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I am wondering if MATLAB has a solver that is able to maximize a multivariate piece-wise constant function.

When I use the naïve optimize function, it gets stuck at the initial value since it's constant within a neighborhood of the initial value.

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before you go trying something different, have you tried just starting with a different initial value.Frederick
I'm not sure that maximizing a piece-wise constant function is really do-able... from an optimization point of view. Maybe some sort of line search or Monte Carlo (random guesses) method would serve you well.Frederick
It might be helpful if you provide your objective function and the accompanying constraints.lightalchemist

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I would follow a more elaborated version of @Frederick suggestion. A simple stochastic approach, but not exactly a linear search because of the multivariate characteristic: simulated annealing