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(Updated)

I am looking to simplify formulation of a MINLP problem with below formulation-

Objective Function -

$$min (\sum_{i=1}^n x_i), x_i \in I, x_i>=0 $$

Below are the constraints -

$$\sum_{i=1}^n (x_i * y_i) = C_1 , y_i \in [0,1], y_i>=0 $$

$$\sum_{i=1}^n (x_i * z_i) = C_2, z_i \in [0,1], z_i>= 0 $$

$$\sum_{i=1}^n (y_i) <= C_3, y_i \in [0,1], y_i>= 0 $$

$$\sum_{i=1}^n (z_i) <= C_4, z_i \in [0,1], z_i>= 0 $$

I am using OR package in SAS which supports MILP/NLP formulation but not MINLP. If the non-linear constraint could be simplified/approximated to linear, I could use MILP solvers for the problem.

Any direction/solution would be helpful.

chandrayan
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  • I have the feeling that you have omitted the definitions of the variables $x_i´s$ and $y_i´s$ – callculus42 Aug 25 '17 at 18:01
  • Both x_i and y_i are continuous variables – chandrayan Aug 26 '17 at 08:02
  • At the moment I don´t see a way to simplify the constraint. Maybe if you add some definitions, constraints, context or the whole model itself I would be able to give an advice. As it stands it has infinitely many solutions, for instance $x_i=y_i=\sqrt{\frac{C}{ n}}$ – callculus42 Aug 26 '17 at 11:54
  • Updated formulation, please note, x_i is positive integer, made an error in previous comment. – chandrayan Aug 26 '17 at 15:34

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