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function [F, dF, d2F] = uctpval(x,par)
%UCTPFGH Evaluate test problem for nconstrained minimization,
% as defined by UCTPGET.
% Call
% F = uctpval(x,par)
% [F, dF] = uctpval(x,par)
% [F, dF, d2F] = uctpval(x,par)
% Input parameters
% x : Argument.
% par : Struct defining the problem.
% Output parameters
% F : The function. If par.p <= 21 and par.pt = 0,
% then F is a vector, otherwise a scalar.
% dF : If par.p <= 21 and par.pt = 0, then dF is the
% Jacobian J(x), otherwize the gradient F'(x) .
% d2F : Presumes par.pt ~= 0. Hessian matrix
% Version 04.04.15. hbn(a)imm.dtu.dk
% Check for Hessian
if nargout > 2, par.pt == 1; end
% Ensure that x is a column vector
x = x(:);
if par.p <= 21 % Least squares problem
difH = 0; % Not difference approximation to Hessian
switch par.p
case {1,2,3} % Linear function
J = par.A; f = J*x - 1;
if nargout > 2, H = J'*J; end
case 4 % Rosenbrock
f = [10*(x(2) - x(1)^2); 1-x(1)];
if nargout > 1
J = [-20*x(1) 10; -1 0];
if nargout > 2, H = J'*J + f(1)*[-20 0; 0 0]; end
end
case 5 % Helical Valley
t = atan(x(2)/x(1))/(2*pi);
if x(1) < 0, t = t + .5; end
nx = norm(x(1:2)); nx2 = nx*nx;
f = [10*(x(3) - 10*t); 10*(nx - 1); x(3)];
if nargout > 1
K1 = 50/pi/nx2; K2 = 10/nx;
J = [K1*x(2) -K1*x(1) 10; K2*x(1:2)' 0; 0 0 1];
if nargout > 2
H = J'*J;
q1 = x(1)^2; q2 = x(2)^2; p = x(1)*x(2); ii = 1:2;
H(ii,ii) = H(ii,ii) + f(1)*K1/nx2*[-2*p q1-q2; q1-q2 2*p] ...
+ f(2)*K2/nx2*[q2 -p; -p q1];
end
end
case 6 % Powell singular
s5 = sqrt(5); s10 = sqrt(10);
d3 = x(2) - 2*x(3); d4 = x(1) - x(4);
f = [x(1)+10*x(2); s5*(x(3) - x(4)); d3^2; s10*d4^2];
if nargout > 1
J = [1 10 0 0; 0 0 s5 -s5; 0 2*d3*[1 -2] 0; 2*s10*d4*[1 0 0 -1]];
if nargout > 2
H = J'*J;
H(2:3,2:3) = H(2:3,2:3) + f(3)*[2 -4;-4 8];
H([1 4],[1 4]) = H([1 4],[1 4]) + f(4)*2*s10*[1 -1;-1 1];
end
end
case 7 % Freudenstein and Roth
x1 = x(1); x2 = x(2);
f = [(x1 - x2*(2 - x2*(5 - x2)) - 13)
(x1 - x2*(14 - x2*(1 + x2)) - 29)];
if nargout > 1
J = [1 (-2 + x2*(10 - 3*x2)); 1 (-14 + x2*(2 + 3*x2))];
if nargout > 2
H = J'*J;
H(2,2) = H(2,2) + f(1)*10-6*x2 + f(2)*(2+6*x2);
end
end
case 8 % Bard
D = par.uvwy(:,2:3)*x(2:3);
f = par.uvwy(:,4) - (x(1) + par.uvwy(:,1)./D);
if nargout > 1
F = -par.uvwy(:,1)./D.^2;
J = -[ones(size(D)) (F*[1 1]).*par.uvwy(:,2:3)];
if nargout > 2
H = J'*J; ii = 2:3; A = zeros(2,2);
for i = 1 : length(f)
u = par.uvwy(i,1); v = par.uvwy(i,2); w = par.uvwy(i,3);
J(i,:) = [-1 [v w]*u/D(i)^2];
A = A + f(i)*[-2*v^2 v*w; v*w -2*w^2]*u/D(i)^3;
end
H(ii,ii) = H(ii,ii) + A;
end
end
case 9 % Kowalik and Osborne
u = par.yu(:,2); x1 = x(1);
N = u.*(u + x(2)); D = u.*(u + x(3)) + x(4);
f = par.yu(:,1) - x1*N./D;
if nargout > 1
F = -x1*N./D.^2;
J = -[N./D x1*u./D F.*u F];
if nargout > 2
H = J'*J;
for i = 1 : 11
Hi = zeros(4,4);
T = x(1)*N(i); ui = u(i); Di = D(i); Ni = N(i); u1 = [ui 1];
