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I have a Matlab code, which use fmincon with some constraints. So that I am able to modify the code I have thought about whether the line position within the condition matrix A makes a difference
I set up a test file so I can change some variables. It turns out that the position of the condition is irrelevant for the result, but the number of rows in A and b plays a role. I´m suprised by that because I would expect that a row with only zeros in A and b just cancel out.
fun = #(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
options1 = optimoptions('fmincon','Display','off');
A=zeros(2,2); %setup A
A(2,2)=1; %x2<0
b=[0 0]'; %setup b
x = fmincon(fun,[-1,2],A,b,[],[],[],[],[],options1);x
%change condition position inside A
A=zeros(2,2);
A(1,2)=1; %x2<0
b=[0 0]';
x = fmincon(fun,[-1,2],A,b,[],[],[],[],[],options1);x
% no change; the position doesn´t influence fmincon
%change row size of A
A=zeros(1,2);
A(1,2)=1; %x2<0
b=[0]';
x = fmincon(fun,[-1,2],A,b,[],[],[],[],[],options1);x
%change in x2
%increase size of A
A=zeros(10,2);
A(1,2)=1; %x2<0
b=[0 0 0 0 0 0 0 0 0 0]';
x = fmincon(fun,[-1,2],A,b,[],[],[],[],[],options1);x
%change in x2
Can someone explain to me why fmincon is influenced by the row number? What is the "right" rownumber in A and b? The number of variables or the number of conditions?
EDIT
For reasons of completeness:
I agree that different values are possible because of the iteration process. Nevertheless I can find situations where the difference is bigger than the tolerance:
Added +log(x(2) to the function:
fun = #(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2+log(x(3));
options1 = optimoptions('fmincon','Display','off');
options = optimoptions('fmincon')
A=zeros(2,3); %setup A
A(2,3)=1; %x2<0
b=[0 0]'; %setup b
x = fmincon(fun,[-1,2,1],A,b,[],[],[],[],[],options1);x
%change row size of A
A=zeros(1,3);
A(1,3)=1; %x2<0
b=[0]';
x = fmincon(fun,[-1,2,1],A,b,[],[],[],[],[],options1);x
%change in x2
%increase size of A
A=zeros(10,3);
A(1,3)=1; %x2<0
b=[0 0 0 0 0 0 0 0 0 0]';
x = fmincon(fun,[-1,2,1],A,b,[],[],[],[],[],options1);x
%change in x2
x =
-0.79876 **0.49156** 2.3103e-11
x =
-0.79921 0.49143 1.1341e-11
x =
-0.80253 **0.50099** 5.8733e-12
Matlab support told me that the A matrix should not have more rows than conditions. Each condition makes it more difficult for the algorithm.
Note that fmincom doesn't necessarily give the exact solution but a good approximation of the solution according to a certain criteria.
The difference in results are plausible since fminconis an iterative algorithm and these matrix multiplications (even if there are mainly zeros) will eventually end with different results. Matlab will actually do these matrix multiplications until he finds the best result. So these results are all correct in the sense they are all close to the solution.
x =
0.161261791015350 -0.000000117317860
x =
0.161261791015350 -0.000000117317860
x =
0.161261838607809 -0.000000077614999
x =
0.161261877075196 -0.000000096088746
The difference in your results is around 1.0e-07 which is decent result considering you don't specify stopping criteria. You can see what you have by default with the command
options = optimoptions('fmincon')
My result is
Default properties:
Algorithm: 'interior-point'
CheckGradients: 0
ConstraintTolerance: 1.0000e-06
Display: 'final'
FiniteDifferenceStepSize: 'sqrt(eps)'
FiniteDifferenceType: 'forward'
HessianApproximation: 'bfgs'
HessianFcn: []
HessianMultiplyFcn: []
HonorBounds: 1
MaxFunctionEvaluations: 3000
MaxIterations: 1000
ObjectiveLimit: -1.0000e+20
OptimalityTolerance: 1.0000e-06
OutputFcn: []
PlotFcn: []
ScaleProblem: 0
SpecifyConstraintGradient: 0
SpecifyObjectiveGradient: 0
StepTolerance: 1.0000e-10
SubproblemAlgorithm: 'factorization'
TypicalX: 'ones(numberOfVariables,1)'
UseParallel: 0
For example, I can reach closer results with the option:
options1 = optimoptions('fmincon','Display','off', 'OptimalityTolerance', 1.0e-09);
Result is
x =
0.161262015455003 -0.000000000243997
x =
0.161262015455003 -0.000000000243997
x =
0.161262015706777 -0.000000000007691
x =
0.161262015313928 -0.000000000234186
You can also try and play with other criteria MaxFunctionEvaluations, MaxFunctionEvaluations etc to see if you can have even closer results...
I have a for-loop in which I want to set the matrix values either to +1 or -1
My code looks like this:
for n=1:512
for m=1:512
A(n,m)= randi([-1 1]);
end
end
But right now the values are either +1,0,-1 and not just +1 or -1.
Also there should be equal probability that it is +1 or -1.
