function fvec = dfovec(m,n,x,nprob) % This is a Matlab version of the subroutine dfovec.f % This subroutine specifies the nonlinear benchmark problems in % % Benchmarking Derivative-Free Optimization Algorithms % Jorge J. More' and Stefan M. Wild % SIAM J. Optimization, Vol. 20 (1), pp.172-191, 2009. % % The latest version of this subroutine is always available at % http://www.mcs.anl.gov/~more/dfo/ % The authors would appreciate feedback and experiences from numerical % studies conducted using this subroutine. % % The data file dfo.dat defines suitable values of m and n % for each problem number nprob. % % This subroutine defines the functions of 22 nonlinear % least squares problems. The allowable values of (m,n) for % functions 1,2 and 3 are variable but with m .ge. n. % For functions 4,5,6,7,8,9 and 10 the values of (m,n) are % (2,2),(3,3),(4,4),(2,2),(15,3),(11,4) and (16,3), respectively. % Function 11 (Watson) has m = 31 with n usually 6 or 9. % However, any n, n = 2,...,31, is permitted. % Functions 12,13 and 14 have n = 3,2 and 4, respectively, but % allow any m .ge. n, with the usual choices being 10,10 and 20. % Function 15 (Chebyquad) allows m and n variable with m .ge. n. % Function 16 (Brown) allows n variable with m = n. % For functions 17 and 18, the values of (m,n) are % (33,5) and (65,11), respectively. % % fvec = ssqfcn(m,n,x,nprob) % fvec is an output array of length m which contains the nprob % function evaluated at x. % m and n are positive integer input variables. n must not % exceed m. % x is an input array of length n. % nprob is a positive integer input variable which defines the % number of the problem. nprob must not exceed 22. % % Argonne National Laboratory % Jorge More' and Stefan Wild. January 2008. % Set lots of constants: c13 = 1.3d1; c14 = 1.4d1; c29 = 2.9d1; c45 = 4.5d1; v = [4.0d0,2.0d0,1.0d0,5.0d-1,2.5d-1,1.67d-1,1.25d-1,1.0d-1,8.33d-2,... 7.14d-2,6.25d-2]; y1 = [1.4d-1,1.8d-1,2.2d-1,2.5d-1,2.9d-1,3.2d-1,3.5d-1,3.9d-1,3.7d-1,... 5.8d-1,7.3d-1,9.6d-1,1.34d0,2.1d0,4.39d0]; y2 = [1.957d-1,1.947d-1,1.735d-1,1.6d-1,8.44d-2,6.27d-2,4.56d-2,3.42d-2,... 3.23d-2,2.35d-2,2.46d-2]; y3 = [3.478d4,2.861d4,2.365d4,1.963d4,1.637d4,1.372d4,1.154d4,9.744d3,... 8.261d3,7.03d3,6.005d3,5.147d3,4.427d3,3.82d3,3.307d3,2.872d3]; y4 = [8.44d-1,9.08d-1,9.32d-1,9.36d-1,9.25d-1,9.08d-1,8.81d-1,8.5d-1,... 8.18d-1,7.84d-1,7.51d-1,7.18d-1,6.85d-1,6.58d-1,6.28d-1,6.03d-1,... 5.8d-1,5.58d-1,5.38d-1,5.22d-1,5.06d-1,4.9d-1,4.78d-1,4.67d-1,... 