########################################################################
##
## Copyright (C) 2005-2020 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or .
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## .
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {[@var{x}, @var{obj}, @var{info}, @var{iter}, @var{nf}, @var{lambda}] =} sqp (@var{x0}, @var{phi})
## @deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g})
## @deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h})
## @deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub})
## @deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}, @var{maxiter})
## @deftypefnx {} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}, @var{maxiter}, @var{tolerance})
## Minimize an objective function using sequential quadratic programming (SQP).
##
## Solve the nonlinear program
## @tex
## $$
## \min_x \phi (x)
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## min phi (x)
## x
## @end group
## @end example
##
## @end ifnottex
## subject to
## @tex
## $$
## g(x) = 0 \qquad h(x) \geq 0 \qquad lb \leq x \leq ub
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## g(x) = 0
## h(x) >= 0
## lb <= x <= ub
## @end group
## @end example
##
## @end ifnottex
## @noindent
## using a sequential quadratic programming method.
##
## The first argument is the initial guess for the vector @var{x0}.
##
## The second argument is a function handle pointing to the objective function
## @var{phi}. The objective function must accept one vector argument and
## return a scalar.
##
## The second argument may also be a 2- or 3-element cell array of function
## handles. The first element should point to the objective function, the
## second should point to a function that computes the gradient of the
## objective function, and the third should point to a function that computes
## the Hessian of the objective function. If the gradient function is not
## supplied, the gradient is computed by finite differences. If the Hessian
## function is not supplied, a BFGS update formula is used to approximate the
## Hessian.
##
## When supplied, the gradient function @code{@var{phi}@{2@}} must accept one
## vector argument and return a vector. When supplied, the Hessian function
## @code{@var{phi}@{3@}} must accept one vector argument and return a matrix.
##
## The third and fourth arguments @var{g} and @var{h} are function handles
## pointing to functions that compute the equality constraints and the
## inequality constraints, respectively. If the problem does not have
## equality (or inequality) constraints, then use an empty matrix ([]) for
## @var{g} (or @var{h}). When supplied, these equality and inequality
## constraint functions must accept one vector argument and return a vector.
##
## The third and fourth arguments may also be 2-element cell arrays of
## function handles. The first element should point to the constraint
## function and the second should point to a function that computes the
## gradient of the constraint function:
## @tex
## $$
## \Bigg( {\partial f(x) \over \partial x_1},
## {\partial f(x) \over \partial x_2}, \ldots,
## {\partial f(x) \over \partial x_N} \Bigg)^T
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## [ d f(x) d f(x) d f(x) ]
