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pnopt_pqn.m
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pnopt_pqn.m
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function [ x, f_x, output ] = pnopt_pqn( smoothF, nonsmoothF, x, options )
% pnopt_pqn : Proximal quasi-Newton methods
%
% [ x, f, output ] = pnopt_pqn( smoothF, nonsmoothF, x, options ) starts at x and
% seeks a minimizer of the objective function in composite form. smoothF is a
% handle to a function that returns the smooth function value and gradient.
% nonsmoothF is a handle to a function that returns the nonsmooth function
% value and proximal mapping. options is a structure created using the
% pnopt_optimset function.
%
REVISION = '$Revision: 0.9.1$';
DATE = '$Date: Dec. 15, 2013$';
REVISION = REVISION(11:end-1);
DATE = DATE(8:end-1);
% ============ Process options ============
debug = options.debug;
desc_param = options.desc_param;
display = options.display;
max_fun_evals = options.max_fun_evals;
max_iter = options.max_iter;
method = options.method;
subprob_solver = options.subprob_solver;
ftol = options.ftol;
optim_tol = options.optim_tol;
xtol = options.xtol;
switch method
case 'bfgs'
case 'Lbfgs'
Lbfgs_mem = options.Lbfgs_mem;
end
% ------------ Set subproblem solver options ------------
switch subprob_solver
case 'sparsa'
sparsa_options = options.sparsa_options;
case 'tfocs'
tfocs_opts = options.tfocs_opts;
if debug
tfocs_opts.countOps = 1;
end
end
% ============ Initialize variables ============
pnopt_flags
iter = 0;
loop = 1;
forcing_term = 0.1;
trace.f_x = zeros( max_iter + 1, 1 );
trace.fun_evals = zeros( max_iter + 1, 1 );
trace.prox_evals = zeros( max_iter + 1, 1 );
trace.optim = zeros( max_iter + 1, 1 );
if debug
trace.forcing_term = zeros( max_iter, 1 );
trace.subprob_iters = zeros( max_iter, 1 );
trace.subprob_optim = zeros( max_iter, 1 );
end
if display > 0
fprintf( ' %s\n', repmat( '=', 1, 64 ) );
fprintf( ' PNOPT v.%s (%s)\n', REVISION, DATE );
fprintf( ' %s\n', repmat( '=', 1, 64 ) );
fprintf( ' %4s %6s %6s %12s %12s %12s \n',...
'','Fun.', 'Prox', 'Step len.', 'Obj. val.', 'Optim.' );
fprintf( ' %s\n', repmat( '-', 1, 64 ) );
end
% ------------ Evaluate objective function at starting x ------------
[ g_x, grad_g_x ] = smoothF( x );
h_x = nonsmoothF( x );
f_x = g_x + h_x;
% ------------ Start collecting data for display and output ------------
fun_evals = 1;
prox_evals = 0;
[ ~, x_prox ] = nonsmoothF( x - grad_g_x, 1 );
optim = norm( x_prox - x, 'inf' );
trace.f_x(1) = f_x;
trace.fun_evals(1) = fun_evals;
trace.prox_evals(1) = prox_evals;
trace.optim(1) = optim;
if display > 0
fprintf( ' %4d | %6d %6d %12s %12.4e %12.4e\n', ...
iter, fun_evals, prox_evals, '', f_x, optim );
end
% ------------ Check if starting x is optimal ------------
if optim <= optim_tol
flag = FLAG_OPTIM;
message = MESSAGE_OPTIM;
loop = 0;
end
% ============ Main Loop ============
while loop
iter = iter + 1;
% ------------ Update Hessian approximation ------------
switch method
% BFGS method
case 'bfgs'
if iter > 1
s = x - x_old;
y = grad_g_x - grad_g_old;
qty1 = cholB' * ( cholB * s );
if s'*y > 1e-9
cholB = cholupdate( cholupdate( cholB, y / sqrt( y' *s ) ), qty1 / ...
sqrt( s' * qty1 ), '-' );
end
H_x = @(x) cholB' * ( cholB * x );
else
cholB = eye( length( x ) );
end
% Limited-memory BFGS method
case 'Lbfgs'
if iter > 1
s = x - x_old;
y = grad_g_x - grad_g_old;
if y'*s > 1e-9
if size( sPrev, 2 ) > Lbfgs_mem
sPrev = [ sPrev(:,2:Lbfgs_mem), s ];
yPrev = [ yPrev(:,2:Lbfgs_mem), y ];
hDiag = ( y' * y ) / ( y' * s );
else
sPrev = [ sPrev, s ]; %#ok<AGROW>
yPrev = [ yPrev, y ]; %#ok<AGROW>
hDiag = ( y' * y ) / ( y' * s );
end
end
H_x = pnopt_bfgs_prod( sPrev, yPrev, hDiag );
else
sPrev = zeros( length(x), 0 );
yPrev = zeros( length(x), 0 );
hDiag = 1;
end
end
% ------------ Solve subproblem for a search direction ------------
if iter > 1
quadF = @(z) pnopt_quad( H_x, grad_g_x, f_x, z - x );
switch subprob_solver
% SpaRSA
case 'sparsa'
sparsa_options = pnopt_optimset( sparsa_options ,...