4.57d-1,4.48d-1,4.38d-1,4.31d-1,4.24d-1,4.2d-1,4.14d-1,4.11d-1,... 4.06d-1]; y5 = [1.366d0,1.191d0,1.112d0,1.013d0,9.91d-1,8.85d-1,8.31d-1,8.47d-1,... 7.86d-1,7.25d-1,7.46d-1,6.79d-1,6.08d-1,6.55d-1,6.16d-1,6.06d-1,... 6.02d-1,6.26d-1,6.51d-1,7.24d-1,6.49d-1,6.49d-1,6.94d-1,6.44d-1,... 6.24d-1,6.61d-1,6.12d-1,5.58d-1,5.33d-1,4.95d-1,5.0d-1,4.23d-1,... 3.95d-1,3.75d-1,3.72d-1,3.91d-1,3.96d-1,4.05d-1,4.28d-1,4.29d-1,... 5.23d-1,5.62d-1,6.07d-1,6.53d-1,6.72d-1,7.08d-1,6.33d-1,6.68d-1,... 6.45d-1,6.32d-1,5.91d-1,5.59d-1,5.97d-1,6.25d-1,7.39d-1,7.1d-1,... 7.29d-1,7.2d-1,6.36d-1,5.81d-1,4.28d-1,2.92d-1,1.62d-1,9.8d-2,5.4d-2]; % Initialize things: fvec = zeros(m,1); sum = 0; switch nprob case 1 % Linear function - full rank. for j = 1:n sum = sum + x(j); end temp = 2*sum/m + 1; for i = 1:m fvec(i) = -temp; if (i <= n) fvec(i) = fvec(i) + x(i); end end case 2 % Linear function - rank 1. for j = 1:n sum = sum + j*x(j); end for i = 1:m fvec(i) = i*sum - 1; end case 3 % Linear function - rank 1 with zero columns and rows. for j = 2:n-1 sum = sum + j*x(j); end for i = 1:m-1 fvec(i) = (i-1)*sum - 1; end fvec(m) = -1; case 4 % Rosenbrock function. fvec(1) = 10*(x(2) - x(1)^2); fvec(2) = 1 - x(1); case 5 % Helical valley function. if (x(1) > 0) th = atan(x(2)/x(1))/(2*pi); elseif (x(1) < 0) th = atan(x(2)/x(1))/(2*pi) + .5; else % x(1)=0 th = .25; end r = sqrt(x(1)^2+x(2)^2); fvec(1) = 10*(x(3) - 10*th); fvec(2) = 10*(r-1); fvec(3) = x(3); case 6 % Powell singular function. fvec(1) = x(1) + 10*x(2); fvec(2) = sqrt(5)*(x(3) - x(4)); fvec(3) = (x(2) - 2*x(3))^2; fvec(4) = sqrt(10)*(x(1) - x(4))^2; case 7 % Freudenstein and Roth function. fvec(1) = -c13 + x(1) + ((5 - x(2))*x(2) - 2)*x(2); fvec(2) = -c29 + x(1) + ((1 + x(2))*x(2) - c14)*x(2); case 8 % Bard function. for i = 1:15 tmp1 = i; tmp2 = 16-i; tmp3 = tmp1; if (i > 8) tmp3 = tmp2; end fvec(i) = y1(i) - (x(1) + tmp1/(x(2)*tmp2 + x(3)*tmp3)); end case 9 % Kowalik and Osborne function. for i = 1:11 tmp1 = v(i)*(v(i) + x(2)); tmp2 = v(i)*(v(i) + x(3)) + x(4); fvec(i) = y2(i) - x(1)*tmp1/tmp2; end case 10 % Meyer function. for i = 1:16 temp = 5*i + c45 + x(3); tmp1 = x(2)/temp; tmp2 = exp(tmp1); fvec(i) = x(1)*tmp2 - y3(i); end case 11 % Watson function. for i = 1:29 div = i/c29; s1 = 0; dx = 1; for j = 2:n s1 = s1 + (j-1)*dx*x(j); dx = div*dx; end s2 = 0; dx = 1; for j = 1:n s2 = s2 + dx*x(j); dx = div*dx; end fvec(i) = s1 - s2^2 - 1; end fvec(30) = x(1); fvec(31) = x(2) - x(1)^2 - 1; case 