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I have some questions about Karush Kuhn Tucker conditions. I am not clear about the theory. There are plenty of theorems and I am confused with it. Let us say we have the following problem: We have a general problem, not a convex problem. $$\min f(x)\\g_{j}(x)\le 0,\forall j\in J\\h_{k}(x)=0,\forall k \in K. $$ I will solve it.....Lagrange function, verifying primal feasibility, dual feasibility and complementary slackness and I will get some KKT points.

If I get no KKT point, can I say that the problem has no solution or what is the conclusion?

glS
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Laura
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1 Answers1

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In answer to your first question: You can't conclude anything without making additional assumptions about the regularity of the problem- these assumptions are called "constraint qualifications."

For example, the problem

$\min x_{2}$

subject to

$(x_{1}-1)^{2}+x_{2}^{2}=1$

$(x_{1}+1)^{2}+x_{2}^{2}=1$

has only one feasible solution at $x_{1}=0$, $x_{2}=0$, but this point doesn't satisfy the KKT conditions. Thus there are no KKT points but $x_{1}=0$, $x_{2}=0$ is the unique optimal solution to the problem.

Brian Borchers
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    Ok, let us say I will solve some nonlinear problem using KKT conditions. I will get no KKT point, so what I am supposed to do next? Verify some constraints qualifications (Slater, Linear independence, Kuhn-Tucker...) for some feasible point and if they are valid the point, the point is a global minimum? And if they are not valid the problem has no solution? – Laura Jan 16 '21 at 20:05