Modeling some constraintsWe have two decision variables x ∈ Z 0 + that is the main decision variable and 0 ⩽ y ⩽ 1 that is an auxiliary decision variable.Now based on the nature of the problem we are studying, we know that y = 1 x , x = 0 ⇔ y = ∅ , x &gt; 0 ⇔ y ⩾ 0.Now the challenge is how to formulate the dynamics of these two decision variables in the constraints of a mathematical program. We have x in the denominator and since at optimality x can actually be zero, this would create a problem. Obviously, we cannot write x y = 1 because if the optimal solution is that x = 0, the equality would be violated.Any suggestions would be much appreciated.

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Modeling some constraintsWe have two decision variables   x  ∈            Z              0      +       that is the main decision variable and   0  ⩽  y  ⩽  1 that is an auxiliary decision variable.Now based on the nature of the problem we are studying, we know that  y  =      1    x    ,    x  =  0  ⇔  y  =  ∅  ,    x  &amp;gt;  0  ⇔  y  ⩾  0.Now the challenge is how to formulate the dynamics of these two decision variables in the constraints of a mathematical program. We have x in the denominator and since at optimality x can actually be zero, this would create a problem. Obviously, we cannot write   x  y  =  1 because if the optimal solution is that   x  =  0, the equality would be violated.Any suggestions would be much appreciated.

dilettato5t1 · Accepted Answer

Step 1Assume we have an a priori upper bound M (integer) on x. Introduce binary variables       z    0    ,      z    1    ,  …  ,      z    M  , where       z    j    =  1 if and only if   x  =  j. Add the constraint  x  =      ∑          j      =      0        M    j  ∗      z    j  to define x in terms of the new variables, and the constraint      ∑          j      =      0        M        z    j    =  1to ensure that x is uniquely determined. Finally, add the constraint  y  =      ∑          j      =      1        M        1    j    ∗      z    j  to define y. Note that   x  =  0 implies   y  =  0 here.Step 2There is no way to formulate a proper optimization model so as to make a variable &quot;undefined&quot; under certain circumstances. If letting   y  =  0 when   x  =  0 causes problems elsewhere in the model, you need to determine a numerical value for y that can function as &quot;undefined&quot; in the remaining constraints.

Modeling some constraints. We have two decision variables x in Z^{0+} that is the main decision variable and 0<= y<=1 that is an auxiliary decision variable.

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