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Some people asked questions about KKT conditions with strict inequality constraints, such as

Kuhn Tucker conditions with strict inequality constraints?

Questions about constraints and KKT conditions

https://www.researchgate.net/post/Is_KKT_condition_applicable_for_optimality_when_the_constraints_is_strictly_less_than_type

Let us see an example.

$\min\quad \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a}$

$\mathrm{s.t.}\quad a>0,\ b>0,\ c>0,\ d>0,\ a^4+b^4+c^4+d^4 = 4.$

The minimum $4$ is attained at $a=b=c=d=1$. The constraints $a,b,c,d>0$ are strict inequalities. How to deal with them in KKT conditions?

Just as some people said (e.g., the 3rd link above), we simply ignore the strict inequality constraints and use KKT conditions. If the minimum is attainable (that is, min not inf), the solution will satisfy the strict inequalities. For this example, it is the Lagrange multiplier method $L = \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} + \lambda (a^4+b^4+c^4+d^4 - 4)$.

I found the lecture note:

https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec6_constr_opt.pdf

In the lecture note (2nd page), the following problem is considered:

$\min\quad f(x)$

$\mathrm{s.t.}\quad h(x) = 0,\ g(x) \le 0, \ x \in X.$

Here $X$ is an open set. Then KKT conditions are used without including $x\in X$.

I want to make sure if this is the case. Thanks.

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Strict inequalities cannot be taken into account in KKT conditions.

To solve the problem, compute stationary points of the problem without these constraints, then throw away all of them that violate these conditions.

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  • $\begingroup$ That's what I mean. I read the book "Nonlinear programming" by Dimitri P. Bertsekas, but the problem including an open set $x\in X$ as constraint is not considered. So I need more textbooks etc. to support this. By the way, in some case, we may replace the strict inequality by the "less than and equal to" and use KKT conditions as usual then choose the desired. $\endgroup$ – River Li Aug 15 '18 at 14:53

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