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function [par, x0, tau0, delta0] = uctpget(p,m,n)
%UCTPGET Define test problem for unconstrained minimization.
% Call: [par, x0, tau0, delta0] = uctpget(p,m,n)
% Input parameters
% p : Problem number, integer in the range [1,22]
% m,n : Number of components in the vectors f(x) and x,
% respectively. Not variable in all problems.
% Output parameters
% par : Struct defining the problem.
% par.p : Problem number
% par.pt = 0 signifies a least squares problem, (p <= 21)
% otherwise a general problem.
% par.xm : Solution. (NaN-s if m or n is free)
% For some problems par has more fields.
% x0 : Standard starting point.
% tau0 : Standard value for the parameter opts(1) in the
% Marquardt functions of the toolbox.
% delta0 : Standard value for the parameter opts(1) in the
% DOGLEG function of the toolbox.
%
% See H.B. Nielsen: "UCTP - Test Problems for Unconstrained
% Optimization", Report IMM-REP-2000-18
% Version 04.01.31. hbn(a)imm.dtu.dk
switch p
case 1 % Linear function. Full rank
par = struct('p',1, 'pt',0, 'A',eye(m,n) - (2/m)*ones(m,n), ...
'xm',-ones(n,1));
x0 = ones(1,n); tau0 = 1e-8; delta0 = 10;
case 2 % Linear function. Rank 1
xm = (6/(2*m+1)/n/(n+1)) * ones(n,1);
par = struct('p',2, 'pt',0, 'A',[1:m]'*[1:n], 'xm',xm);
x0 = ones(1,n); tau0 = 1e-8; delta0 = 10;
case 3 % Linear function. Rank 1. Zero cols. and rows
xm = (6/(2*m-3)/(n-2)/(n+1)) * ones(n,1);
par = struct('p',3, 'pt',0, 'A',[0 1:m-2 0]'*[0 2:n-1 0], 'xm',xm);
x0 = ones(1,n); tau0 = 1e-8; delta0 = 10;
case 4 % Rosenbrock
par = struct('p',4, 'pt',0, 'xm',[1;1]);
x0 = [-1.2 1]; tau0 = 1; delta0 = 1;
case 5 % Helical Valley
par = struct('p',5, 'pt',0, 'xm',[1;0;0]);
x0 = [-1 0 0]; tau0 = 1; delta0 = 1;
case 6 % Powell singular
par = struct('p',6, 'pt',0, 'xm',zeros(4,1));
x0 = [3 -1 0 1]; tau0 = 1e-8; delta0 = 1;
case 7 % Freudenstein and Roth
xm = ([53;2] - sqrt(22)*[4;1])/3;
par = struct('p',7, 'pt',0, 'xm',xm);
x0 = [.5 -2]; tau0 = 1; delta0 = 1;
case 8 % Bard
Bard = [0.14 0.18 0.22 0.25 0.29 0.32 0.35 0.39 0.37 ...
0.58 0.73 0.96 1.34 2.10 4.39]';
u = [1:15]'; v = 16 - u; w = min([u'; v']).';
xm = [8.241055996e-2; 1.133036099; 2.343695172];
par = struct('p',8, 'pt',0, 'uvwy',[u v w Bard], 'xm',xm);
x0 = ones(1,3); tau0 = 1e-8; delta0 = 1;
case 9 % Kowalik and Osborne
xm = [0.1928069346; 0.1912823287; 0.1230565069; 0.1360623307];
y = [0.1957 0.1947 0.1735 0.1600 0.0844 0.0627 ...
0.0456 0.0342 0.0323 0.0235 0.0246]';
u = [4.0000 2.0000 1.0000 0.5000 0.2500 0.1670 ...
0.1250 0.1000 0.0833 0.0714 0.0625]';
par = struct('p',9, 'pt',0, 'yu',[y u], 'xm',xm);
x0 = [.25 .39 .415 .39]; tau0 = 1; delta0 = .1;
case 10 % Meyer
xm = [5.60963646990603e-003; 6.181346346e+003; 3.452236346e+002];
par = struct('p',10, 'pt',0, 'ty',[45+5*(1:16)' Meyer], 'xm',xm);
x0 = [.02 4e3 250]; tau0 = 1e-6; delta0 = 100;
case 11 % Watson
t = [1:29]'/29; B = ones(29,n);
for j = 2 : n, B(:,j) = t.*B(:,j-1); end
A = B(:,1:n-1) * diag(1:n-1); xm = [];
switch n
case 6, xm = [-1.572508640e-2; 1.012434869; -2.329916260e-1;
1.260430088; -1.513728923; 9.929964324e-1];
case 9, xm = [-1.530703649e-5; 9.997897039e-1; 1.476396369e-2
0.1463423283; 1.000821103; -2.617731141;
4.104403164; -3.143612279; 1.052626408];
case 12, xm = [-6.638060573e-9; 1.000001647; -5.639322310e-4
0.3478205409; -0.1567315046; 1.052815176
-3.247271154; 7.288434898; -10.27184824
9.074113648; -4.541375467; 1.012011889];
otherwise, xm = repmat(NaN,n,1);
end
par = struct('p',11, 'pt',0, 'A',A, 'B',B, 'xm',xm);
x0 = zeros(1,n); tau0 = 1e-8; delta0 = 1;
case 12 % Box 3-d
