Solve the following NLP: { min − 3 x + y − z 2 s . t g ( x , y , z ) = x + y + z ≤ 0 h ( x , y , z ) = − x + 2 y + z 2 z = 0 My attemptUsing kkt conditions, we have 2 possibles situations:1) If g &lt; 0: ∇ f + λ ∇ h = 0 , h = 0 { − 3 − λ = 0 − 3 y 2 + 1 + 2 λ = 0 2 λ z = 0 − x + 2 y + z 2 = 0 From first and second, we see it is impossible**2)** g = 0: ∇ f + λ ∇ h + μ ∇ g = 0 , μ ≥ 0 { − 3 − λ + μ = 0 − 3 y 2 + 1 + 2 λ + μ = 0 2 λ z + μ = 0 − x + 2 y + z 2 = 0 x + y + z = 0 I couldnt solve this last one. Any idea? I've tried to put y,z im function of μ and use the last 2 eq., but didnt work (it became complicated).Thanks in advance!

Question

Solve the following NLP:      {                            min                          −          3          x          +          y          −                      z            2                                                s          .          t                          g          (          x          ,          y          ,          z          )          =          x          +          y          +          z          ≤          0                                              h          (          x          ,          y          ,          z          )          =          −          x          +          2          y          +                      z            2                    z          =          0                        My attemptUsing kkt conditions, we have 2 possibles situations:1) If   g  &amp;lt;  0:   ∇  f  +  λ  ∇  h  =  0  ,  h  =  0      {                            −          3          −          λ                          =                          0                                      −          3                      y            2                    +          1          +          2          λ                          =                          0                                      2          λ          z                          =                          0                                      −          x          +          2          y          +                      z            2                                    =                          0                        From first and second, we see it is impossible**2)**  g  =  0:   ∇  f  +  λ  ∇  h  +  μ  ∇  g  =  0  ,    μ  ≥  0      {                            −          3          −          λ          +          μ                          =                          0                                      −          3                      y            2                    +          1          +          2          λ          +          μ                          =                          0                                      2          λ          z          +          μ                          =                          0                                      −          x          +          2          y          +                      z            2                                    =                          0                                      x          +          y          +          z                          =                          0                        I couldnt solve this last one. Any idea? I&#039;ve tried to put y,z im function of   μ and use the last 2 eq., but didnt work (it became complicated).Thanks in advance!

Alexis Fields · Accepted Answer

x  =  2  y  +      z    2      ⟹    3  y  +  z  +      z    2    ≤  0    ⟹    −  y  ≥                              z          2                +        z            3        ,  −  3  x  +  y  −      z    2    =  −  6  y  −  3      z    2    +  y  −      z    2    =  −  5  y  −  4      z    2    ≥                    5        (                  z          2                +        z        )            3        −  4      z    2    =  −                    7                  z          2                −        5        z            3       it seems no min when   z  →  ∞

Solve the following NLP: { <mtable rowspacing="4pt" columnspacing="1em"> <m

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