I have a noisy picture that I want to denoise, with specific energy function, my function have 3 free variable which I can change them until the energy function converge to the minimum value,I found values of these 3 variable by testing different value and run the program, but I want to know how can I find them by a good optimization algorithm. as far as I know I can use fminunc function in matlab but it gives me an error:
fminunc stopped because the objective function value is less than or equal to the default value of the objective function limit.
h,beta,v is the 3 variable which initialize by h=0.8,beta=4,v=0.9
here is the enrgy formula
and also my code:
load Input_images.mat;
X=noisyImg();
Z=X;
[n,m]=size(Z);
c=1;
for c=1:5
c
for i=1:n
for j=1:m
[neighbours] = neighbour(i,j,n,m,Z);
sumneighpos=sum(Z(i,j)*neighbours(1,:)');
inputnoisypos=(Z(i,j)*X(i,j));
noisyzpos=h*Z(i,j);
sumpos=noisyzpos-(beta*sumneighpos)-(v*inputnoisypos);
sumneg=-sumpos;
if sumneg<sumpos
Z(i,j)=-Z(i,j);
end
end
end
end
code for the optimization part.
f = #(q,w) h*Z(i,j)-q*sumneighpos-w*inputnoisypos;
fun = #(x) f(x(1),x(2));
x0 = [0.9; 4];
options = optimoptions('fminunc','Algorithm','quasi-newton');
[x, fval, exitflag, output] = fminunc(fun,x0,options);
here I have 2 question how can I fix the error and second one, is there any better algorithm?
Related
I'm trying to solve a system of non linear odes in Matlab as follows.
editparams %parameters
Tend = 50;
Nt = 100;
% Define RHS functions
RHS = #(t,x) ModelRHS(t,x,param); %anonymous function defining the set of equations
%Execution-----------------------------------------------------------------
x0 =[0.04;0.75;0.85]; %Initial condition
t = linspace(0,Tend,Nt); %TSPAN
[t x] = ode45(RHS, t, x0);
Now, I need to find the steady state of the system and I'm trying to create a function for this. I thought I'd calculate the steady state using the Jacobian. My equations are in an anonymous function which is defined as f in the code below. However, I realised that jacobian does not work for anonymous functions (or may be there is a way to do with anonymous functions) . so I thought I would convert the anonymous function to a symbolic function and try it. But
i still have a difficulty in getting that done. So any help is really appreciated!
function SS = SteadyState(param, E)
f = #(t,x)ModelRHS(t,x,param,E); %set of odes
SymbolicSystem = sym(f); %trying to make the anonymous function symbolic
SymbolicJacobian = jacobian(SymbolicSystem',x); %jacobian
Jacob = matlabFunction(SymbolicJacobian,x);
end
Also if there is any other way apart from finding the Jacobian, kindly let me know about that too.
I tried using 'fsolve' to calculate the steady-state as follows:
f = #(t,x)ModelRHS(t,x,param);
x0 =[0.04;0.75;0.85]';
options = optimoptions('fsolve','Display','iter'); % Option to display output
SS = fsolve(f,x0,options); % Call solver
but it returned an error
Not enough input arguments.
Error in #(t,x)ModelRHS(t,x,param)
Error in fsolve (line 242)
fuser = feval(funfcn{3},x,varargin{:});
Caused by:
Failure in initial objective function evaluation. FSOLVE cannot continue.
I want to fit two parameters using lsqnonlin. I have set up my system of ODE and want to solve them with my new parameters that are found after I performed the lsqnonlin.
function [dcdt] = CSTRSinSeries(t,c,par)
for i=1:par.n
if i==1
dcdt(i) = 1/par.tau_per_tank * (par.c0-c(i))+par.kf*(par.c0_meth-c(i))- ...
par.kb*c(i).^2;
else
dcdt(i) = 1/par.tau_per_tank * (c(i-1)-c(i))+par.k*(par.c0_meth-c(i))- ...
par.kb*c(i).^2;
end
end
dcdt = dcdt';
end
% fitcrit function
function error = fitcrit(curve_fit_parameters,time_exp,conc_exp, par, init)
[time_model_fit,conc_model_fit] = ode45(#(t,c) CSTRSinSeries(t,c,par),time_exp,init,[]);
error = (conc_exp-conc_model_fit);
end
I think the problem has to do with the fact that my parameters are in parameter struct par and that I don't want to fit on all of these parameters, but just on basis of two of those.
This is my main script for performing the curve fitting:
% initial guesses for model parameters, no. of indeces is number of fitted % parameters
k0 = [0.028 0.002];
% lower and upper bounds for model parameters, this can be altered
LB = [0.00 0.00];
UB = [Inf Inf];
% Set up fitting options
options = optimset('TolX',1.0E-6,'MaxFunEvals',1000);
% Perform nonlinear least squares fit (note that we store much more
% statistics than just the final fitted parameters)
[curve_fit_parameters,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA,JACOBIAN] = ...
lsqnonlin(#(k) fitcrit(k,time_exp, conc_exp, par, init),k0,LB,UB,options);
The code now does not give an error, however the output of my curve_fit_parameters is now the same as my initial values (also when I change my initial value, it stays the same).
