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I am looking for the method or idea to solve the following optimization problem:

min f(x)

s.t. g(xi, yi) <= f(x), i=1,...,n

where x, y are variables in R^n. f(x) is convex function with respect to x. g(xi, yi) is a bunch of convex functions with respect to (xi, yi).

It is the problem of difference of convex functions (DC) optimization due to the DC structure of the constraints. Since I am fairly new to 'DC programming', I hope to know the global optimality condition of DC programs and the efficient and popular approaches for global optimization.

In my specific problem, it is already verified that the necessary optimality condition is g(xi*, yi*)=f(x*) for i=1,...,n.

Any ideas or solution would be appreciated, thanks.

Zezu Liang
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  • While "optimization" is often used by compilers and therefore connected to programming, it can also be used for other non-programming tasks. So can you please edit your question to elaborate on how this question is related to programming? And please take some time to read [the help pages](http://stackoverflow.com/help), [take the tour](http://stackoverflow.com/tour) and [read about how to ask good questions](http://stackoverflow.com/help/how-to-ask). – Some programmer dude Feb 27 '18 at 12:17
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    I don't think this question is suitable for StackOverflow – Jarek Przygódzki Feb 27 '18 at 12:51

1 Answers1

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For global methods, I would suggest looking into Branch and Bound, Branch and Cut, and Cutting Plane methods. These methods may be notoriously slow though depending on the problem size. It's because it is non-convex. It would be difficult to get efficient algorithms for global optimization for this problem.

For local methods, look into the convex-concave procedure. Actually, any heuristic might work.

Miguel Alberto C Mercado
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