## transpose ( [ ------ ----- ... ------ ] )
## [ dx_1 dx_2 dx_N ]
## @end group
## @end example
##
## @end ifnottex
## The fifth and sixth arguments, @var{lb} and @var{ub}, contain lower and
## upper bounds on @var{x}. These must be consistent with the equality and
## inequality constraints @var{g} and @var{h}. If the arguments are vectors
## then @var{x}(i) is bound by @var{lb}(i) and @var{ub}(i). A bound can also
## be a scalar in which case all elements of @var{x} will share the same
## bound.
##
## The seventh argument @var{maxiter} specifies the maximum number of
## iterations. The default value is 100.
##
## The eighth argument @var{tolerance} specifies the tolerance for the stopping
## criteria. The default value is @code{sqrt (eps)}.
##
## The value returned in @var{info} may be one of the following:
##
## @table @asis
## @item 101
## The algorithm terminated normally.
## All constraints meet the specified tolerance.
##
## @item 102
## The BFGS update failed.
##
## @item 103
## The maximum number of iterations was reached.
##
## @item 104
## The stepsize has become too small, i.e.,
## @tex
## $\Delta x,$
## @end tex
## @ifnottex
## delta @var{x},
## @end ifnottex
## is less than @code{@var{tol} * norm (x)}.
## @end table
##
## An example of calling @code{sqp}:
##
## @example
## function r = g (x)
## r = [ sumsq(x)-10;
## x(2)*x(3)-5*x(4)*x(5);
## x(1)^3+x(2)^3+1 ];
## endfunction
##
## function obj = phi (x)
## obj = exp (prod (x)) - 0.5*(x(1)^3+x(2)^3+1)^2;
## endfunction
##
## x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];
##
## [x, obj, info, iter, nf, lambda] = sqp (x0, @@phi, @@g, [])
##
## x =
##
## -1.71714
## 1.59571
## 1.82725
## -0.76364
## -0.76364
##
## obj = 0.053950
## info = 101
## iter = 8
## nf = 10
## lambda =
##
## -0.0401627
## 0.0379578
## -0.0052227
## @end example
##
## @seealso{qp}
## @end deftypefn
function [x, obj, info, iter, nf, lambda] = sqp (x0, objf, cef, cif, lb, ub, maxiter, tolerance)
globals = struct (); # data and handles, needed and changed by subfunctions
if (nargin < 2 || nargin == 5)
print_usage ();
endif
if (! isvector (x0))
error ("sqp: X0 must be a vector");
endif
if (rows (x0) == 1)
x0 = x0';
endif
have_grd = 0;
have_hess = 0;
if (iscell (objf))
switch (numel (objf))
case 1
obj_fun = objf{1};
obj_grd = @(x, obj) fd_obj_grd (x, obj, obj_fun);
case 2
obj_fun = objf{1};
obj_grd = objf{2};
have_grd = 1;
case 3
obj_fun = objf{1};
obj_grd = objf{2};
obj_hess = objf{3};
have_grd = 1;
have_hess = 1;
otherwise
error ("sqp: invalid objective function specification");
endswitch
else
obj_fun = objf; # No cell array, only obj_fun set
obj_grd = @(x, obj) fd_obj_grd (x, obj, obj_fun);
endif
ce_fun = @empty_cf;
ce_grd = @empty_jac;
if (nargin > 2)
if (iscell (cef))
switch (numel (cef))
case 1
ce_fun = cef{1};
ce_grd = @(x) fd_ce_jac (x, ce_fun);
case 2
ce_fun = cef{1};
ce_grd = cef{2};
otherwise
error ("sqp: invalid equality constraint function specification");
endswitch
elseif (! isempty (cef))
ce_fun = cef; # No cell array, only constraint equality function set
ce_grd = @(x) fd_ce_jac (x, ce_fun);
endif
endif
ci_fun = @empty_cf;
ci_grd = @empty_jac;
if (nargin > 3)
## constraint function given by user with possible gradient
globals.cif = cif;
## constraint function given by user without gradient
globals.cifcn = @empty_cf;
if (iscell (cif))
if (length (cif) > 0)
globals.cifcn = cif{1};
endif
elseif (! isempty (cif))
globals.cifcn = cif;
endif
if (nargin < 5 || (nargin > 5 && isempty (lb) && isempty (ub)))
## constraint inequality function only without any bounds
ci_grd = @(x) fd_ci_jac (x, globals.cifcn);
if (iscell (cif))
switch (length (cif))
case 1
ci_fun = cif{1};
case 2
ci_fun = cif{1};
ci_grd = cif{2};
otherwise
error ("sqp: invalid inequality constraint function specification");
endswitch