The error is:
>> PackedBed
Initial point is a local minimum.
Optimization completed because the size of the gradient at the initial point
is less than the default value of the optimality tolerance.
<stopping criteria details>
The stopping criteria gives a relative first-order optimality 0.00+00, so I think that the parameters lsqnonlin changes have no influence on my error.
I think that the mistake is the lsqnonlin function, where I refer to 'k' instead of 'par.kb and par.kf'. However, I don't know how to refer to these as these are nested functions. Replacing 'k' by 'par.kb, par.kf' gives me the error: Unexpected matlab operator.
Could anyone help me with my problem?
As suggested by Adriaan in the comments, you can easily make a copy of the par struct in your fitcrit function, and assign the parameters to optimize to this par_tmp struct. The fitcrit function then becomes:
function error = fitcrit(curve_fit_parameters,time_exp,conc_exp, par, init)
par_tmp = par;
par_tmp.kf = curve_fit_parameters(1); % change to parameters you need to fit
par_tmp.kb = curve_fit_parameters(2);
[time_model_fit,conc_model_fit] = ode45(#(t,c) odefun(t,c,par_tmp),time_exp,init,[]); % pass par_tmp to ODE solver
error = (conc_exp-conc_model_fit);
end
In the script calling lsqnonlin, you will have to change the par struct to include the converged solution of the parameter subset to optimize:
% initial guesses for model parameters, no. of indeces is number of fitted % parameters
k0 = [0.028 0.002];
% lower and upper bounds for model parameters, this can be altered
LB = [0.00 0.00];
UB = [Inf Inf];
% Set up fitting options
options = optimset('TolX',1.0E-6,'MaxFunEvals',1000);
% Perform nonlinear least squares fit (note that we store much more
% statistics than just the final fitted parameters)
[curve_fit_parameters,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA,JACOBIAN] = ...
lsqnonlin(#(k) fitcrit(k,time_exp, conc_exp, par, init),k0,LB,UB,options);
% make par_final
par_final = par;
par_final.kf = curve_fit_parameters(1);
par_final.kb = curve_fit_parameters(2);
```
I am working on an optimization problem where I want to maximize utility while searching over variables lx_init and kx_init. Each time I solve an issue with fmincon another one pops up. The error right now is..
"Not enough input arguments."
"Failure in initial objective function evaluation. FMINCON cannot continue."
I have tried to trace back the error in debug mode but tracing back my error is proving difficult. The function is able to be evaluated on its own and spit back a value so I know the error is in the fmincon function. My Matlab experience is very minimal. I am sure there are other syntax errors I am guilty of and too naive to see. I will provide all variables so the code can be replicated.
A second question: I want to maximize my utility value here but also find the values of x and y given once the utility is maximized. However, we are also maximizing over lx_init and kx_init. How do I get the function to return x and y? Right now it is just returning the utility value.
Here is my function I am optimizing
function [utility, x, y] = utility_with_prod(gamma_X, gamma_Y, alpha_LX, alpha_KX, alpha_LY, alpha_KY, alpha_uX, sigma_U, sigma_X, sigma_Y, L_bar, K_bar, lx_init, kx_init)
% X's production function
x = gamma_X*((alpha_LX*lx_init^((sigma_X-1)/sigma_X)) + ((alpha_KX)*kx_init^((sigma_X-1)/sigma_X)))^(sigma_X/(sigma_X-1));
% Y's production function
y = gamma_Y*((alpha_LY*(L_bar-lx_init)^((sigma_Y-1)/sigma_Y)) + ((alpha_KY)*(K_bar-kx_init)^((sigma_Y-1)/sigma_Y)))^(sigma_Y/(sigma_Y-1));
% utility function with nested production function
utility = -(((alpha_uX*x^((sigma_U-1)/sigma_U)) + ((1-alpha_uX)*y^((sigma_U-1)/sigma_U)))^(sigma_U/(sigma_U-1)));
end
Here are my initial values
sigma_U= 0.5038;
sigma_X= 0.5029;
sigma_Y= 0.5029;
alpha_uX= 0.000236017865342454;
alpha_LX= 0.180813595922536;
alpha_KX= 0.819186404077464;
gamma_X= 1.768587207836113;
alpha_LY= 0.505368332690592;
alpha_KY= 0.494631667309408;
gamma_Y= 1.999942066647923;
lx_init = 1;
kx_init = 2;
L_bar = 3;
K_bar = 3;
x0 = [lx_init, kx_init];
and the optimization
utility = #(lx_init, kx_init)utility_with_prod(gamma_X, gamma_Y, alpha_LX, alpha_KX, alpha_LY, alpha_KY, alpha_uX, sigma_U, sigma_X, sigma_Y, L_bar, K_bar, lx_init, kx_init)
[optimal_lx_kx, utility_max, exitflag] = fmincon(utility, x0, [],[])
x0 in fmincon is a vector, that's an n by 1 matrix or 1 by n,
here 1 by 2 ---> x0 = [lx_init, kx_init];
Function handle #(lx_init, kx_init) is different from #([lx_init, kx_init]).
