minimization problem -using simplex method

sir i want to implement minimization problem using simplx method can i use matrix for this and how can i iterate each time the matrix as per simplex method rule, plz any one if know tell me (my question simply tells processing of simplex method )

Answer 1

[James Tursa]

Which Simplex Method? Dantzig or Nelder-Mead?

[Walter Roberson]

Nelder-Mead is intended for continuous functions, not for discrete matrices, so it does not appear that would be appropriate.

[muthu kumar]

ya ur idea is exacly correct thanks

[Walter Roberson]

Muthu, I'm not sure who that was addressed to?

[James Tursa]

@Walter: It was not clear to me from original post what problem was being solved and what the "matrix" was. I would also have assumed a Linear Programming problem as you did (which, btw, is also a "continuous" function problem ... not sure what you mean by "discrete") but was hoping for more clarification from OP. (Side Note: Neldor-Mead could also be used to solve a LP "matrix" problem, albeit I would expect not very efficiently)

[Walter Roberson]

With the question as given, I did not consider a Linear Programming problem: it looked to me like perhaps the poster might just be looking for a way to find the minima of an array (number of dimensions not specified.) Why they would want that, I don't know.

Answer 2

[antonio espejo]

Hello,

My name is Antonio, and I try to solve an optimization proble.

I have a mathematical model, Lines 1-82 of code, that calculate values of 'xp' and 'Pp' in time 'bucle j', assumming an initial values A,B, Pa(1) y Q, that I call with a vector 'x'. Finally, I calculate in each time 'L', and 'Ls', where Ls=sum(L(j)).

Now, I want minimice 'Ls' and to obtain the optimal values of 'x' vector. The state of the art indicates that use the multidimensional simplex method fo Nelder-M.

can anyone help?

I know that the optimal values should be approximately:

 % A= 1.24 * 10-8
 % B= 1.625
 % Q= 2.413 * 10-2
 % Pa(1)= 1.504 * 10-2

Thanks in advance,

Antonio, [e-mail]

clear all
close all
clc

%Data of paper: 
%Katul y col. 1993. Estimation of in Situ Hydraulic Conductivity Function from Nonlinear Filtering Theory. Water Resources Research, 29: 1063-70.

%%Computational scheme (A Appendix)
%Observed measured
t=[0.01;0.09;0.1;0.1745;0.1745;0.4322;0.5194;0.8644;0.9516;1.3838;1.8993;2.4187;2.8509;3.4536;3.8858;4.318;5.0119;5.5274;5.9596;6.3045;6.8239;7.3434;7.9461;8.291;8.8977;9.5876;10.1071;10.5393;11.0547;11.4869;12.0936;12.4385;12.958;13.3862;13.9929;14.5956;14.8573;15.6344;16.4116;17.0143;17.4465;17.8787;18.4853;18.8303;19.5202;19.9524;20.4718;20.9873;21.5067;22.0262;22.2839;22.9738;23.4933;23.9255;24.5282;25.0476;25.3925;26.0825;26.4314;26.9469;27.4663;27.8985;28.4179;29.0206;29.3656;29.885;30.4877;31.0071;31.3521;31.8715;32.391;32.9937;33.5131;33.9453;34.6352;35.0674;35.4124;35.9318;36.709;36.9667;37.6566;38.0888;38.6955;39.1277;39.4726;40.0753;40.7653;41.027;41.6297;42.1491]; %time, days
z=[0.4765;0.4707;0.4691;0.4609;0.466;0.4495;0.4445;0.4484;0.4456;0.4362;0.4339;0.4339;0.4343;0.4284;0.4269;0.4202;0.4249;0.4175;0.4202;0.4131;0.4131;0.4135;0.4147;0.4081;0.4061;0.4061;0.4026;0.4061;0.3959;0.3999;0.3959;0.394;0.3995;0.3983;0.3967;0.3963;0.3928;0.3944;0.3959;0.3971;0.3936;0.3862;0.3944;0.3916;0.3905;0.392;0.3815;0.3823;0.3862;0.3795;0.3913;0.3838;0.3791;0.3811;0.3838;0.3807;0.3873;0.3779;0.3819;0.3799;0.3803;0.3779;0.3823;0.3756;0.3826;0.3815;0.3807;0.3693;0.3799;0.3736;0.3854;0.3725;0.3826;0.374;0.3748;0.3721;0.3748;0.374;0.3768;0.3752;0.376;0.3709;0.3811;0.3662;0.3658;0.3662;0.3697;0.3721;0.3701;0.3713]; %average soil moisture content, v/v
zmax=max(z);
H= [-1.0517;-2.0047;-2.3471;-1.9071;-2.2988;-2.1271;-2.2494;-2.1024;-2.0294;-2.4941;-1.9564;-0.9047;-2.0541;-2.4447;-1.9071;-2.2988;-1.9071;-2.3235;-1.9564;-2.2494;-2.0294;-2.1518;-1.9318;-1.9811;-2.0047;-1.9071;-2.2247;-1.9811;-1.9811;-2.0047;-1.4671;-2.1271;-2.15;-2.0788;-2.1024;-1.9564;-2.2247;-2.0047;-2.2011;-2.1024;-1.5647;-1.5647;-1.2471;-1.4917;-1.5894;-1.4917;-1.54;-1.9071;-1.7118;-1.7118;-1.3447;-1.8094;-1.9564;-1.7611;-0.9294;-1;-1.7847;-1.6871;-1.54;-1.6624;-1.54;-1.7118;-1.4671;-1.7611;-1.4188;-1.7611;-1.54;-1.6871;-1.4424;-1.5164;-1.3447;-1.29;-1.5647;-1.2471;-1.8588;-1.1988;-1.3447;-1.0517;-1.0764;-1.1988;-1.1741;-0.95;-0.9047;-1.0271;-0.807;-0.7824;-0.66;-0.8564;-0.4164;-0.587];%hidráulic gradient, mbar/cm
R=0.02; %calculated value, constant
%Create arrays of zeros
n=length(t);
xp=zeros(length(t),1); %Predicted value of x (xp is the value of z modeled with the additional application of Kalman filter)
xa=zeros(length(t),1); %Updated value of x, with use of observation 'z' and the gain of Kalman filter

