Certain convex minimization problem

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SUMMARY

The discussion centers on the optimization problem of minimizing a convex function f0(x) subject to the constraint f1(x) ≥ 0. It confirms that the inequality sign does not affect the application of the Lagrangian method and KKT conditions. The key takeaway is that f1(x) ≥ 0 can be transformed into the standard form by redefining the function as -f1(x) ≤ 0, allowing for the use of conventional KKT conditions. The solution process remains valid regardless of the direction of the inequality.

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nikozm
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Hi, I would like to know if the inequality sign plays any role to the following optimization problem:

minimize f0(x)
subject to f1(x)>=0

where both f0(x) and f1(x) are convex. The standard form of these problems require a constraint such as: f1(x)<=0, but i am interested in the opposite condition (f1(x)>=0) and solving it using the standard Lagrangian method and KKT conditions. Is this right?

Any help would be useful. Thanks in advance!
 
The method essentially says that the solution occurs when the inequality is strict, and you can just find points where the gradient is zero, or the inequality is an equality, and you can use Lagrangian multipliers. It doesn't matter whether the inequality is less than or greater than, everything I said above still holds.
 
f1(x)>=0 is the same as -f1(x)<=0, so just define f1 appropriately and you get the 'normal' KKT conditions.
 

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