#([lx_init, kx_init])accepts only one input.
#(lx_init, kx_init) accepts only two inputs, no more, no less
Also input variable should not be predefined value
Change #(lx_init, kx_init) to #(x) where x(1) = lx_init and x(2) = kx_init
To sum up
utility = #(lx_init, kx_init)utility_with_prod(gamma_X, gamma_Y, alpha_LX,alpha_KX, alpha_LY, alpha_KY, alpha_uX, sigma_U, sigma_X, sigma_Y, L_bar, K_bar, lx_init, kx_init)
is replaced by
utility = #(x)utility_with_prod(gamma_X, gamma_Y, alpha_LX, alpha_KX, alpha_LY, alpha_KY, alpha_uX, sigma_U, sigma_X, sigma_Y, L_bar, K_bar, x(1), x(2))
How do I solve the following system of equations on MATLAB when one of the elements of the variable vector is a constant? Please do give the code if possible.
More generally, if the solution is to use symbolic math, how will I go about generating large number of variables, say 12 (rather than just two) even before solving them?
For example, create a number of symbolic variables using syms, and then make the system of equations like below.
syms a1 a2
A = [matrix]
x = [1;a1;a2];
y = [1;0;0];
eqs = A*x == y
sol = solve(eqs,[a1, a2])
sol.a1
sol.a2
In case you have a system with many variables, you could define all the symbols using syms, and solve it like above.
You could also perform a parameter optimization with fminsearch. First you have to define a cost function, in a separate function file, in this example called cost_fcn.m.
function J = cost_fcn(p)
% make sure p is a vector
p = reshape(p, [length(p) 1]);
% system of equations, can be linear or nonlinear
A = magic(12); % your system, I took some arbitrary matrix
sol = A*p;
% the goal of the system of equations to reach, can be zero, or some other
% vector
goal = zeros(12,1);
% calculate the error
error = goal - sol;
% Use a cost criterion, e.g. sum of squares
J = sum(error.^2);
end
This cost function will contain your system of equations, and goal solution. This can be any kind of system. The vector p will contain the parameters that are being estimated, which will be optimized, starting from some initial guess. To do the optimization, you will have to create a script:
% initial guess, can be zeros, or some other starting point
p0 = zeros(12,1);
% do the parameter optimization
p = fminsearch(#cost_fcn, p0);
In this case p0 is the initial guess, which you provide to fminsearch. Then the values of this initial guess will be incremented, until a minimum to the cost function is found. When the parameter optimization is finished, p will contain the parameters that will result in the lowest error for your system of equations. It is however possible that this is a local minimum, if there is no exact solution to the problem.
Your system is over-constrained, meaning you have more equations than unknown, so you can't solve it. What you can do is find a least square solution, using mldivide. First re-arrange your equations so that you have all the constant terms on the right side of the equal sign, then use mldivide:
>> A = [0.0297 -1.7796; 2.2749 0.0297; 0.0297 2.2749]
A =
0.029700 -1.779600
2.274900 0.029700
0.029700 2.274900
>> b = [1-2.2749; -0.0297; 1.7796]
b =
-1.274900
-0.029700
1.779600
>> A\b
ans =
-0.022191
0.757299
I have tried many times taking different initial values of c. I did not get optimum values of c using function fminsearch. I have found more error between simulated and measured values of sigma. Please help me: how can optimize my function?
clc
close all
clear all
syms c sigma est_bio mea_bio
sigma=[-15.1015 -13.7879 -13.0576 -12.7818 -12.3839 -11.7587 -11.1756 -10.6291 -9.9176];
mea_bio=[0.181 0.204 0.529 0.632 1.059 1.533 1.934 1.977 1.861];
%%% create model function q with parameters
q = #(c, mea_bio) ((c(1)/(-2*c(2))).*(1-exp(2*c(2).*mea_bio))+c(3).*exp(2*c(2).*mea_bio))
%// create the desired error-functions for minimization
h = #(c) sum((q(c, mea_bio) - sigma).^2); %// default minimizaton function
c= [-.05 -.0500 -.0500]; % an initial guess
[p_fit_e, r_e] = fminsearch(h, c) % Optimize
est_q = ((c(1)/(-2*c(2))).*(1-exp(2*c(2).*mea_bio))+c(3).*exp(2*c(2).*mea_bio))
err=est_q-sigma
Why do you define c, sigma est_bio and mea_bio as symbolic variables? It doesn't make sense. This is a numerical optimization, no need to involve symbolic computations. Remove that line from your code and it should work:
syms c sigma est_bio mea_bio