Hello, I have a basic question,
I know that the convex optimization problem is the one in which its objective and inequality functions are convex and its equality function is affine. My question is which set the objective and inequality functions have to be convex on? Is it their entire domain? or just the feasible set of the problem?
For example: let the problem be minimizing f0(x),
s.t.:
fi(x)
Dear Ali Zaher Assaygh,
Insuggest you to see links and attached files on topic.
https://www.princeton.edu/~aaa/Public/Teaching/ORF523/S16/ORF523_S16_Lec7_gh.pdf
https://www.solver.com/convex-optimization
https://www.sciencedirect.com/topics/mathematics/convex-programming-problem
http://www.ifp.illinois.edu/~angelia/L5_exist_optimality.pdf
Best regards
The answer is yes if the feasible set is convex. Otherwise, the convexity of a function on a nonconvex set has no sens
Most results and algorithms of convex optimization would not hold if the functions involved are only convex on the feasible set (provided the latter is convex - as Mounir Haddou pointed out). For example, the Lagrangian will not be convex spoiling most of the duality theory; if we are talking about nonsmooth problems, subgradiens may not exist, etc etc. One can of course rectify this by defining the functions involved to be +infinity outside of the (convex, closed) feasible set thereby restoring their convexity, but this will ruin any smoothness your original problem may have had.
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