q ) ~= 1 || q <= 0,

error( 'Argument 1 must be positive.' );

end

if isempty(y)

op = tfocs_prox( @(x)hinge(x,r,q), @(x,t)prox_hinge(x,t,r,q) );

else

ry = r./y;

op = tfocs_prox( @(x)hingeY(x,r,q,y), @(x,t)prox_hingeY(x,t,r,q,ry,y) );

end

% -- the actual functions --

function v = hingeY(x,r,q,y)

if ~isscalar(r), assert( numel(r) == numel(x),'r is wrong size' ); end

if ~isscalar(y), assert( numel(y) == numel(x),'y is wrong size'); end

v = q*sum( max( r(:) - y(:).*x(:), 0 ) );

end

function v = hinge(x,r,q)

if ~isscalar(r), assert( numel(r) == numel(x),'r is wrong size' ); end

v = q*sum( max( r(:) - x(:), 0 ) );

end

% PROX_F( Y, t ) = argmin_X F(X) + 1/(2*t)*|| X - Y ||^2

function x = prox_hinge(x,t,r,q)

tq = t * q;

x = r + (x-r).*( x > r ) + (x + tq - r).*( x + tq < r );

end

%{

in the q = r = t = 1 case, the prox is:

prox(x) = { x, if x > 1

1, if x <= 1, and x > 0

x + q, if x <= 0

%}

function x = prox_hingeY(x,t,r,q,ry,y)

tq = t * q;

x = ry + (x-ry).*( y.*x > r ) + (x + tq*y - ry).*( y.*(x + tq*y) < r );

end

end

% Added Feb, 2011; modified Dec, 2011

% TFOCS v1.3 by Stephen Becker, Emmanuel Candes, and Michael Grant.

% Copyright 2013 California Institute of Technology and CVX Research.

% See the file LICENSE for full license information.