My Matlab program has multiple inputs as a struct (in.a, in.b, etc.)
and multiple outputs (out.a, out.b, etc.)
I would like to use the genetic algorithm solver from teh optimization toolbox to find the best input in.a, while all the other inputs are constant. The fitness is one of the outputs, e.g. out.b(2,3).
How do I "tell" the solver this?
Thanks
Daniel
It is not uncommon in programming to have a situation where what is most convenient for your function and what some library call expects of it don't agree. The normal resolution to such a problem is to write a small layer in between that allows the two to talk; an interface.
From help ga:
X = GA(FITNESSFCN,NVARS) finds a local unconstrained minimum X to the
FITNESSFCN using GA. [...] FITNESSFCN accepts a vector X of size
1-by-NVARS, and returns a scalar evaluated at X.
So, ga expects vector input, scalar output, whereas you have a structure going in and out. You would have to write the following (sub)function:
function Y = wrapper_Objfun(X, in)
in.a = X; %# variable being optimized
out = YOUR_REAL_FUNCTION(in); %# call to your actual function
Y = out.b(2,3); %# objective value
end
and then the call to ga will look like
X = ga(#(x) wrapper_Objfun(x,in), N);
where N is however large in.a should be.
Also have a read about it in Matlab's own documentation on the subject.
Related
I have a task of writing an M script that would set up (and finally solve) a set of 19 differential equations and all of the equation coefficients. I am not sure what is the best way to input those equations.
Example of some equations:
Example of coefficients:
I don't think that this would be suited for Simulink. Other examples use a function in the form of #(t,x) where t is time and x the vector of all variables.
There are loads of "examples" online, but they don't seem to be appropriate for such a large set of large equations.
For example this exmple of solving 3 equatons
Even for 3 simple equations as they have solved, the functions are getting messy:
f = #(t,x) [-x(1)+3*x(3);-x(2)+2*x(3);x(1)^2-2*x(3)];
Using this notation and getting to the x(19) and cross-referencing all instances of x would be a mess.
I would like your help, and a simple example of how could I write these equations line by line, maybe using the symbolic toolbox, and then put them in an array that I can then forward to the solver.
As I said, I know there are examples online, but they tackle only the most basic system, and really they don't scale well if you want a clean and easily readable code.
I would like to have a similar to Wolfram alpha, where you type variable names as they are (instead od x(1), x(2), ... m x(19)) and if it's possible to get all solution vectors with their variable names.
You don't have to use an anonymous function handle as an ode function, you can create a separate function file (as shown in the odefun section of ode45.
For example, your odefun can look like:
function dy = myode(t,y)
% first unpack state variables
i_d2 = y(1);
i_q2 = y(2);
...
gamma2 = y(end-1);
omega2 = y(end);
% determine all constants
c34 = expression_for_c34;
...
c61 = expression_for_61;
% determine state derivative
i_d2_dot = expression;
...
omega2_dot = expression;
% pack state derivative based on order in state vector
dy(1) = i_d2_dot;
...
dy(end) = omega2_dot;
end
From this myode function you can also call other functions to e.g. determine the value for some coefficients based on the current state. Next, integrate the system using a suitable ode solver:
[t,y] = ode45(#myode,tspan,y0);
I want to minimize a function like below:
Here, n can be 5,10,50 etc. I want to use Matlab and want to use Gradient Descent and Quasi-Newton Method with BFGS update to solve this problem along with backtracking line search. I am a novice in Matlab. Can anyone help, please? I can find a solution for a similar problem in that link: https://www.mathworks.com/help/optim/ug/unconstrained-nonlinear-optimization-algorithms.html .
But, I really don't know how to create a vector-valued function in Matlab (in my case input x can be an n-dimensional vector).
You will have to make quite a leap to get where you want to be -- may I suggest to go through some basic tutorial first in order to digest basic MATLAB syntax and concepts? Another useful read is the very basic example to unconstrained optimization in the documentation. However, the answer to your question touches only basic syntax, so we can go through it quickly nevertheless.
The absolute minimum to invoke the unconstraint nonlinear optimization algorithms of the Optimization Toolbox is the formulation of an objective function. That function is supposed to return the function value f of your function at any given point x, and in your case it reads
function f = objfun(x)
f = sum(100 * (x(2:end) - x(1:end-1).^2).^2 + (1 - x(1:end-1)).^2);
end
Notice that
we select the indiviual components of the x vector by matrix indexing, and that
the .^ notation effects that the operand is to be squared elementwise.
For simplicity, save this function to a file objfun.m in your current working directory, so that you have it available from the command window.
Now all you have to do is to call the appropriate optimization algorithm, say, the quasi Newton method, from the command window:
n = 10; % Use n variables
options = optimoptions(#fminunc,'Algorithm','quasi-newton'); % Use QM method
x0 = rand(n,1); % Random starting guess
[x,fval,exitflag] = fminunc(#objfun, x0, options); % Solve!
fprintf('Final objval=%.2e, exitflag=%d\n', fval, exitflag);
On my machine I see that the algorithm converges:
Local minimum found.
