fmincon exitflag 2 and different results with different initial points

I am solving a nonlinear optimization problem using fmincon, in which I have only 2 decision variables, with the following lower and upper bounds:

lb = [1228.527,837.942]

ub = [1728.527,1337.942]

and the equality constraint: x1 + x2 = 2566.469

my objective function has no closed form solution and calls a program that returns values depending on the values of x1 and x2.

Here is a plot of the values of the objective function for different values of x1 and x2:

However, this picture may be a bit misleading, since the objective function has several small ups and downs when zoomed in:

If I use fmincon default options, the analysis ends at the initial point as a local minimum.

so I have tried using:

options = optimoptions('fmincon','FiniteDifferenceStepSize',1e-3);

The problem I have is that using different initial points, such as:

x0(1) = lb(1) + 0.15*500

x0(2) = lb(2) + 0.85*500

and

x0(1) = lb(1) + 0.85*500

x0(2) = lb(2) + 0.15*500

these result in very different results, such as (1317,1249) and (1653,913)

also, the exitflag I get is 2. I hoped I could get an exitflag of 1 and, at least, similar results with different initial points

Does anyone have any suggestions on how to calibrate fmincon options in this case or which algorithm should I use?

I am running patternsearch now, but I am pretty sure that different initial points will give different results...

Answer 1

[Matt J]

You should use your equality constraint to eliminate one of the variables, e.g.,

x2 = 2566.469-x1

lb=max( [1228.527, 2566.469-1337.942] ); %lower bound on x1
ub=min( [1728.527, 2566.469-837.942]  ); %upper bound on x1

Once you have things in the form of a 1-dimensional problem, you can just evaluate your function on a linspace of x1 values to get the minimum, or at least to get an initial guess very close to the global solution. Once you have a good initial guess, you can refine it using fminbnd. Note that unlike fmincon, fminbnd will handle non-differentiable objective functions, which your objective appears to be.

[Conrado Neto]

Those are great suggestions for this problem specifically

I think I am doing it wrong by asking for help here with simplified versions of my actual problems...

I am solving this simple version first, but my actual problem will have 8 decision variables and still just 1 constraint equation.

[Matt J]

If you have a general non-convex, non-differentiable objective in 8 unknowns, my recommendation would be to try ga(). It is clear, in any case, why a local optimizer like fmincon may give you different solutions for different initial guesses - that's what local optimizers do when you initialize too close to local minima. It is also clear that if your function is not differentiable, you may not be able to drive the first order optimality measures to zero, which is what exitflag=1 requires.

[Conrado Neto]

thanks I will try ga then..

I thought fmincon options could be adjusted for this case