Can you please help me how can solve the following nonlinear program by indicating some useful reference or videos relevant to this problem? With my limited knowledge in this field, I tried to solve it using the Karush-Kuhn-Tucker (KKT) conditions, but it is not working.

\begin{alignat*}{2} \text{maximize } \qquad & f(x_1, x_2 , x_3 , x_4, x_5, x_6)=\Big[8\left(x_1^{3} + x_2^{3} + x_3^{3} + x_4^{3} + x_5^{3} + x_6^{3}\right) \Big]^{\frac{2}{3}} \\ \text{subject to } \qquad & 4\left(x_1^{2} + x_2^{2} + x_3^{2} + x_4^{2}\right) \le 1\\ & 4\left(x_3^{2} + x_4^{2} + x_5^{2} + x_6^{2}\right) \le 1\\ & 4\left(x_1^{2} + x_2^{2} + x_5^{2} + x_6^{2}\right) \le 1\\ & x_1, x_2, x_3, x_4, x_5, x_6 \ge 0. \end{alignat*}

  • $\begingroup$ Show us your trial to see where it fails $\endgroup$ – paf Jun 16 '18 at 10:35
  • $\begingroup$ Are you sure this is convex? WA suggests it is not. $\endgroup$ – Rodrigo de Azevedo Jun 16 '18 at 10:37
  • $\begingroup$ Isn't it $f(x_1, x_2 , x_3 , x_4, x_5, x_6)=\Big[8\left(x_1^{2} + x_2^{2} + x_3^{2} + x_4^{2} + x_5^{2} + x_6^{2}\right) \Big]^{\frac{2}{3}}$ instead? $\endgroup$ – Rodrigo de Azevedo Jun 16 '18 at 10:43
  • $\begingroup$ Prof. Rodrigo de Azevedo thank you very much for your answer, but I am interesting for the above conjecture object function is the aim, but I am not checking if it is convex or not? so building upon your answer, is this means that my nonlinear problem is not working due to the objective function is not convex? $\endgroup$ – Marcus Jun 16 '18 at 14:06
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    $\begingroup$ Note that you are maximizing the objective, so you don't want it to be convex, you want it to be concave (which it isn't as it is convex quadratic when you set all but one variable to 0). Also note that the power $2/3$ is completely redundant as that is a monotonic increasing operation. $\endgroup$ – Johan Löfberg Jun 16 '18 at 14:25

$$f=\left(8\sum_{k=1}^6x_k^2\right)^{\frac{2}{3}}\leq(4\cdot\frac{3}{4})^{\frac{2}{3}}=\sqrt[3]{9}.$$ The equality occurs for $x_k=\frac{1}{4},$ which says that we got a maximal value.

  • $\begingroup$ That is, of course, correct. But I believe @Marcus would like to know what lead you to this solution, and not only a proof of why it is a solution :) $\endgroup$ – Alex Shtof Jun 16 '18 at 14:17
  • $\begingroup$ @Alex I just saw conditions and it gives a solution. $\endgroup$ – Michael Rozenberg Jun 16 '18 at 14:21
  • $\begingroup$ Has it been confirmed that it should be quadratics in the objective? $\endgroup$ – Johan Löfberg Jun 16 '18 at 14:34
  • $\begingroup$ thank you for all these useful comments. Actually I am working on some open problems and after a series of complex mathematical computations I got this mathematical modelling ( nonlinear optimization problem), but I am still confusing about the ideal answer depend on some above answer I think the first step to check the objective function is concave then what is the second step? just I want to know if there are some references that contains some examples similar to my problem? or someone have experiment about this field to tell me my nonlinear optimization problem is not correct. $\endgroup$ – Marcus Jun 16 '18 at 15:28
  • $\begingroup$ The statement "to tell me my nonlinear optimization problem is not correct" does not make sense. It is a well-defined optimization problem, but whether it is the one you want or not is impossible for us to know. And is it supposed to be squares or cubes? (as this answer is only relevant if you have a typo in your question, i.e. it does not answer your current problem formulation) $\endgroup$ – Johan Löfberg Jun 16 '18 at 15:43

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