elseif (! isempty (cif))
ci_fun = cif; # No cell array, only constraint inequality function set
endif
else
## constraint inequality function with bounds present
lb_idx = ub_idx = true (size (x0));
ub_grad = - (lb_grad = eye (rows (x0)));
## if unspecified set ub and lb to +/-Inf, preserving single type
if (isempty (lb))
if (isa (x0, "single"))
lb = - inf (size (x0), "single");
else
lb = - inf (size (x0));
endif
endif
if (isvector (lb))
lb = lb(:);
lb_idx(:) = (lb != -Inf);
globals.lb = lb(lb_idx, 1);
lb_grad = lb_grad(lb_idx, :);
else
error ("sqp: invalid lower bound");
endif
if (isempty (ub))
if (isa (x0, "single"))
ub = inf (size (x0), "single");
else
ub = inf (size (x0));
endif
endif
if (isvector (ub))
ub = ub(:);
ub_idx(:) = (ub != Inf);
globals.ub = ub(ub_idx, 1);
ub_grad = ub_grad(ub_idx, :);
else
error ("sqp: invalid upper bound");
endif
if (any (globals.lb > globals.ub))
error ("sqp: upper bound smaller than lower bound");
endif
bounds_grad = [lb_grad; ub_grad];
ci_fun = @(x) cf_ub_lb (x, lb_idx, ub_idx, globals);
ci_grd = @(x) cigrad_ub_lb (x, bounds_grad, globals);
endif
endif # if (nargin > 3)
iter_max = 100;
if (nargin > 6 && ! isempty (maxiter))
if (isscalar (maxiter) && maxiter > 0 && fix (maxiter) == maxiter)
iter_max = maxiter;
else
error ("sqp: invalid number of maximum iterations");
endif
endif
tol = sqrt (eps);
if (nargin > 7 && ! isempty (tolerance))
if (isscalar (tolerance) && tolerance > 0)
tol = tolerance;
else
error ("sqp: invalid value for TOLERANCE");
endif
endif
## Initialize variables for search loop
## Seed x with initial guess and evaluate objective function, constraints,
## and gradients at initial value x0.
##
## obj_fun -- objective function
## obj_grad -- objective gradient
## ce_fun -- equality constraint functions
## ci_fun -- inequality constraint functions
## A == [grad_{x_1} cx_fun, grad_{x_2} cx_fun, ..., grad_{x_n} cx_fun]^T
x = x0;
obj = feval (obj_fun, x0);
globals.nfun = 1;
if (have_grd)
c = feval (obj_grd, x0);
else
c = feval (obj_grd, x0, obj);
endif
## Choose an initial NxN symmetric positive definite Hessian approximation B.
n = length (x0);
if (have_hess)
B = feval (obj_hess, x0);
else
B = eye (n, n);
endif
ce = feval (ce_fun, x0);
F = feval (ce_grd, x0);
ci = feval (ci_fun, x0);
C = feval (ci_grd, x0);
A = [F; C];
## Choose an initial lambda (x is provided by the caller).
lambda = 100 * ones (rows (A), 1);
qp_iter = 1;
alpha = 1;
info = 0;
iter = 0;
## report (); # Called with no arguments to initialize reporting
## report (iter, qp_iter, alpha, __sqp_nfun__, obj);
while (++iter < iter_max)
## Check convergence. This is just a simple check on the first
## order necessary conditions.
nr_f = rows (F);
lambda_e = lambda((1:nr_f)');
lambda_i = lambda((nr_f+1:end)');
con = [ce; ci];
t0 = norm (c - A' * lambda);
t1 = norm (ce);
t2 = all (ci >= 0);
t3 = all (lambda_i >= 0);
t4 = norm (lambda .* con);
## Normal convergence. All constraints are satisfied
## and objective has converged.
if (t2 && t3 && max ([t0; t1; t4]) < tol)
info = 101;
break;
endif
## Compute search direction p by solving QP.
g = -ce;
d = -ci;
old_lambda = lambda;
[p, obj_qp, INFO, lambda] = qp (x, B, c, F, g, [], [], d, C,
Inf (size (d)), struct ("TolX", tol));
info = INFO.info;
## FIXME: check QP solution and attempt to recover if it has failed.
## For now, just warn about possible problems.
id = "Octave:SQP-QP-subproblem";
switch (info)
case 2
warning (id, "sqp: QP subproblem is non-convex and unbounded");
case 3
warning (id, "sqp: QP subproblem failed to converge in %d iterations",
INFO.solveiter);
case 6
warning (id, "sqp: QP subproblem is infeasible");
lambda = old_lambda; # The return value was size 0x0 in this case.
endswitch
## Choose mu such that p is a descent direction for the chosen
## merit function phi.
[x_new, alpha, obj_new, globals] = ...