12 % Box 3-dimensional function. for i = 1:m temp = i; tmp1 = temp/10; fvec(i) = exp(-tmp1*x(1)) - exp(-tmp1*x(2))+ ... (exp(-temp) - exp(-tmp1))*x(3); end case 13 % Jennrich and Sampson function. for i = 1:m temp = i; fvec(i) = 2 + 2*temp - exp(temp*x(1)) - exp(temp*x(2)); end case 14 % Brown and Dennis function. for i = 1:m temp = i/5; tmp1 = x(1) + temp*x(2) - exp(temp); tmp2 = x(3) + sin(temp)*x(4) - cos(temp); fvec(i) = tmp1^2 + tmp2^2; end case 15 % Chebyquad function. for j = 1:n t1 = 1; t2 = 2*x(j) - 1; t = 2*t2; for i = 1:m fvec(i) = fvec(i) + t2; th = t*t2 - t1; t1 = t2; t2 = th; end end iev = -1; for i = 1:m fvec(i) = fvec(i)/n; if (iev > 0) fvec(i) = fvec(i) + 1/(i^2 - 1); end iev = -iev; end case 16 % Brown almost-linear function. sum1 = -(n+1); prod1 = 1; for j = 1:n sum1 = sum1 + x(j); prod1 = x(j)*prod1; end for i = 1:n-1 fvec(i) = x(i) + sum1; end fvec(n) = prod1 - 1; case 17 % Osborne 1 function. for i = 1:33 temp = 10*(i-1); tmp1 = exp(-x(4)*temp); tmp2 = exp(-x(5)*temp); fvec(i) = y4(i) - (x(1) + x(2)*tmp1 + x(3)*tmp2); end case 18 % Osborne 2 function. for i = 1:65 temp = (i-1)/10; tmp1 = exp(-x(5)*temp); tmp2 = exp(-x(6)*(temp-x(9))^2); tmp3 = exp(-x(7)*(temp-x(10))^2); tmp4 = exp(-x(8)*(temp-x(11))^2); fvec(i) = y5(i) - (x(1)*tmp1 + x(2)*tmp2 + ... x(3)*tmp3 + x(4)*tmp4); end case 19 % Bdqrtic % n>=5, m = (n-4)*2 for i=1:n-4 fvec(i)=(-4*x(i)+3.0); fvec(n-4+i)=(x(i)^2+2*x(i+1)^2+3*x(i+2)^2+4*x(i+3)^2+5*x(n)^2); end case 20 % Cube % n=2; m=n; fvec(1) = (x(1)-1.0); for i=2:n fvec(i) = 10*(x(i)-x(i-1)^3); end case 21 % Mancino % n >=2; m=n for i=1:n ss=0; for j=1:n v2 = sqrt (x(i)^2 +i/j); ss = ss+v2*((sin(log(v2)))^5 + (cos(log(v2)))^5); end fvec(i)=1400*x(i) + (i-50)^3 + ss; end case 22 % Heart8ls % m=n=8 fvec(1) = x(1) + x(2) + 0.69; fvec(2) = x(3) + x(4) + 0.044; fvec(3) = x(5)*x(1) + x(6)*x(2) - x(7)*x(3) - x(8)*x(4) + 1.57; fvec(4) = x(7)*x(1) + x(8)*x(2) + x(5)*x(3) + x(6)*x(4) + 1.31; fvec(5) = x(1)*(x(5)^2-x(7)^2) - 2.0*x(3)*x(5)*x(7) + ... x(2)*(x(6)^2-x(8)^2) - 2.0*x(4)*x(6)*x(8) + 2.65; fvec(6) = x(3)*(x(5)^2-x(7)^2) + 2.0*x(1)*x(5)*x(7) + ... x(4)*(x(6)^2-x(8)^2) + 2.0*x(2)*x(6)*x(8) - 2.0; fvec(7) = x(1)*x(5)*(x(5)^2-3.0*x(7)^2) + ... x(3)*x(7)*(x(7)^2-3.0*x(5)^2) + ... x(2)*x(6)*(x(6)^2-3.0*x(8)^2) + ... x(4)*x(8)*(x(8)^2-3.0*x(6)^2) + 12.6; fvec(8) = x(3)*x(5)*(x(5)^2-3.0*x(7)^2) - ... x(1)*x(7)*(x(7)^2-3.0*x(5)^2) + ... x(4)*x(6)*(x(6)^2-3.0*x(8)^2) - ... x(2)*x(8)*(x(8)^2-3.0*x(6)^2) - 9.48; end