'optim_tol', max( 0.1 * optim_tol, forcing_term * optim ) ...
);
[ x_prox, ~, sparsa_out ] = ...
pnopt_sparsa( quadF, nonsmoothF, x, sparsa_options );
% ------------ Collect data from subproblem solve ------------
subprob_iters = sparsa_out.iters;
subprob_prox_evals = sparsa_out.trace.prox_evals;
subprob_optim = sparsa_out.optim;
% TFOCS
case 'tfocs'
tfocs_opts.stopFcn = @(f, x) tfocs_stop( x, nonsmoothF,...
max( 0.1 * optim_tol, forcing_term * optim ) );
[ x_prox, tfocs_out ] = ...
tfocs( quadF, [], nonsmoothF, x, tfocs_opts );
subprob_iters = tfocs_out.niter;
if isfield( tfocs_opts, 'countOps' ) && tfocs_opts.countOps
subprob_prox_evals = tfocs_out.counts(end,5);
else
subprob_prox_evals = tfocs_out.niter;
end
subprob_optim = tfocs_out.err(end);
end
p = x_prox - x;
else
subprob_iters = 0;
subprob_prox_evals = 0;
subprob_optim = 0;
p = - grad_g_x;
end
% ------------ Conduct line search ------------
x_old = x;
f_old = f_x; %#ok<NASGU>
grad_g_old = grad_g_x;
optim_old = optim;
if iter > 1
[ x, f_x, grad_g_x, step, backtrack_flag, backtrack_iters ] = ...
pnopt_backtrack( x, p, 1, f_x, h_x, grad_g_x' * p, smoothF, nonsmoothF, ...
desc_param, xtol, max_fun_evals - fun_evals ); %#ok<ASGLU>
else
[ x, f_x, grad_g_x, step, backtrack_flag, backtrack_iters ] = ...
pnopt_curvtrack( x, p, max( min( 1, 1 / norm( grad_g_x ) ), xtol ), f_x, ...
grad_g_x'*p, smoothF, nonsmoothF, desc_param, xtol, max_fun_evals - fun_evals ); %#ok<ASGLU>
end
% ------------ Select safeguarded forcing term ------------
if iter > 1
forcing_term = min( 0.1 , norm( optim - subprob_optim ) / norm( optim_old ) );
end
% ------------ Collect data for display and output ------------
fun_evals = fun_evals + backtrack_iters ;
prox_evals = prox_evals + backtrack_iters + subprob_prox_evals;
[ ~, x_prox ] = nonsmoothF( x - grad_g_x, 1 );
optim = norm( x_prox - x, 'inf' );
trace.f_x(iter+1) = f_x;
trace.fun_evals(iter+1) = fun_evals;
trace.prox_evals(iter+1) = prox_evals;
trace.optim(iter+1) = optim;
if debug
trace.forcing_term(iter) = forcing_term;
trace.backtrack_iters(iter) = backtrack_iters;
trace.subprob_iters(iter) = subprob_iters;
trace.subprob_optim(iter) = subprob_optim;
end
if display > 0 && mod( iter, display ) == 0
fprintf( ' %4d | %6d %6d %12.4e %12.4e %12.4e\n', ...
iter, fun_evals, prox_evals, step, f_x, optim );
end
pnopt_stop
end
% ============ Clean up and exit ============
trace.f_x = trace.f_x(1:iter+1);
trace.fun_evals = trace.fun_evals(1:iter+1);
trace.prox_evals = trace.prox_evals(1:iter+1);
trace.optim = trace.optim(1:iter+1);
if debug
trace.forcing_term = trace.forcing_term(1:iter);
trace.backtrack_iters = trace.backtrack_iters(1:iter);
trace.subprob_iters = trace.subprob_iters(1:iter);
trace.subprob_optim = trace.subprob_optim(1:iter);
end
if display > 0 && mod( iter, display ) > 0
fprintf( ' %4d | %6d %6d %12.4e %12.4e %12.4e\n', ...
iter, fun_evals, prox_evals, step, f_x, optim );
fprintf( ' %s\n', repmat( '-', 1, 64 ) );
end
output = struct( ...
'flag' , flag ,...
'fun_evals' , fun_evals ,...
'iters' , iter ,...
'optim' , optim ,...
'options' , options ,...
'prox_evals' , prox_evals ,...
'trace' , trace ...
);
clear global subprob_Dg_y subprob_optim
function H_x = pnopt_bfgs_prod( S, Y, de )
% pnopt_bfgs_prod : L-BFGS Hessian approximation
%
l = size( S, 2 );
L = zeros( l );
for k = 1:l;
L(k+1:l,k) = S(:,k+1:l)' * Y(:,k);
end
d1 = sum( S .* Y );
d2 = sqrt( d1 );
R = chol( de * ( S' * S ) + L * ( diag( 1 ./ d1 ) * L' ), 'lower' );
R1 = [ diag( d2 ), zeros(l); - L*diag( 1 ./ d2 ), R ];
R2 = [- diag( d2 ), diag( 1 ./ d2 ) * L'; zeros( l ), R' ];
YdS = [ Y, de * S ];
H_x = @(x) de * x - YdS * ( R2 \ ( R1 \ ( YdS' * x ) ) );