linear equality constraint writing

I am facing difficulty to write linear inequality constraint given below. Here Φ, 48 element long vector, is decision variable where as P and μ are vectors of 48 elements each and λ is a constant. I shall appreciate any help.

    λ * P  ≤  μ * Φ

Answer 1

[Walter Roberson]

lambda .* P <= mu * phi so
lambda .* P / mu <= phi

With lambda, P, and mu all being constants, this does not need to be written as a linear constraint. Instead, it can be written as lower bound: phi >= lambda .* P / mu so

x0 = ....    %starting point

A = []; b = [];
Aeq = []; beq = [];
LB = lamda .* P ./ mu; UB = inf(1,48);
fmincon( fun, x0, A, b, Aeq, beq, LB, UB )

If you prefer to implement it as a linear constraint (perhaps because you are using a different minimizer) then

A = -eye(48); B = lambda .* P ./ mu;

[KSSV]

Ohh...was my answer is not related to the contest of question?

[Walter Roberson]

No, there was an fmincon tag, so this is about creating appropriate A, b matrices for the various MATLAB optimizers.

[Saifullah Khalid]

sir, thanks for response, I am actually trying to optimize using fmincon. But, I have other constraints also where LB = 0 and UB = 1; so if take LB = lamda .* P ./ mu; it goes beyond UB in many cases. Your second option would be good for me. Secondly, shouldn't B would also be negative like A? I am sorry about being stuck in basics.

A = -eye(48); B = - lambda .* P ./ mu;

[Walter Roberson]

Good point, it should be negative.

You can use

LB = max(0,  lamda .* P ./ mu);
UB = ones(1,48);
if any(LB > UB); error('Non-feasible lower bound'); end

[Saifullah Khalid]

I am grateful for prompt support, it solved my problem. I have used the the last suggested approach of UB and LB given above.

[Saifullah Khalid]

Once more a little query, after running optimization, the value of phi (optimized) is set to all 1s, meaning 100% resource utilization which is not practically true. For example, if we have a cloud with 48 servers and 30 processing tasks to be executed, they would be assigned to any of the 30 servers whose phi may be between 0 to 1 but for rest of the servers, phi should not be 1. If you can kindly comments on this? Is my understanding correct?

Answer 2

[KSSV]

% take some random data for test
lambda = rand ;
P = rand(48,1) ;
mu = rand(48,1) ;
phi = rand(48,1) ;
% inequality 
idx = lambda*P<=mu.*phi ;
%%check result
R = [lambda*P(idx) mu(idx).*phi(idx)]

[Saifullah Khalid]

sir, thanks for response, I am actually trying to optimize using fmincon, the guide says constraints should be Ax-b < 0. I am not sure I can use this as it is?