Optimization completed because the size of the gradient is less than
the default value of the optimality tolerance.
Final objval=5.57e-11, exitflag=1
I'm using octave 3.8.1 which works like matlab.
I have an array of thousands of values I've only included three groupings as an example below:
(amp1=0.2; freq1=3; phase1=1; is an example of one grouping)
t=0;
amp1=0.2; freq1=3; phase1=1; %1st grouping
amp2=1.4; freq2=2; phase2=1.7; %2nd grouping
amp3=0.8; freq3=5; phase3=1.5; %3rd grouping
The Octave / Matlab code below solves for Y so I can plug it back into the equation to check values along with calculating values not located in the array.
clear all
t=0;
Y=0;
a1=[.2,3,1;1.4,2,1.7;.8,5,1.5]
for kk=1:1:length(a1)
Y=Y+a1(kk,1)*cos ((a1(kk,2))*t+a1(kk,3))
kk
end
Y
PS: I'm not trying to solve for Y since it's already solved for I'm trying to solve for Phase
The formulas located below are used to calculate Phase but I'm not sure how to put it into a for loop that will work in an array of n groupings:
How would I write the equation / for loop for finding the phase if I want to find freq=2.5 and amp=.23 and the phase is unknown I've looked online and it may require writing non linear equations which I'm not sure how to convert what I'm trying to do into such an equation.
phase1_test=acos(Y/amp1-amp3*cos(2*freq3*pi*t+phase3)/amp1-amp2*cos(2*freq2*pi*t+phase2)/amp1)-2*freq1*pi*t
phase2_test=acos(Y/amp2-amp3*cos(2*freq3*pi*t+phase3)/amp2-amp1*cos(2*freq1*pi*t+phase1)/amp2)-2*freq2*pi*t
phase3_test=acos(Y/amp3-amp2*cos(2*freq2*pi*t+phase2)/amp3-amp1*cos(2*freq1*pi*t+phase1)/amp3)-2*freq2*pi*t
Image of formula below:
I would like to do a check / calculate phases if given a freq and amp values.
I know I have to do a for loop but how do I convert the phase equation into a for loop so it will work on n groupings in an array and calculate different values not found in the array?
Basically I would be given an array of n groupings and freq=2.5 and amp=.23 and use the formula to calculate phase. Note: freq will not always be in the array hence why I'm trying to calculate the phase using a formula.
Ok, I think I finally understand your question:
you are trying to find a set of phase1, phase2,..., phaseN, such that equations like the ones you describe are satisfied
You know how to find y, and you supply values for freq and amp.
In Matlab, such a problem would be solved using, for example fsolve, but let's look at your problem step by step.
For simplicity, let me re-write your equations for phase1, phase2, and phase3. For example, your first equation, the one for phase1, would read
amp1*cos(phase1 + 2 freq1 pi t) + amp2*cos(2 freq2 pi t + phase2) + amp3*cos(2 freq3 pi t + phase3) - y = 0
Note that ampX (X is a placeholder for 1, 2, 3) are given, pi is a constant, t is given via Y (I think), freqX are given.
Hence, you are, in fact, dealing with a non-linear vector equation of the form
F(phase) = 0
where F is a multi-dimensional (vector) function taking a multi-dimensional (vector) input variable phase (comprised of phase1, phase2,..., phaseN). And you are looking for the set of phaseX, where all of the components of your vector function F are zero. N.B. F is a shorthand for your functions. Therefore, the first component of F, called f1, for example, is
f1 = amp1*cos(phase1+...) + amp2*cos(phase2+...) + amp3*cos(phase3+...) - y = 0.
Hence, f1 is a one-dimensional function of phase1, phase2, and phase3.
The technical term for what you are trying to do is find a zero of a non-linear vector function, or find a solution of a non-linear vector function. In Matlab, there are different approaches.
For a one-dimensional function, you can use fzero, which is explained at http://www.mathworks.com/help/matlab/ref/fzero.html?refresh=true
For a multi-dimensional (vector) function as yours, I would look into using fsolve, which is part of Matlab's optimization toolbox (which means I don't know how to do this in Octave). The function fsolve is explained at http://www.mathworks.com/help/optim/ug/fsolve.html
If you know an approximate solution for your phases, you may also look into iterative, local methods.
In particular, I would recommend you look into the Newton's Method, which allows you to find a solution to your system of equations F. Wikipedia has a good explanation of Newton's Method at https://en.wikipedia.org/wiki/Newton%27s_method . Newton iterations are very simple to implement and you should find a lot of resources online. You will have to compute the derivative of your function F with respect to each of your variables phaseX, which is very simple to compute since you're only dealing with cos() functions. For starters, have a look at the one-dimensional Newton iteration method in Matlab at http://www.math.colostate.edu/~gerhard/classes/331/lab/newton.html .
