Optimization under constraints - matlab

Question

I have a question regarding optimization.
I have a matrix x with 3 columns and a certain number of rows (max 200). Each row represents a candidate. The column one contains a score (between 0 and 1) , the column 2 contains the kind of candidate (there are 10 kinds in total labeled from 1 to 10) and the column 3 contains the amount of each candidate. There is one thing to take into consideration: the amount can be NEGATIVE
What I would like to do is to select max 35 elements among these candidates which would maximize the function which sum over their respective score (column 1) under the constraints that there can be a maximum of 10% of each kind computed in the following way: percenteage of kind 1: sum amount of kind 1 divided by sum all amount.
At the end, I would like to have a set of max 35 candidates which satisfy the constraints and optimize the sum of their scores.
Here is a the code I have come up with so far but I am struggling on the 10% constraint as it seems not to be taken into account:
rng('default');
clc;
clear;
n = 100;
maxSize = 35;
%%%TOP BASKET
nbCandidates = 100;
score = rand(100,1)/10+0.9;
quantity = rand(100,1)*100000;
type = ceil(rand(100,1)*10)
typeMask = zeros(n,10);
for i=1:10
typeMask(:,i) = type(:,1) == i;
end
fTop = -score;
intconTop = [1:1:n];
%Write the linear INEQUALITY constraints:
A = [ones(1,n);bsxfun(#times,typeMask,quantity)'/sum(type.*quantity)];
b = [maxSize;0.1*ones(10,1)];
%Write the linear EQUALITY constraints:
Aeq = [];
beq = [];
%Write the BOUND constraints:
lb = zeros(n,1);
ub = ones(n,1); % Enforces i1,i2,...in binary
x = intlinprog(fTop,intconTop,A,b,Aeq,beq,lb,ub);
I would be grateful to some advice where I m doing it wrong!

Answer 1

A linear program for your model might look something like this:
n is the number of candidates.
S[x] is candidate x's score.
A[i][x] is the amount of candidate x for kind i (A[i][x] can be positive or negative, like you said).
T[i] is the total amount of all candidates for kind i.
I[x] is 1 if element x is to be included, and 0 if element x is to be excluded.
The function f which you want to optimize is a function of S[x] and I[x]. You can think of S and I as n-dimensional vectors, so the function you want to optimize is their dot-product.
f() = DotProduct(I, S)
This is equivalent to the linear function I1 * S1 + I2 * S2 + ... + In * Sn.
We can formulate all of the constraints in this way to get a set of linear functions whose coeffecients are the components in an n dimensional vector that we can dot with I, the parameters to optimize.
For the constraint that we can only take 35 elements at most, let C1() be a function which computes the total number of elements.
Then the first constraint can be formalized as C1() <= 35 and C1() is a linear function which can be computed thusly:
Let j be an n dimensional vector with each component equal to 1: j = <1,1,...,1>.
C1() = DotProduct(I, j)
So C1() <= 35 is a linear inequality equivalent to:
I1 * 1 + I2 * 1 + ... + In * 1 <= 35
I1 + I2 + ... + In <= 35
We need to add a slack variable x1 here to turn this into and equivalence relation:
I1 + I2 + ... + In + x1 = 35
For the constraint that we can only take 10% of each kind, we will have a function C2[i]() for each kind i (you said there are 10 in all). C2[i]() Computes the amount of students taken for kind i given the students we have selected:
C21() <= .1 * T1
C22() <= .1 * T2
...
C210() <= .1 * T10
We compute C2[i]() like this:
Let k be an n dimensional vector equal to <A[i]1, A[i]2, ..., A[i]n>, each component is the amount of each candidate for kind i.
Then DotProduct(I, k) = I1 * A[i]1 + I2 * A[i]2 + ... + In * A[i]n, is the total amount we are taking of kind i given I, the vector which captures what elements we are including.
So C2[i]() = DotProduct(I, k)
Now that we know how to compute C2[i](), we need to add a slack variable to turn this into an equality relation:
C2[i]() + x[i + 1] = .1 * T[i]
Here x's subscript is [i + 1] because x1 is already used as a slack variable for the previous constraint.
In summary, the linear program would look like this (adding 11 slack variables x1, x2, ..., x11 for each constraint that is an inequality):
Let:
V = <I1, I2, ..., In, x1, x2, ..., x11> (variables)
|S1|
|S2|
|. |
|. |
|. |
P = |Sn| (parameters of objective function)
|0 |
|0 |
|. |
|. |
|. |
|0 |
|35 |
|.1*T1 |
C = |.1*T2 | (right-hand sides of constraining equality relations)
|... |
|.1*T10|
|1 |1 |...|1 |1|0|...|0|
|A1,1 |A1,2 |...|A1,n |0|1|...|0|
CP = |A2,1 |A2,2 |...|A2,n |0|0|...|0| (parameters of constraint functions)
|... |... |...|... |0|0|...|0|
|A10,1|A10,2|...|A10,n|0|0|...|1|
Maximize:
V x P
Subject to:
CP x Transpose(V) = C
Hopefully this is clear, sorry for terrible formatting.

Answer 2

I believe the MIP model can look like:
Here i are the data points and j indicates the type. For simplicity I assumed here every type has the same number of data points (i.e. Amount(i,j), Score(i,j) are matrices). It is easy to handle the more irregular case by restricting the summations.
The 10% rule is simply applied on the sum of the amounts. I hope that is the correct interpretation. Not sure if this is true if we have negative sums.