par = struct('p',12, 'pt',0, 't', .1*[1:m]', 'xm',[1; 10; 1]);
x0 = [0 10 20]; tau0 = 1e-8; delta0 = 1;
case 13 % Jennrich and Sampson
if m == 10, xm = .25782521367*[1;1]; else, xm = [NaN; NaN]; end
par = struct('p',13, 'pt',0, 't', [1:m]', 'xm',xm);
x0 = [.3 .4]; tau0 = 1; delta0 = .05;
case 14 % Brown and Dennis
if m == 20
xm = [-11.59443981; 13.20363001; -0.4034393057; 0.2367787344];
else, xm = repmat(NaN,4,1); end
par = struct('p',14, 'pt',0, 't', [1:m]'/5, 'xm',xm);
x0 = [25 5 -5 -1]; tau0 = 1e-3; delta0 = .5;
case 15 % Chebyquad
y = zeros(m,1);
if (n == 8) & (m == 8)
xm = [4.315283152e-2; 0.1930908899; 0.2663289000; 0.4999999000
0.5000001000; 0.7336711000; 0.8069090000; 0.9568470101];
elseif (n == 8) & (m == 16)
xm = [0.07974310852; 0.1783549801; 0.3101200148; 0.4412049499
0.5587950200; 0.6898800200; 0.8216450649; 0.9202569150];
elseif (n == 9) & (m == 9)
xm = [0.0442053010; 0.1994910050; 0.2356188900; 0.4160470950
0.5000001100; 0.5839528850; 0.7643811500; 0.8005091500; 0.9557948250];
elseif (n == 9) & (m == 18)
xm = [0.1018907415; 0.1894831408; 0.2944820487; 0.3970994868;
0.5000000000; 0.6029005132; 0.7055179513; 0.8105168592; 0.8981092585];
else
xm = repmat(NaN,n,1);
end
for i = 2 : 2 : m, y(i) = -1/(i^2 - 1); end
par = struct('p',15, 'pt',0, 'y', y, 'xm',xm);
x0 = [1:n]/(n+1); tau0 = 1; delta0 = 1/(n+1);
case 16 % Brown almost linear
A = eye(n-1,n) + ones(n-1,n);
par = struct('p',16, 'pt',0, 'A', A, 'xm',ones(n,1));
x0 = .5*ones(1,n); tau0 = 1; delta0 = 1;
case 17 % Osborne 1
xm = [0.3754100521; 1.935846917;
-1.464687141; 0.01286753465; 0.02212269965];
par = struct('p',17, 'pt',0, 'ty',Osborne1, 'xm',xm);
x0 = [.5 1.5 -1 .01 .02]; tau0 = 1e-8; delta0 = .1;
case 18 % Exponential fit. n = 4
xm = [-4.000026378; -4.999964833; 4.000242976; -4.000242571];
par = struct('p',18, 'pt',0, 'ty',hbnefit, 'xm',xm);
x0 = [-1 -2 1 -1]; tau0 = 1e-3; delta0 = 1;
case 19 % Exponential fit. n = 2
xm = [-4.000026708; -4.999964374];
par = struct('p',19, 'pt',0, 'ty',hbnefit, 'xm',xm);
x0 = [-1 -2]; tau0 = 1e-3; delta0 = 1;
case 20 % Scaled Meyer
xm = [2.481778299; 6.181346346; 3.452236346];
par = struct('p',20, 'pt',0, 'ty',[.45+.05*(1:16)' 1e-3*Meyer], 'xm',xm);
x0 = [8.85 4 2.5]; tau0 = 1; delta0 = 1;
case 21 % Separated Meyer
xm = [6.181346324e+3; 3.452236339e+2];
par = struct('p',21, 'pt',0, 'ty',[45+5*(1:16)' Meyer], 'xm',xm);
x0 = [4e3 250]; tau0 = 1; delta0 = 100;
case 22 % Exp and squares
nn = [1:n].^2; A = sum(1./nn);
y = .75; more = 1;
while more
yo = y; e = A*exp(-y); y = y - (y-e)/(1+e);
more = abs(y-yo) > 1e-12;
end
xm = exp(-y)./nn;
par = struct('p',22, 'pt',1, 'xm',xm(:));
x0 = zeros(1,n); tau0 = 1e-3; delta0 = 1;
otherwise
error(['Problem no ' int2str(p) ' is not available'])
end
function y = Meyer
y = [34780 28610 23650 19630 16370 13720 11540 ...
9744 8261 7030 6005 5147 4427 3820 ...
3307 2872]';
function ty = Osborne1
y = [0.844 0.908 0.932 0.936 0.925 0.908 0.881 ...
0.850 0.818 0.784 0.751 0.718 0.685 0.658 ...
0.628 0.603 0.580 0.558 0.538 0.522 0.506 ...
0.490 0.478 0.467 0.457 0.448 0.438 0.431 ...
0.424 0.420 0.414 0.411 0.406]';
t = 10 * (0 : 32); ty = [t(:) y];
function ty = hbnefit
y =[0.090542 0.124569 0.179367 0.195654 0.269707 ...
0.286027 0.289892 0.317475 0.308191 0.336995 ...
0.348371 0.321337 0.299423 0.338972 0.304763 ...
0.288903 0.300820 0.303974 0.283987 0.262078 ...
0.281593 0.267531 0.218926 0.225572 0.200594 ...
0.197375 0.182440 0.183892 0.152285 0.174028 ...
0.150874 0.126220 0.126266 0.106384 0.118923 ...
0.091868 0.128926 0.119273 0.115997 0.105831 ...
0.075261 0.068387 0.090823 0.085205 0.067203]';
t = .02 * [1:45]; ty = [t(:) y];