I am trying to solve the following optimization problem in Matlab:
MPC problem
k is the time with N being the amount of timesteps.
linprog(c_k,F_uk,f_k) solves elements 7.1a and 7.1e of the above problem description. However, the output of the model/problem, x, needs to be constrained within bounds. In the image, x is first converted to y and then y is constrained, but directly constraining x would also work.
For context, u is the (decision variable) input to various radiators in a building and x the resulting temperatures of rooms and wall, which need to be constrained between eg 20 and 25 degrees. v are external factors such as outside temperature.
Is there a way to incorporate the constraint on x in the linprog function? Or should I use another optimization method altogether?
What I tried:
what linprog solves
I've thought a lot about how to rewrite x/u and/or how to use one of the three constraint methods as seen in the image to constrain x. Note that vector u in my problem description is the x to be solved in matlab, while x in my problem is a different variable.
I've thought about adding the states x to the decision variable u, but the problem is also that x depends on x in the previous timestep. u is currently a long vector with input variables for each timestep.
Perhaps I should use a heuristic algorithm, but a low computation time is important for my research.
Related
I have a problem detecting outliers in a set of data. Let's say I have two arrays x and y, and y is a quadratic function of x. Some of the values of y do not follow this function. How can I detect them?
I tried the rmoutliers function, but it doesn't seem to solve this problem since it only deals with normally distributed data.
Basically, I am trying to study some material behavior. The behavior is represented by y. I use an optimization method to get the different values of y as a function of x. Because sometimes the optimization doesn't yield accurate results, I get outliers.The relationship I am expecting should follow some nearly quadratic function, but the coefficients of this function are variable based one the provided set of data, so I can't use a certain function of x and use it to detect the outliers in the array of y values.
I have a function X_t which I have defined anonymously to take as variables t which is a scalar time and z which is a vector of potentially arbitrary dimension.
That is,
X_t = #(t,z) fun(t,z).
I want to find the integral L2 norm of this object over the space of z's. That is, I want to find
X_t_norm = #(t) integral(#(z) abs(X_t(t,z))^2, -infinity,infinity).
Now clearly two things are a problem here.
Firstly, taking the limits as infinite isn't going to work, but I think I should be fine to take the limits as just large numbers (100 would certainly be enough for my purposes).
However, the real problem comes with taking this integral. My variable z is a vector of somewhat arbitrary (even) dimension and as a result I've gotten stuck figuring out how to pass z into the integral in such a way that I can compute this n-dimensional integral. (Basically I want to vary the length of z for different cases and compute the L2 norm for each of these cases).
In particular, I'm not sure how to tell the integral function to compute the integral over every component of z between the limits discussed above.
Any help would be greatly appreciated!
I want to implement densities of probability measures in Matlab. For that I define density as a function handle such that the integral of some function f (given as a function handle) on the interval [a,b] can be computed by
syms x
int(f(x)*density(x),x,a,b)
When it comes to the Dirac measure the problem is that
int(dirac(x),x,0,b)
delivers the value 1/2 instead of 1 for all b>0. However if I type
int(dirac(x),x,a,b)
where a<0 and b>0 the returned value is 1 as it should be. For this reason multiplying by 2 will not suffice as I want my density to be valid for all intervals [a,b]. I also dont want to distinct cases before integrating so that the code remains valid for a large class of densities.
Does someone know how I can implement the Dirac probability measure (as defined here) in Matlab?
There's no unique, accepted definition for int(delta, 0, b). The problem here isn't that you're getting the "wrong" answer as that you want to impose a different convention somehow on what the delta function is than what was provided by Matlab. (Their choice is defensible but not unique.) If you evaluate this in Wolfram Alpha, for example, it will give you theta(0) - which is not defined as anything in particular. Here theta is the Heaviside function. If you want to impose your own convention, implement your own delta function.
EDIT
I see you wrote a comment on the question, while I was writing this answer, so.... Keep in mind that the Dirac measure or Dirac delta function is not a function at all. The problem you're having, along with what's described below, all relate to trying to give a functional form to something that is inherently not a function. What you are doing is not well-defined in the framework that you have in Matlab.
END OF EDIT
To put the point about conventions in context, the delta function can be defined by different properties. One is that int(delta(x) f(x), a, b) = f(0) when a < 0 < b. This doesn't tell you anything about the integral that you want. Another, which probably leads to an answer as you're getting from Matlab, is to define it as a limit. One (but not the only choice) is the limit of the zero-mean Gaussian as the variance goes to 0.
If you want to use a convention int(delta(x) f(x), a, b) = f(0) when a <= 0 < b, that probably won't get you in much trouble, but just keep in mind that it's a convention you've chosen more than a "right" or
"wrong" answer relative to what you got from Matlab.
As related note, there is a similar choice to be made on the step function (Heaviside function) at x=0. There are conventions in which is is (a) undefined, (b) -1, (c) +1, and (d) 1/2. None is "wrong." This probably corresponds roughly to the choice on the Dirac function since the Heaviside is (roughly) the integral of the Dirac.
I would like to find the maximin solution of a function f in Matlab (below is the definition of maximin)
x and y are both real vectors and f is smooth but 'quite complex to calculate' (it is formed from the output of a neural network).
I tried an alternating approach of holding x constant and minimising for y and then holding y constant and maximising for x but this did not converge and instead oscillated.
I believe you can use genetic algorithms to solve the problem but firstly could not see how to do it in Matlab and secondly thought it may be a waste of the fact that f is smooth.
I have both the Optimization and Global Optimization toolbox. What is the best way to solve this problem in Matlab?
I'm not too familiar with MATLAB or computational mathematics so I was wondering how I might solve an equation involving the sum of squares, where each term involves two vectors- one known and one unknown. This formula is supposed to represent the error and I need to minimize the error. I think I'm supposed to use least squares but I don't know too much about it and I'm wondering what function is best for doing that and what arguments would represent my equation. My teacher also mentioned something about taking derivatives and he formed a matrix using derivatives which confused me even more- am I required to take derivatives?
The problem that you must be trying to solve is
Min u'u = min \sum_i u_i^2, u=y-Xbeta, where u is the error, y is the vector of dependent variables you are trying to explain, X is a matrix of independent variables and beta is the vector you want to estimate.
Since sum u_i^2 is diferentiable (and convex), you can evaluate the minimal of this expression calculating its derivative and making it equal to zero.
If you do that, you find that beta=inv(X'X)X'y. This maybe calculated using the matlab function regress http://www.mathworks.com/help/stats/regress.html or writing this formula in Matlab. However, you should be careful how to evaluate the inverse (X'X) see Most efficient matrix inversion in MATLAB