cvx is normally used as a package to support optimization problems with a specified function. However, I find that it can be used to solve some easy equations as long as the feasible set is convex. For example, the following code will give the answer $x=2$.
clear all; clc;
cvx_begin
variable x;
x >= 0;
3*x == 6;
cvx_end
So I wonder what's the default objective function of cvx? I did not specify the objective function. Thanks in advance!
Related
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 have been working on solving some equation in a more complicated context. However, I want to illustrate my question through the following simple example.
Consider the following two functions:
function y=f1(x)
y=1-x;
end
function y=f2(x)
if x<0
y=0;
else
y=x;
end
end
I want to solve the following equation: f1(x)=f2(x). The code I used is:
syms x;
x=solve(f1(x)-f2(x));
And I got the following error:
??? Error using ==> sym.sym>notimplemented at 2621
Function 'lt' is not implemented for MuPAD symbolic objects.
Error in ==> sym.sym>sym.lt at 812
notimplemented('lt');
Error in ==> f2 at 3
if x<0
I know the error is because x is a symbolic variable and therefore I could not compare x with 0 in the piecewise function f2(x).
Is there a way to fix this and solve the equation?
First, make sure symbolic math is even the appropriate solution method for your problem. In many cases it isn't. Look at fzero and fsolve amongst many others. A symbolic method is only needed if, for example, you want a formula or if you need to ensure precision.
In such an old version of Matlab, you may want to break up your piecewise function into separate continuous functions and solve them separately:
syms x;
s1 = solve(1-x^2,x) % For x >= 0
s2 = solve(1-x,x) % For x < 0
Then you can either manually examine or numerically compare the outputs to determine if any or all of the solutions are valid for the chosen regime – something like this:
s = [s1(double(s1) >= 0);s2(double(s2) < 0)]
You can also take advantage of the heaviside function, which is available in much older versions.
syms x;
f1 = 1-x;
f2 = x*heaviside(x);
s = solve(f1-f2,x)
Yes, the Heaviside function is 0.5 at zero – this gives it the appropriate mathematical properties. You can shift it to compare values other than zero. This is a standard technique.
In Matlab R2012a+, you can take advantage of assumptions in addition to the normal relational operators. To add to #AlexB's comment, you should convert the output of any logical comparison to symbolic before using isAlways:
isAlways(sym(x<0))
In your case, x is obviously not "always" on one side or the other of zero, but you may still find this useful in other cases.
If you want to get deep into Matlab's symbolic math, you can create piecewise functions using MuPAD, which are accessible from Matlab – e.g., see my example here.
Help. I am trying to solve this system of nonlinear equations in MATLAB for a homework assignment. I have tried wolfram alpha and this online equation solver, and neither of them work.
I have tried my graphing calculator and it keeps saying non algebraic variable or expression.
These are my two equations in two unknowns:
.75*(1100)= x*10^(6.82485-943.453/(T+239.711))
25*1100=(1-x)*10^(6.88555-1175.817/(T+224.887)
I don't quite understand how to use MATLAB to solve this system. Please help.
You want the function fsolve in Matlab. Define a function myfun that returns [0,0] at the solution, then run fsolve(myfun,x0). x0 is a guess for the solution.
Define myfun:
function F = myfun(x)
F = [<put modified eqt1 here>;
<put modified eqt2 here>;];
Save it. Then solve:
x0 = [1,1];
options = optimoptions('fsolve','Display','iter');
[x,fval] = fsolve(#myfun,x0,options) % Call solver
Recently I have run into a Quadratically constrainted quadratic programming (QCQP) problem in my research. I have found something useful in MATLAB optimization toolbox, i.e. 'fmincon' function (general nonlinear optimization with nonlinear constraints), it use 'interior point algorithm' to solve my problem, which contains 8 variables, 1 equality quadratic constraint and 1 inequality quadratic constraint. 'fmincon' with or without 'Hessian' and 'Gradient' provide quite good solution, the only thing I am not satisfied is the efficiency, since I need to call it like million times in my main code. I need to find something which may be more specific to QCQP, possibly efficiency may improved. However I have found a lot of information from netlib and wiki, but I have no judgement on which one I should use, and it would be tedious to try things one by one, I really need some suggestions. By the way, I am mostly programming in MATLAB for this problem, but suitable c/fortran are also useful.
-Yan
An alternative is to use CVX, available here, which works nicely for QCQPs (amongst many other types of problems). Here is a code snippet which solves a QCQP:
close all; clear; clc
n = 10;
H = rand(n); H = H*H'; % make spsd
f = -rand(n,1);
Q = rand(n); Q = Q*Q'; % make spsd
g = -rand(n,1);
cvx_begin
variable x(n)
0.5*x'*Q*x+g'*x <=0
x >= 0
minimize(0.5*x'*H*x + f'*x)
cvx_end
I have to analyze 802.11 saturation throughput using matlab, and here is my problem. I'm trying to solve parametric equations below (parameters are m,W,a) using solve function and i get
Warning: Explicit solution could not be found
How could I solve above equations using matlab?
I guess you were trying to find an analytical solution for tau and p using symbolic math. Unless you're really lucky with your parameters (e.g. m=1), there won't be an analytical solution.
If you're interested in numerical values for tau and p, I suggest you manually substitue p in the first equation, and then solve an equation of the form tau-bigFraction=0 using, e.g. fzero.
Here's how you'd use fzero to solve a simple equation kx=exp(-x), with k being a parameter.
k = 5; %# set a value for k
x = fzero(#(x)k*x-exp(-x),0); %# initial guess: x=0