Formulation help

Answered

nick wilson

• September 06, 2024 23:43

I have a midsized stochastic program that's experience longer runtimes than I'd like. I'm simplifying some notation, but the crux of my problem is: I have a known entity, d, and nonnegative variables, x and l. I have some simple constraints x <=d and x<= l, and then x = min(l, d). The minimum is there to keep things consistent across scenarios in my model, so it's needed. In effect what it does is when x << d, it sets l = x. I was toying around with using a quadratic constraint to approximate this as follows: l <= x + 1/(d-x) or equivalently (l-x)(d-x) <= 1, so when x << d the constraint relaxes and allows l > x. Arguably, we could replace the 1 with an arbitrary constant for tuning. Unfortunately, this is nonconvex. I was wondering if your team might have a suggestion on something else to try.

Related to

• Big-M
• Stochastic

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• 

  Riley Clement

  Gurobi Staff

  • September 09, 2024 04:08

  Hi Nick,

  Using a quadratic expression is likely a step in the wrong direction.  You could use Gurobi's "MIN" function - in gurobipy this is addGenConstrMin, or you could introduce a binary variable y and use the following set of constraints

  x <= l
  x <= d   # better as an upper bound, rather than constraint
  x >= d*y
  x >= l - y*l.ub

  where l.ub is an upper bound on l.

  Note that if your objective function naturally causes the x variable to want to be as large as possible then you will only need the first two of these constraints.

  - Riley

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