linesearch_L1 (x, p, obj_fun, obj_grd, ce_fun, ci_fun, lambda, ...
obj, c, globals);
delx = x_new - x;
## Check if step size has become too small (indicates lack of progress).
if (norm (delx) < tol * norm (x))
info = 104;
break;
endif
## Evaluate objective function, constraints, and gradients at x_new.
if (have_grd)
c_new = feval (obj_grd, x_new);
else
c_new = feval (obj_grd, x_new, obj_new);
endif
ce_new = feval (ce_fun, x_new);
F_new = feval (ce_grd, x_new);
ci_new = feval (ci_fun, x_new);
C_new = feval (ci_grd, x_new);
A_new = [F_new; C_new];
## Set
##
## s = alpha * p
## y = grad_x L (x_new, lambda) - grad_x L (x, lambda})
y = c_new - c;
if (! isempty (A))
t = ((A_new - A)'*lambda);
y -= t;
endif
if (have_hess)
B = feval (obj_hess, x);
else
## Update B using a quasi-Newton formula.
delxt = delx';
## Damped BFGS. Or maybe we would actually want to use the Hessian
## of the Lagrangian, computed directly?
d1 = delxt*B*delx;
t1 = 0.2 * d1;
t2 = delxt*y;
if (t2 < t1)
theta = 0.8*d1/(d1 - t2);
else
theta = 1;
endif
r = theta*y + (1-theta)*B*delx;
d2 = delxt*r;
## Check if the next BFGS update will work properly.
## If d1 or d2 vanish, the BFGS update will fail.
if (d1 == 0 || d2 == 0)
info = 102;
break;
endif
B = B - B*delx*delxt*B/d1 + r*r'/d2;
endif
x = x_new;
obj = obj_new;
c = c_new;
ce = ce_new;
F = F_new;
ci = ci_new;
C = C_new;
A = A_new;
## report (iter, qp_iter, alpha, __sqp_nfun__, obj);
endwhile
## Check if we've spent too many iterations without converging.
if (iter >= iter_max)
info = 103;
endif
nf = globals.nfun;
endfunction
function [merit, obj, globals] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, ...
x, mu, globals)
ce = feval (ce_fun, x);
ci = feval (ci_fun, x);
idx = ci < 0;
con = [ce; ci(idx)];
if (isempty (obj))
obj = feval (obj_fun, x);
globals.nfun += 1;
endif
merit = obj;
t = norm (con, 1) / mu;
if (! isempty (t))
merit += t;
endif
endfunction
function [x_new, alpha, obj, globals] = ...
linesearch_L1 (x, p, obj_fun, obj_grd, ce_fun, ci_fun, lambda, obj, c, globals)
## Choose parameters
##
## eta in the range (0, 0.5)
## tau in the range (0, 1)
eta = 0.25;
tau = 0.5;
delta_bar = sqrt (eps);
if (isempty (lambda))
mu = 1 / delta_bar;
else
mu = 1 / (norm (lambda, Inf) + delta_bar);
endif
alpha = 1;
ce = feval (ce_fun, x);
[phi_x_mu, obj, globals] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, ...
mu, globals);
D_phi_x_mu = c' * p;
d = feval (ci_fun, x);
## only those elements of d corresponding
## to violated constraints should be included.
idx = d < 0;
t = - norm ([ce; d(idx)], 1) / mu;
if (! isempty (t))
D_phi_x_mu += t;
endif
while (1)
[p1, obj, globals] = phi_L1 ([], obj_fun, ce_fun, ci_fun, ...
x+alpha*p, mu, globals);
p2 = phi_x_mu+eta*alpha*D_phi_x_mu;
if (p1 > p2)
## Reset alpha = tau_alpha * alpha for some tau_alpha in the
## range (0, tau).
tau_alpha = 0.9 * tau; # ??
