# Transforming an Optimization Problem into a multi-vector Optimization Problem

I am sitting on an MultiAgent optimization problem trying to solve for a quadruple of variables

{e,l,b,c}


In a single agent (n=1) environment the optimization problem looks like this which works:

Objective Function:

f[e_?NumericQ, l_?NumericQ, b_?NumericQ] := (l*1.04 + e*1.057 - b*0.8632^(-1))/(-1)


Maximization:

NMaximize[{f[e, l, b],
c >= 0.2*d[[1]] && (c + e + l - d[[1]] - b)/(e + 0.2*l) >= 0.09 &&
c + l + e == b + W[[1]] + d[[1]] && c >= 0 && e >= 0 && l >= 0 &&
b >= 0}, {e, l, b, c}]


Where d[[1]] and W[[1]] are Values drawn from NormalDistributions:

W = RandomVariate[NormalDistribution[65, 10], {n, 1}]

d = RandomVariate[NormalDistribution[300, 100], {n, 1}]


Output:

{-0.00165118, {e -> 532.54, l -> 626.225, b -> 702.162, c -> 54.2522}}

Problem:

Now I want to expand that problem to a MulitAgent problem, let's say n=10. I don't know how to designate Vector properties to the variables and how to index them properly.

I tried this after clearing all variables:

Objective Function multiple Agents

F[e_?VectorQ, l_?VectorQ, b_?VectorQ] := (l*1.04 + e*1.057 - b*0.8632^(-1))/(-1);


Maximization Multiple Agents

Table[NMaximize[{f[e[[i]], l[[i]], b[[i]], c[[i]]],
c[[i]] >= 0.2*d[[i]] && (c[[i]] + e[[i]] + l[[i]] - d[[i]] - b[[i]])/(e[[i]] + 0.2*l[[i]]) >= 0.09 &&
c[[i]] + l[[i]] + e[[i]] == b[[i]] + W[[i]] + d[[i]] && c[[i]] >= 0 && e[[i]] >= 0 && l[[i]] >= 0 &&
b[[i]] >= 0}, {e[[i]], l[[i]], b[[i]], c[[i]]}],{i,1,n}]


Which is basically the same Maximization as above, only trying to index it, since i hoped that Mathematica would calculate Variables {e[[i]], l[[i]], b[[i]], c[[i]]}, which is for every Agent i different due to drawings of W and d out of the Normal Distribution.

However, the indexation is not proper according to the error message:

Part::partd: "Part specification e[[1]] is longer than depth of object" etc.

How can I transform this scenario in a multiple Vector Scenario?

• Creating a Table won't help you. You need to create arrays for your individual variables like eVars = Array[e,n] where n is some fixed integer. Then you need to create an appropriate objective function, that mixes all the components together in the way you want them to interact and yield a real value, not some kind of vector. Essentially you can just optimize in ordered spaces (you need to know whats bigger and smaller). Jun 23 '15 at 18:30
• Thanks! It worked to define the variable as arrays to create the necessary depth. The Error Message Part::partd: is no problem anymore Jun 23 '15 at 21:38
• ResidentStiefel or @Wizard would either one of you consider writing this solution up as an answer, so the question does not show up as unanswered anymore? Self-answers are also encouraged on StackExchange. Jun 25 '15 at 3:52

eVars = Array[e,n]

where n is some fixed integer.