Discretized/Vector Nonlinear Optimization using fmincon (Optimal Scheduling)

I have a discretized nonlinear optimization problem. I want to find the optimal schedule of a decision variable (X) over 3 sampling instants. I am attempting to use the fmincon function. The objective function (J) for minimization is nonlinear and the only constraints are the upper (ub) and lower bounds (lb) on the decision variable. The solution should be of the form X = [x(1) x(2) x(3)]. Variables T, V_max, & QGg_max are known.

%Known parameters
V_max = 3.15;
Qg_max = 200;
T = [2 4 6];

%Constraints
A = [];
b = [];
Aeq = [];
beq = [];

%Boundary limits
lb = transpose((T/(3.6*Qg_max)).*(ones(1,3)));
ub = V_max*ones(3,1);

%Initial guess
x0 = [1 2 3];

%Optimization
X = fmincon(@myFUN,x0,A,b,Aeq,beq,lb,ub);

The myFUN.m function contains the following code:

function J = myFUN(X)
%Nonlinear objective function
T = [2 4 6];
J = (X.*T.^2) + (X) + ((T.^2)./X) + (T) + (T.*X.^2);
end

But when attempting to run the script, I get the following error:

Error using fmincon (line 619)
Supplied objective function must return a scalar value.

Error in fminconHELP (line 20)
X = fmincon(@myFUN,x0,A,b,Aeq,beq,lb,ub);

I cannot find any help on addressing this issue or an example of using fmincon for discrete nonlinear optimization. So I'm not sure if fmincon can be used for this purpose? In that case, any suggestions on alternative techniques?

Answer 1

[Matt J]

As the error message says, the J returned by myFUN is not a scalar, so it is not clear to MATLAB what it means to minimize or maximize it.

In any case, it appears that in your problem each sampling time is independent of the others, so there is no need to approach this as a multi-variable optimization problem. You could just apply fmincon (or even just use fminbnd) to each sampling instant separately:

x0 = [1 2 3];
T=[2,4,6];

for i=1:3
  X(i) = fmincon(@(X) (X.*T(i).^2) + (X) + ((T(i).^2)./X) + (T(i)) + (T(i).*X.^2),x0(i),...
      [],[],[],[],lb(i),ub(i));
end

[Shane Loser]

Another problem I'm experiencing is how to apply this to a multi-variable optimization problem, i.e. both the vectors X and T are the decision vectors and subject to certain upper and lower bounds. The solution would thus be of the form X = [x(1) x(2) x(3)] and T = [t(1) t(2) t(3)]. How can fmincon be used for this multi-variable scheduling problem?

[Torsten]

And what would be your objective function ?

[Shane Loser]

I want to minimize objective function J(X,T) = (X*T^2) + (X) + ((T^2)/X) + (T) + (T*X^2)

[Torsten]

But the result will be a vector, not a scalar value as it is necessary for optimization ...

[Shane Loser]

J is a function of X and T where optimization of X and T will result in a scalar-valued J at each sampling instant.

[Torsten]

And these scalars J_i at each sampling instat i are added to get the overall value J of the objective ?

[Matt J]

If I understand correctly, the revised problem is still separable across the sampling instants. At sampling instant i, you know have a 2-variable optimization with objective

J(X,T) = (X*T^2) + (X) + ((T^2)/X) + (T) + (T*X^2)

Here X and T (and hence also J) are scalars.

[Shane Loser]

Yes, that is correct. My attempt thus far is as follows in the code (which I believe to be correct).

%Known parameters
V_max = 3.15;
Qg_max = 200;
numVars = 2;
numSamples = 3;

%Constraints
A = [];
b = [];
Aeq = [];
beq = [];

%Boundary limits
lbV = transpose((T/(3.6*Qg_max)).*(ones(1,numSamples)));
lbT = 500*ones(numSamples,1);
lb = [lbV,lbT];
ubV = V_max*ones(numSamples,1);
ubT = 2000*ones(numSamples,1);
ub = [ubV,ubT];

% Initial guess
% Initial guess for belt speed
V0 = ones(numSamples,1);
% Initial guess for feed rate
T0 = 600*ones(numSamples,1);
% Matrix form of intial guess
x0 = [V0,T0];

% Form of the solution
    % X = [x(1) x(2)];
    % X = [V(1) T(1)
    %      V(2) T(2)
    %      V(3) T(3)]

% Initialise arrays
X = zeros(numSamples,numVars);
fval = zeros(numSamples,1);

% Optimization
for i=1:numSamples
  [X(i,:),fval(i)] = fmincon(@(X) (X(1).*X(2).^2) + (X(1)) + ((X(2).^2)./X(1)) + (X(2)) + (X(2).*X(1).^2),x0(i,:),[],[],[],[],lb(i,:),ub(i,:));
end

% Display solution
V_optimal = transpose(X(:,1))
T_optimal = transpose(X(:,2))
fval = transpose(fval)

My next issue is to implement a constraint such that the summation of the elements of T (i.e. T(1) + T(2) + T(3)) exceeds a certain value Tsum. I am unable to implement the correct A and b vectors to satisfy AX < b for this multi-variable problem.

