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Does anyone have an idea of how to tackle the following maximization problem?

Maximize the function $ f(x,y,z) = x - y - \alpha z^2 $, $ \alpha > 0 $, under the following constraints:

  • C1: $ x>0 $, $\:\: y>0 $, $\:\: 0<z<1 $
  • C2: $ y - \frac{1}{2} x^2 + c_1x \geq 0 $, $\:\: c_1>0 $
  • C3: $ y - \frac{1}{2} x^2 + \frac{1}{2} (c_2 - A z )^2 < 0 $, $\:\: c_2,A>0 $

I would be grateful for any help/suggestions! :)

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  • $\begingroup$ What is $\alpha$ ? $\endgroup$ – Ken Mar 1 at 10:56
  • $\begingroup$ If you keep all the inequality strict, then there is no maximum clearly. $f$ does not have any critical points. $\endgroup$ – Pebeto Mar 1 at 15:45
  • $\begingroup$ @Pebeto Thank you! $\endgroup$ – Apollo13 Mar 2 at 12:03

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