alpha = tau_alpha * alpha;
else
break;
endif
endwhile
x_new = x + alpha * p;
endfunction
function grd = fdgrd (f, x, y0)
if (! isempty (f))
nx = length (x);
grd = zeros (nx, 1);
deltax = sqrt (eps);
for i = 1:nx
t = x(i);
x(i) += deltax;
grd(i) = (feval (f, x) - y0) / deltax;
x(i) = t;
endfor
else
grd = zeros (0, 1);
endif
endfunction
function jac = fdjac (f, x)
nx = length (x);
if (! isempty (f))
y0 = feval (f, x);
nf = length (y0);
nx = length (x);
jac = zeros (nf, nx);
deltax = sqrt (eps);
for i = 1:nx
t = x(i);
x(i) += deltax;
jac(:,i) = (feval (f, x) - y0) / deltax;
x(i) = t;
endfor
else
jac = zeros (0, nx);
endif
endfunction
function grd = fd_obj_grd (x, obj, obj_fun)
grd = fdgrd (obj_fun, x, obj);
endfunction
function res = empty_cf (x)
res = zeros (0, 1);
endfunction
function res = empty_jac (x)
res = zeros (0, length (x));
endfunction
function jac = fd_ce_jac (x, ce_fun)
jac = fdjac (ce_fun, x);
endfunction
function jac = fd_ci_jac (x, cifcn)
## cifcn = constraint function without gradients and lb or ub
jac = fdjac (cifcn, x);
endfunction
function res = cf_ub_lb (x, lbidx, ubidx, globals)
## function returning constraint evaluated at x, and distance between ub/lb
## and x, when they are specified (otherwise return empty)
## inequality constraints
if (! isempty (globals.cifcn))
ci = feval (globals.cifcn, x);
else
ci = [];
endif
## lower bounds
if (! isempty (globals.lb))
lb = x(lbidx, 1) - globals.lb;
else
lb = [];
endif
## upper bounds
if (! isempty (globals.ub))
ub = globals.ub - x(ubidx, 1);
else
ub = [];
end
res = [ci; lb; ub];
endfunction
function res = cigrad_ub_lb (x, bgrad, globals)
cigradfcn = @(x) fd_ci_jac (x, globals.cifcn);
if (iscell (globals.cif) && length (globals.cif) > 1)
cigradfcn = globals.cif{2};
endif
if (isempty (cigradfcn))
res = bgrad;
else
res = [feval(cigradfcn,x); bgrad];
endif
endfunction
## Utility function used to debug sqp
function report (iter, qp_iter, alpha, nfun, obj)
if (nargin == 0)
printf (" Itn ItQP Step Nfun Objective\n");
else
printf ("%5d %4d %8.1g %5d %13.6e\n", iter, qp_iter, alpha, nfun, obj);
endif
endfunction
################################################################################
## Test Code
%!function r = __g (x)
%! r = [sumsq(x)-10;
%! x(2)*x(3)-5*x(4)*x(5);
%! x(1)^3+x(2)^3+1 ];
%!endfunction
%!
%!function obj = __phi (x)
%! obj = exp (prod (x)) - 0.5*(x(1)^3 + x(2)^3 + 1)^2;
%!endfunction
%!
%!test
%!
%! x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];
%!
%! [x, obj, info, iter, nf, lambda] = sqp (x0, @__phi, @__g, []);
%!
%! x_opt = [-1.717143501952599;
%! 1.595709610928535;
%! 1.827245880097156;
%! -0.763643103133572;
%! -0.763643068453300];
%!
%! obj_opt = 0.0539498477702739;
%!
%! assert (x, x_opt, 8*sqrt (eps));
%! assert (obj, obj_opt, sqrt (eps));
## Test input validation
%!error sqp ()
%!error sqp (1)
%!error sqp (1,2,3,4,5)
%!error sqp (ones (2,2))
%!error sqp (1, cell (4,1))
%!error sqp (1, cell (3,1), cell (3,1))
%!error sqp (1, cell (3,1), cell (2,1), cell (3,1))
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1), ones (2,2),[])
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[], ones (2,2))
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1),1,-1)
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[], ones (2,2))
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],-1)
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],1.5)
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],[], ones (2,2))
%!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],[],-1)