Solving Min-Max (Minimax) problem

[Akbar Abdillah]

Hello everyone. I have some problem with my optimization problem.

I'm trying to solve minimax optimization to find x value where y value bring the objective function value to the max (worst case).
Here is my problem. Alpha and Beta are scalar.

Please help me by explain  the idea to solve that problem.

Thank you.

Akbar

[John Hedengren]

Akbar,

There is additional information on solving minimax and maximin problems at http://apmonitor.com/me575/index.php/Main/MiniMax

Below is a version of the model that should solve your problem. You can plug it into the online interface at http://apmonitor.com/online/view_pass.php or else use the freely available MATLAB, Python, or Julia programming interfaces.

Best regards,

John Hedengren

Parameters

  a = 1.0  ! alpha

  b = 2.0  ! beta

 

Variables

  x[1:3] >= 0 <= 10

  x[4:6] >= 0 <= 5

  x[7:9] >= 0 <= 10

  y[1]   >= 2 <= 5

  y[2]   >= 3 <= 6

  z

 

Equations

  !min(x) max(y) a * (x[7]+x[8]+x[9]) + b * (x[4]+x[5]+x[6])

  minimize z

  z >= a * x[7:9]

  z >= b * x[4:6]

 

  x[7] - x[1] - x[4] + x[5] + x[6] = 0

  x[8] - x[2] - x[5] + y[1]        = 0

  x[9] - x[3] - x[6] + y[2]        = 0

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John Hedengren

APMonitor Modeling Language

jo...@apmonitor.com