Is there a function in Matlab where you can determine that?
first of all randi([imin,imax]) returns a integer drawn from the discrete uniform distribution on the interval [imin,imax], so, in other words, it return -1 or 0 or 1 so that is why you get 0 too.
second you can omit 0 by changing your code like this:
for n=1:512
for m=1:512
A(n,m)= randi([0 1])*2-1;
end
end
use randsample to sample from population (in your case [-1,1]):
% this is sampling from uniform distribution
k = 1;
y = randsample([-1 1],k,true);
and in your case:
k = 512*512;
A = reshape(randsample([-1 1],k,true), [512 512]);
you can use y = randsample([-1 1],k,true,w) to sample from arbitrary distribution:
% this is sampling from distribution where p(x=-1) = 0.2 and p(x=1) = 0.8
k = 1;
w = [0.2,0.8];
y = randsample([-1 1],k,true,w);
To generate a random number that equals -1 or 1 with equal probabilities:
Generate a number uniformly distributed on the interval (0,1) (using rand) and compare with 0.5. This gives false (0) or true (1) with the same probability.
Multiply by 2 and subtract 1 to convert 0, 1 to -1, 1.
So:
result = 2*(rand<0.5)-1;
Or, if you want to generate the whole matrix at once:
A = 2*(rand(512,512)<0.5)-1;
I am trying to create a single column vector (out), which is comprised of a sequence of ones and zeros. These should occur in sets of length B and C respectively, which are repeated A number of times. For example:
out=[1
0
0
1
0
0
1
0
0]
It is currently set up as:
out=[0]; %not ideal, but used to initially define 'out'
A=3;
B=1;
C=2;
for i = 1:length(A)
for ii = 1:length(B)
out(end+1,1) = ones(ii,1);
end
for iii = 1:length(C)
out(end+1,1) = zeros(iii,1);
end
end
This is not working - current output:
out=[0
1
0]
How can I correct these loops to get the desired output? Also, is there a better way of achieving this with the given the inputs?
Many thanks.
1) You do not need to use length as this returns the length of an array type, so A,B,C will all be length of 1.
2) Just directly use the values as shown below. Also you can initialize an empty array with empty brackets []
3) If you're using the zeros and ones commands, these generate whole arrays/matrices and do not need to be in a loop. If you want to keep your loop version, just use =1 or =0
out=[]; %--> you can use this instead
A=3;
B=1;
C=2;
for i = 1:A
out(end+1:end+B,1) = ones(B,1);
out(end+1:end+C,1) = zeros(C,1);
end
... or of course to be more "Matlaby" just do what David said in the comments repmat([ones(B,1);zeros(C,1)],A,1), but the above is there to help you on your way.
How about some modulo arithmetic?
result = double(mod(0:(B+C)*A-1, B+C)<B).';
Example:
>> B = 2; %// number of ones in each period
>> C = 4; %// number of zeros in each period
>> A = 3; %// number of periods
>> result = double(mod(0:(B+C)*A-1, B+C)<B).'
result =
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
I can suggest 2 ways:
a)Using for loop-
A=3;
B=2;
C=3;
OneVector=ones(1,B); % B is the length of ones.
zeroVector=zeros(1,C); % C is the length of zeros.
combinedVector=cat(2,OneVector,zeroVector);
Warehouse=[]; % to save data
for(i=1:A)
Warehouse=cat(2,Warehouse,combinedVector);
end
b)using repmat:
OneVector=ones(1,B); % B is the length of ones.
zeroVector=zeros(1,C); % C is the length of zeros.
combinedVector=cat(2,OneVector,zeroVector);
Warehouse=repmat(combinedVector, [A,1]);
I hope, this will solve your problem.
Below is my code, I want to calculate the hamming distance between my "inp" and each row in the a matrix and save these hamming distances in different variables.:
a=[1 1 1 1;1 0 1 1;1 1 0 0; 1 0 0 1]
inp=[0 1 0 1]
for i = 1:4
D=a(i,:);
('dist%d',i)=pdist2(inp,D,'hamming')
fprintf('\n')
i=i+1;
end
This code is not working and I know that the ('dist%d',i) is the part that is wrong. However, I can't solve it. What i want to do is get the results as follows: dist1= , dist2= , dist3= , dist4= .And this is why I tied it with the "i" because it is my loop. Any ideas how can this be solved.
It appears you confused printing with assignment of variables. In general: evaluate, assign to a variable, then print.
Although Matlab does not require this, it's a good practice to initialize the place where you will store the distances. I do it with dist = zeros(4,1) below.
Store each distance in dist(i), the ith element of the array.
After that, print a formatted string with i and dist(i).
You don't need i=i+1, the for loop does incrementing for you.
a=[1 1 1 1;1 0 1 1;1 1 0 0; 1 0 0 1];
inp=[0 1 0 1];
dist = zeros(4,1);
for i = 1:4
D=a(i,:);
dist(i)=pdist2(inp,D,'hamming');
fprintf('dist%d = %f \n', i, dist(i))
end
Note that if the printout was the only goal, storing the results in dist would be unnecessary. You could just do
fprintf('dist%d = %f \n', i, pdist2(inp,D,'hamming'))
then, without introducing the array dist.
