Hi Harsh,
From your description I understand that your cost function f depends on (x,y), namely f = f(x,y) and that f(10, 20) = 1.33 is your initial guess.
If the objective function is non-convex, then it may contain multiple minima/maxima. In such a case constraining your design space might yield only one of these minima/maxima ignoring the rest of the design space. This can help convexify the problem at hand, but it may yield not the most optimal solution in terms of the objective value.
The algorithms you mentioned, namely, the genetic algorithm, the particle swarm, and the simulated annealing are all algorithms of the Global Optimization Toolbox and best suited for problems where the global minimum/maximum is sought (oftentimes one seeks only local minima/maxima where other algorithms are better suited).
My guess would be that if all these algorithms yield the same result, then it is very likely that the solution found is indeed the global optimum. The reason for this argumentation is that these algorithms are inherently different and thus having all of them yield the same result would be a good indication that the converged solution is indeed an optimum.
Since your problem is two dimensional, you can select a set of points (x_i,y_j), compute the value of the objective function at these points f(x_i,y_j) and then plot the objective function as a surface using for instance function surf. You can then plot also the solution from the latter algorithms on this graph and visually verify that the point (x,y) found by these algorithms is a stationary point. You can also compute the gradient of the converged point at that location by means of Finite Differences or so (you would need some points around the converged solution for this and then use function diff for instance) to verify that this is a stationary point.
I hope this information helps.
Kind regardds,
Andreas