[Torsten]

Again the question:

If

X = [V(1) T(1) V(2) T(2) V(3) T(3)]

is your decision vector, what is your objective ?

[Matt J]

My next issue is to implement a constraint such that the summation of the elements of T (i.e. T(1) + T(2) + T(3)) exceeds a certain value Tsum.

In that case, the problem is no longer decomposable across sampling instants. As Torsten says, we need you to write down for us the 6-variable function that you intend to minimize.

[Shane Loser]

I do not know how to adapt my objective function to account for this summation inequality constraint.

[Matt J]

The objective function isn't something that "accounts" for the constraints. They are separate things.

You must know the thing you are trying to minimize (subject to the constraints). If you don't know, how can anyone else?

[Shane Loser]

I want to minimize J = (X*T^2) + (X) + ((T^2)/X) + (T) + (T*X^2) at each sampling instant such that the summation of the elements of T exceed a particular value. I have no further information on the objective function. So I'm not sure how I would write down the required 6-variable objective function.

[Matt J]

I will give you some examples to help guide your thinking. First, define

   f(a,b)=(a*b^2) + (a) + ((b^2)/a) + (b) + (b*a^2)

where a and b are scalars. You have 6 unknowns X(i),T(i), i.=1,2,3.

You could minimize

J = f(X(1),T(1)) + f(X(2),T(2)) +  f(X(3),T(3))

subject to sum(T)>=Tsum.

Alernatively, you could minimize

J = 10*f(X(1),T(1)) + f(X(2),T(2)) +  f(X(3),T(3))

imposing the same constraints sum(T)>=Tsum.

Yet another alternative is to minimize

J = 3*f(X(1),T(1)) + pi*f(X(2),T(2)) +  75*f(X(3),T(3))

In all of these choices of J (and infinitely many more), the optimization algorithm has an incentive to make each f(X(i),T(i)) as small as possible, because reducing f(X(i),T(i)) also reduces J. However, they will all give different solutions when your constraint sum(T)>=Tsum is imposed.

So, your problem is not fully specified (and we cannot help you) until you are able to say more than "I want f(X(i),T(i)) as small as possible at all sampling instants i".

[Shane Loser]

Apologies, I understand now what you mean. I wish to minimize:

J = f(X(1),T(1)) + f(X(2),T(2)) +  f(X(3),T(3))

Where:

f(X,T) = (X*T^2) + (X) + ((T^2)/X) + (T) + (T*X^2)

Subject to sum(T) >= Tsum; and lower- and upper-bounds on X and T

[Matt J]

Very good. That would be,

X0=[1,2,3];
T0=[2,4,6];

A=[0 0 0 -1 -1 -1];
b=-Tsum;

lb=[lb1,lb2,lb3,lb4,lb5,lb6];
ub=[ub1,ub2,ub3,ub4,ub5,ub6];

psol=fmincon(@myObjective, [X0,T0], A,b,[],[],lb,ub)

X=psol(1:3);
T=psol(4:6);

function J = myObjective(p)

    X=p(1:3);
    T=p(4:6);

        f = @(a,b)  (a*b^2) + (a) + ((b^2)/a) + (b) + (b*a^2);

     J = f(X(1),T(1)) + f(X(2),T(2)) +  f(X(3),T(3));

end

[Shane Loser]

One last problem is incorporating 2 nonlinear constraints. At each sampling instant i, the following constraints should be met:

-T(i)/X(i) <= 0

and

T(i)/X(i) - 100 <= 0

My approach is to define a function myCON as follows:

function [c,ceq] = myCON(q)
%Nonlinear constraints
c(1) = -(q(2))/(q(1));
c(2) = ((q(2))/(q(1))) - 100;
ceq = [];
end

Then I try to pass the nonlinear constraint function handle as a parameter of the fmincon function, but erroneous results are produced (the script does not crash however).

psol=fmincon(@myObjective, [X0,T0], A,b,[],[],lb,ub,@myCON)

I suspect that q in the myCON is not properly linked to X and T in the code I have written. I do not understand how to properly apply nonlinear constraints in a multi-variable scheduling problem. Thank you for the assistance thus far.

[Matt J]

What are the upper and lower bounds lb,ub on X(i)? Are the X(i) always positive? If so, your nonlinear constraints are the same as the following linear constraints and it would be better to pose it that way instead,

T(i)>=0
T(i)-100*X(i)<=0

A similar thing can be done if X(i) are constrainted to be negative.