a = [-1 u1*(Ni/Di)]*(ui/Di);
Hi(1,2:4) = a; Hi(2:4,1) = a';
Hi(2,3:4) = u1*(x1/Di^2); Hi(3:4,2) = Hi(ii(2),3:4)';
a = -2*T/Di^3;
Hi(3,3:4) = u1*(ui*a); Hi(4,3) = Hi(ii(3),4);
Hi(4,4) = a; H = H + f(i)*Hi;
end
end
end
case 10 % Meyer
D = par.ty(:,1) + x(3); q = x(2)./D; e = exp(q);
f = x(1)*e - par.ty(:,2);
if nargout > 1
J = [e x(1)*e./D -x(1)*x(2)*e./D.^2];
if nargout > 2
H = J'*J;
for i = 1 : m
p = x(1)/D(i); a = q(i); b = p*a; c = -p*(1+a);
H = H + f(i)*(e(i)/D(i))*[0 1 -a;1 p c;-a c b*(2+a)];
end
end
end
case 11 % Watson
m = 31; n = length(x);
A = [zeros(29,1) par.A]; B = par.B;
g = B*x; x1 = x(1); x2 = x(2);
f = [(par.A*x(2:end) - g.^2 - 1); x1; x2-x1^2-1];
if nargout > 1
J = zeros(31,n); g = -2*g;
J(:,1) = [g; 1; -2*x1]; J(31,2) = 1;
for j = 2 : n
J(1:29,j) = par.A(:,j-1) + g.*par.B(:,j);
end
if nargout > 2
H = J'*J;
for i = 1 : 29
bi = B(i,:);
H = H - (2*f(i)*bi')*bi;
end
H(1,1) = H(1,1) - 2*f(end);
end
end
case 12 % Box 3-d
t = par.t; E = exp(-t*[x(1:2)' 1 10]);
c = [1; -1; -x(3)*[1; -1]];
f = E*c;
if nargout > 1
J = [-t.*E(:,1) t.*E(:,2) E(:,3:4)*[-1;1]];
if nargout > 2
H = J'*J; ii = 1:2;
for i = 1 : length(t)
H(ii,ii) = H(ii,ii) + f(i)*t(i)*diag(J(i,ii));
end
end
end
case 13 % Jennrich and Sampson
t = par.t; E = exp(t*x');
f = 2*(1 + t) - E*[1; 1];
if nargout > 1
J = -(t*ones(1,2)).*E;
if nargout > 2
H = J'*J;
for i = 1 : length(t)
H = H + f(i)*t(i)*diag(J(i,:));
end
end
end
case 14 % Brown and Dennis
t = par.t; st = sin(t);
d1 = x(1) + x(2)*t - exp(t); d2 = x(3) + x(4)*st - cos(t);
f = d1.^2 + d2.^2;
if nargout > 1
J = 2*[d1 d1.*t d2 d2.*st];
if nargout > 2
H = J'*J;
for i = 1 : length(t)
a = [1 t(i) 1 st(i)];
H = H + a'*(2*f(i)*a);
end
end
end
case 15 % Chebyquad
z = real(acos(2*x' - 1)); f = -par.y;
m = length(f); n = length(x);
for r = 1 : m, f(r) = sum(cos(r*z))/n + f(r); end
if nargout > 1
J = zeros(m,n); d = sqrt(1 - (2*x' - 1).^2);
nz = find(d); iz = find(d == 0);
for r = 1 : m
J(r,nz) = 2/n*r*sin(r*z(nz)) ./ d(nz);
J(r,iz) = 2/n*r^2;
end
J = real(J);
if nargout > 2, difH = 1; end
end
case 16 % Brown almost linear
n = length(x); p = prod(x);
f = [par.A*x-(n+1); p-1];
if nargout > 1
pp = zeros(1,n);
for j = 1 : n, pp(j) = prod(x([1:j-1 j+1:n])); end
J = [par.A; pp];
if nargout > 2 % Hessian
if n == 1, H = 0;
elseif n == 2, H = [0 1;1 0];
else
for i = 1 : n-1
for j = i+1 : n
H(i,j) = prod(x([1:i-1 i+1:j-1 j+1:n]));
H(j,i) = H(i,j);
end
end
end
end
end
case 17 % Osborne 1
t = par.ty(:,1); E = exp(-t*x(4:5)');
f = par.ty(:,2) - x(1) - E*x(2:3);
m = length(t); n = length(x);
if nargout > 1
J = -ones(length(t),5);
J(:,2:5) = -[E -x(2)*t.*E(:,1) -x(3)*t.*E(:,2)];
if nargout > 2
A4 = zeros(5,5);
A1 = A4; A1(2,4) = 1; A1(4,2) = 1;
A2 = A4; A2(3,5) = 1; A2(5,3) = 1;
A3 = A4; A3(4,4) = -x(2); A4(5,5) = -x(3);
H = J'*J;
for i = 1 : length(t)
H = H + f(i)*t(i)*(E(i,1)*(A1 + t(i)*A3) ...