Finally, if you want to dig deeper, I found a textbook on this topic from the society for industrial and applied math: https://www.siam.org/books/textbooks/fr16_book.pdf .
As you can see, this is a very large field; Newton's method should be able to help you out, though.
Good luck!
As the title says I'd like to know how to retrieve the 2nd order derivative from a Matlab ode.
The system i'm trying to simulate is described by 12 1st order differential equations.
Let's explain:
...
[T,Y] = ode113(#sixdofsolver,time,Y0,options,settings);
T = 1700x1 vector
Y = 1700x12 matrix
Now if i do:
[dY] = sixdofsolver(T,Y,settings)
dY = 12x1 vector
I'd have expected to have a same-size matrix like Y.
What am I doing wrong?
You're on the right track. Your integration function sixdofsolver is likely designed first for used by Matlab's ODE solvers. These functions evaluate the function at one point in time (and a single state value) rather than across a range of times.
You either need to rewrite your sixdofsolver function so that it can handle more than a single time or your need to create a new function based on it that does that. In other words you need to vectorize the integration function. You probably have variable like y(1), y(2), ..., y(12). Well, now the input state vector is a matrix so you need to use something likey(:,1),y(:,2), ..., y(:,12). You may need to do other things like switch to element-wise operators. I can't help you more from what you've provided.
I'd like to find the principal components of a data matrix X in Matlab by solving the optimization problem min||X-XBB'||, where the norm is the Frobenius norm, and B is an orthonormal matrix. I'm wondering if anyone could tell me how to do that. Ideally, I'd like to be able to do this using the optimization toolbox. I know how to find the principal components using other methods. My goal is to understand how to set up and solve an optimization problem which has a matrix as the answer. I'd very much appreciate any suggestions or comments.
Thanks!
MJ
The thing about Optimization is that there are different methods to solve a problem, some of which can require extensive computation.
Your solution, given the constraints for B, is to use fmincon. Start by creating a file for the non-linear constraints:
function [c,ceq] = nonLinCon(x)
c = 0;
ceq = norm((x'*x - eye (size(x))),'fro'); %this checks to see if B is orthonormal.
then call the routine:
B = fmincon(#(B) norm(X - X*B*B','fro'),B0,[],[],[],[],[],[],#nonLinCon)
with B0 being a good guess on what the answer will be.
Also, you need to understand that this algorithms tries to find a local minimum, which may not be the solution you ultimately want. For instance:
X = randn(1,2)
fmincon(#(B) norm(X - X*B*B','fro'),rand(2),[],[],[],[],[],[],#nonLinCon)
ans =
0.4904 0.8719
0.8708 -0.4909
fmincon(#(B) norm(X - X*B*B','fro'),rand(2),[],[],[],[],[],[],#nonLinCon)
ans =
0.9864 -0.1646
0.1646 0.9864
So be careful, when using these methods, and try to select a good starting point
The Statistics toolbox has a built-in function 'princomp' that does PCA. If you want to learn (in general, without the optimization toolbox) how to create your own code to do PCA, this site is a good resource.
Since you've specifically mentioned wanting to use the Optimization Toolbox and to set this up as an optimization problem, there is a very well-trusted 3rd-party package known as CVX from Stanford University that can solve the optimization problem you are referring to at this site.
Do you have the optimization toolbox? The documentation is really good, just try one of their examples: http://www.mathworks.com/help/toolbox/optim/ug/brg0p3g-1.html.
But in general the optimization function look like this:
[OptimizedMatrix, OptimizedObjectiveFunction] = optimize( (#MatrixToOptimize) MyObjectiveFunction(MatrixToOptimize), InitialConditionsMatrix, ...optional constraints and options... );
You must create MyObjectiveFunction() yourself, it must take the Matrix you want to optimize as an input and output a scalar value indicating the cost of the current input Matrix. Most of the optimizers will try to minimise this cost. Note that the cost must be a scalar.
fmincon() is a good place to start, once you are used to the toolbox you and if you can you should choose a more specific optimization algorithm for your problem.
To optimize a matrix rather than a vector, reshape the matrix to a vector, pass this vector to your objective function, and then reshape it back to the matrix within your objective function.
For example say you are trying to optimize the 3 x 3 matrix M. You have defined objective function MyObjectiveFunction(InputVector). Pass M as a vector:
MyObjectiveFunction(M(:));
And within the MyObjectiveFunction you must reshape M (if necessary) to be a matrix again:
function cost = MyObjectiveFunction(InputVector)
InputMatrix = reshape(InputVector, [3 3]);
%Code that performs matrix operations on InputMatrix to produce a scalar cost
cost = %some scalar value
end