Mathematical programming is the best optimization tool with many years of strong theoretical background. Also, it is demonstrated that it can solve complex optimization problems on the scale of one million design variables, efficiently. Also, the methods are so reliable! Besides, there is mathematical proof for the existence of the solution and the globality of the optimum.

However, in some cases in which there are discontinuities in the objective function, there would be some problems due to non-differentiable problem. Some methods such as sub-gradients are proposed to solve such problems. However, I cannot find many papers in the state-of-the-art of engineering optimization of discontinuous optimization using mathematical programming. Engineers mostly use metaheuristics for such cases.

Can all problems with discontinuities be solved with mathematical programming? Is it easy to implement sub-gradients for large scale industrial problems? Do they work in non-convex problems?

A simple simple example of such a function is attached here.

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