+ E(i,2)*(A2 + t(i)*A4));
end
end
end
case 18 % Exponential fit. n = 4
t = par.ty(:,1); E = exp(t*x(1:2)');
f = par.ty(:,2) - E*x(3:4);
if nargout > 1
J = -[E E];
J(:,1:2) = -[x(3)*t.*E(:,1) x(4)*t.*E(:,2)];
if nargout > 2
H = J'*J;
j1 = [1 3]; j2 = [2 4]; x3 = -x(3); x4 = -x(4);
for i = 1 : length(t)
H(1,j1) = H(1,j1) + f(i)*t(i)*E(i,1)*[x3*t(i) 1];
H(2,j2) = H(2,j2) + f(i)*t(i)*E(i,2)*[x4*t(i) 1];
end
H(3,1) = H(1,3); H(4,2) = H(2,4);
end
end
case 19 % Exponential fit. n = 2
t = par.ty(:,1); E = exp(t*x'); c = E\par.ty(:,2);
f = par.ty(:,2) - E*c;
if nargout > 1
A = E'*E; H = [t t] .* E;
G = A\ (diag(H'*f) - H'*E*diag(c));
J = -(E*G + H*diag(c));
if nargout > 2, difH = 1; end
end
case 20 % Modified Meyer
D = par.ty(:,1) + x(3); e = exp(10*x(2)./D - 13);
f = x(1)*e - par.ty(:,2);
if nargout > 1
q = (10*x(1)./D) .* e;
J = [e q -(x(2)./D).*q];
if nargout > 2
H = J'*J; A = zeros(3,3);
x1 = x(1); x2 = x(2);
for i = 1 : length(f)
Di = D(i); A(1,:) = [0 Di -1];
A(2,:) = [Di 10*x1^2 -x1*(1 +10*x2/Di)];
A(3,:) = [-1 -x1*(1 +10*x2/Di) 2*x1*(1 +5*x2/Di)];
H = H + f(i)*e(i)/Di^2*A;
end
end
end
case 21 % Separated Meyer
x2 = par.ty(:,1) + x(2); a = x(1) ./ x2;
F = exp(a); E = F'*F; c = (F' * par.ty(:,2))/E;
f = F*c - par.ty(:,2);
if nargout > 1
d1 = x(1) ./ x2; d2 = c ./ x2;
by = (f + F*c) ./ x2; by = [by -by.*d1];
dc = -(F'*by)/E;
J = [F F] .* [dc(1)+d2 dc(2)-d1.*d2];
if nargout > 2, difH = 1; end
end
end
% Reformulate to scalar problem
if par.pt == 0 % Vector function
F = f;
if nargout > 1, dF = J; end % Jacobian
else % Scalar function
F = .5 * norm(f)^2;
if nargout > 1 % Gradient
dF = J'*f;
if nargout > 2 % Hessian
if difH % difference approximation
n = length(x); H = zeros(n,n); u = x;
for j = 1 : n
d = 1e-5*(abs(x(j)) + 1e-5); % step length
fj = x(j) + d; bj = x(j) - d;
u(j) = fj; [dum gf] = uctpval(u,par);
u(j) = bj; [dum gb] = uctpval(u,par);
H(:,j) = (gf - gb)/(fj - bj);
end
end
H = (H + H')/2; % symmetrize
d2F = H;
end
end
end
else % 'Born' scalar function
switch par.p
case 22 % Exp and squares
n = length(x);
e = exp(-sum(x)); ii = (1:n).^2;
F = e + .5*sum(ii(:) .* x.^2);
if nargout > 1
dF = -e + ii(:) .* x;
if nargout > 2
d2F = diag(ii) + e;
end
end
end
end