Let d and p be two integers. I need to generate a large matrix A of integers, having d columns and N=nchoosek(d+p,p) rows. Note that nchoosek(d+p,p) increases quickly with d and p, so it's very important that I can generate A quickly. The rows of A are all the multi-indices with components from 0 to p, such that the sum of the components is less than or equal to p. This means that, if d=3 and p=3, then A is an [N=nchoosek(3+3,3)=20x3] matrix with the following structure:
A=[0 0 0;
1 0 0;
0 1 0;
0 0 1;
2 0 0;
1 1 0;
1 0 1;
0 2 0;
0 1 1;
0 0 2;
3 0 0;
2 1 0;
2 0 1;
1 2 0;
1 1 1;
1 0 2;
0 3 0;
0 2 1;
0 1 2;
0 0 3]
It is not indispensable to follow exactly the row ordering I used, although it would make my life easier (for those interested, it's called graded lexicographical ordering and it's described here:
http://en.wikipedia.org/wiki/Monomial_order).
In case you are curious about the origin of this weird matrix, let me know!
Solution using nchoosek and diff
The following solution is based on this clever answer by Mark Dickinson.
function degrees = monomialDegrees(numVars, maxDegree)
if numVars==1
degrees = (0:maxDegree).';
return;
end
degrees = cell(maxDegree+1,1);
k = numVars;
for n = 0:maxDegree
dividers = flipud(nchoosek(1:(n+k-1), k-1));
degrees{n+1} = [dividers(:,1), diff(dividers,1,2), (n+k)-dividers(:,end)]-1;
end
degrees = cell2mat(degrees);
You can get your matrix by calling monomialDegrees(d,p).
Solution using nchoosek and accumarray/histc
This approach is based on the following idea: There is a bijection between all k-multicombinations and the matrix we are looking for. The multicombinations give the positions, where the entries should be added. For example the multicombination [1,1,1,1,3] will be mapped to [4,0,1], as there are four 1s, and one 3. This can be either converted using accumarray or histc. Here is the accumarray-approach:
function degrees = monomialDegrees(numVars, maxDegree)
if numVars==1
degrees = (0:maxDegree).';
return;
end
degrees = cell(maxDegree+1,1);
degrees{1} = zeros(1,numVars);
for n = 1:maxDegree
pos = nmultichoosek(1:numVars, n);
degrees{n+1} = accumarray([reshape((1:size(pos,1)).'*ones(1,n),[],1),pos(:)],1);
end
degrees = cell2mat(degrees);
And here the alternative using histc:
function degrees = monomialDegrees(numVars, maxDegree)
if numVars==1
degrees = (0:maxDegree).';
return;
end
degrees = cell(maxDegree+1,1);
degrees(1:2) = {zeros(1,numVars); eye(numVars);};
for n = 2:maxDegree
pos = nmultichoosek(1:numVars, n);
degrees{n+1} = histc(pos.',1:numVars).';
end
degrees = cell2mat(degrees(1:maxDegree+1));
Both use the following function to generate multicombinations:
function combs = nmultichoosek(values, k)
if numel(values)==1
n = values;
combs = nchoosek(n+k-1,k);
else
n = numel(values);
combs = bsxfun(#minus, nchoosek(1:n+k-1,k), 0:k-1);
combs = reshape(values(combs),[],k);
end
Benchmarking:
Benchmarking the above codes yields that the diff-solution is faster if your numVars is low and maxDegree high. If numVars is higher than maxDegree, then the histc solution will be faster.
Old approach:
This is an alternative to Dennis' approach of dec2base, which has a limit on the maximum base. It is still a lot slower than the above solutions.
function degrees = monomialDegrees(numVars, maxDegree)
Cs = cell(1,numVars);
[Cs{:}] = ndgrid(0:maxDegree);
degrees = reshape(cat(maxDegree+1, Cs{:}),(maxDegree+1)^numVars,[]);
degrees = degrees(sum(degrees,2)<=maxDegree,:);
I would solve it this way:
ncols=d;
colsum=p;
base=(0:colsum)';
v=#(dm)permute(base,[dm:-1:1]);
M=bsxfun(#plus,base,v(2));
for idx=3:ncols
M=bsxfun(#plus,M,v(idx));
end
L=M<=colsum;
A=cell(1,ncols);
[A{:}]=ind2sub(size(L),find(L));
a=cell2mat(A);
%subtract 1 because 1 based indexing but base starts at 0
a=a-1+min(base);
It builds up a p-dimensional matrix which contains the sum. The efficiency of this code depends on sum(L(:))/numel(L), this quotient tells you how much of the created matrix is actually used for solutions. If this gets low for your intput, there probably exits a better solution.
Here is a very easy way to do it:
L = dec2base(0:4^3-1,4);
idx=sum(num2str(L)-'0',2)<=3;
L(idx,:)
I think the first line can be very time efficient for creating a list of candidates, but unfortunately I don't know how to reduce the list in an efficient way after that.
So the second line works, but could use improvement performance wise.