How can we check the convexity of the two variables function?

Asked Apr 07 '19 at 15:15

Active Apr 09 '19 at 21:04

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I am working on the problem of KKT conditions with inequality constraints and at the last stage, it needs to check if the point in question satisfies Slater's constraint qualification.

According to the Slater's CQ, $f$ should be the concave function for maximization.

$f(x)=x_1^3 + 3x_2$ and the point in question is $(x_1,x_2) = (6,2)$. And the answer sheet says that $f$ is NOT CONCAVE, so it does not satisfy the Slater's CQ and therefore doesn't have global maxima.

How do we check the convexity of two variables function?

E.g., $x_1^3 + 3x_2$ (or can be written as $x^3 + 2y$) and $-x^2 + y \leq 0$.

edited Apr 09 '19 at 21:04

Rodrigo de Azevedo

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asked Apr 07 '19 at 15:15

DS_Veritas

Welcome to Math.SE! There are various ways to check if a function is convex. Try searching around on this website for other users who have asked about this (This post, for example). – David M. Apr 07 '19 at 18:23
Why not compute the Hessian and check where it is positive semidefinite? – Rodrigo de Azevedo Apr 09 '19 at 21:05

How can we check the convexity of the two variables function?

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