Pp=zeros(length(t),1); %Predicted value of P (error)
Pa=zeros(length(t),1); %Updated value of P with use of observation 'z' and the gain of Kalman filter
L=zeros(length(t),1); 
Kg=zeros(length(t),1); %Kalman gain obtained in each time with step 5

% h (meaurement time increment)neccesary to resolution of differential equations with Runge Kutta method (step 3)
for i=1:n-1
    t0(i)=t(i);
    tf(i)=t(i+1); 
    h(i)=tf(i)-t0(i);
end

% 1. initial conditions
x=[1,1,1,1];
x0=x;
% 1.a) xp;
xa(1)=z(1);
xp0(1)=xa(1);
xp(1)=xp0(1);
% 1.b) Pp;
Pa(1)=x(4); %Assume this value as A, B, Q, and after step 9 is to adjust A, B, Q, Pa (1) to be minimal Ls
Pp0(1)=Pa(1);
Pp(1)=Pp0(1);

% 2. Assume [A,B,Q]
A= x(1);
B= x(2);
Q= x(3); % I start assuming A, B, Q, Pa(1;

% 3. Numerically integrate  dxp/dt and dPp/dt, Runge Kutta method with 'for' bucle in all times
  % Press, W.H.; W.T. Vettering; S.A. Teukolsky and B.P. Flannery. 1992. Numerical recipes in Fortran. 963 págs. Cambridge University Press, New York. 
    % Page 701-740: Integration of ordinary differential equations
for j=2:n
%3.a) Resolution of dxp/dt=-A*(Xp^B)*H
xp0(j)=xa(j-1);
kx1(j)=-A*(xp0(j)^B)*H(j-1);  
kx2(j)=-A*((xp0(j)+kx1(j)/2)^B)*H(j-1);
kx3(j)=-A*((xp0(j)+kx2(j)/2)^B)*H(j-1);
kx4(j)=-A*((xp0(j)+kx3(j))^B)*H(j-1);
xp(j)=xp0(j)+h(j-1)*((kx1(j)+2*(kx2(j)+kx3(j))+kx4(j))/6);

%3.b) Resolution of dPp/dt=-2*(h1*A*B*(xp1^(B-1)))*Pp+Q
Pp0(j)=Pa(j-1);   
kP1(j)=-2*(H(j-1)*A*B*(xp(j)^(B-1)))*Pp0(j)+Q;    
kP2(j)=-2*(H(j-1)*A*B*((xp(j)+kP1(j)/2)^(B-1)))*Pp0(j)+Q;   
kP3(j)=-2*(H(j-1)*A*B*((xp(j)+kP2(j)/2)^(B-1)))*Pp0(j)+Q;  
kP4(j)=-2*(H(j-1)*A*B*((xp(j)+kP3(j))^(B-1)))*Pp0(j)+Q;  
Pp(j)=Pp0(j)+h(j-1)*((kP1(j)+2*(kP2(j)+kP3(j))+kP4(j))/6);

% 4. Estimate of L
% L(j)=((z(j)-xp(j))^2)/Pp(j);
%  
% sumrosen=@(x)sum(L);

% 5. Estimate of Kg
Kg(j)=Pp(j)/(Pp(j)+R);

% 6. Updated: xa(t), Pa(t)
xa(j)=xp(j)+Kg(j)*(z(j)-xp(j));
Pa(j)=Pp(j)*(1-Kg(j));
end
% 7. Repeat steps 3-6 with the initial conditions xa1 y Pa1 (I do it for all time by introducing a loop)

% 8. Computo ahora el valor de Ls
%   % FUNCIÓN A MINIMIZAR, paso 9

% 9. Repeat steps 1.b) - 8) until Ls is minimal. Thereby adjusting the initial values ??of A, B, Q y Pa(1).
 % Simplex method for multidimensional cases, lsqnonlin matlab function
% Parameters obtained after optimization by the authors using multidimensional simplex method (Katul y col. 1993)
 % A= 1.24 * 10-8
 % B= 1.625
 % Q= 2.413 * 10-2
 % Pa(1)= 1.504 * 10-2

for i=2:length(xp)
    var1=xp(i);
    var2=z(i);
    var3=Pp(i);
    var4=H(i);
Ls=@(x)(((z(1)-(-x(1)*(xp(1)^x(2))*H(1)))^2)/(-2*(H(1)*x(1)*x(2)*(xp(1)^(x(2)-1)))*x(3)+x(4)))+(sum(((var2-(-x(1)*(var1^x(2))*var4))^2)/(-2*(var4*x(1)*x(2)*(var1^(x(2)-1)))*var3+x(4))));
end
% unconstrained fminsearch solution 
[xopt,Ls]=fminsearch(Ls,x0);
Ls
Aopt=xopt(1)
Bopt=xopt(2)
Pa1opt=xopt(3)
Qopt=xopt(4)

[Walter Roberson]

Antonio, I recommend that you create a new Question for this.