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Currently I have this linear programming model:

Max X

such that:

Max_a(Min_b(F(a,b,X))) <= some constant

*Max_a meaning to maximize the following equation by just changing a, and the same applies to Min_b

Now, the problem becomes how to linearize the constraint part. Most of the current Minmax linearization papers talks about Minmax as an objective. But how to linearize it if it was an constraint??

Thanks

user3013013
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    Please show the function. Also, this is more apt for math.stackexchange.com. – Abhishek Bansal Nov 20 '13 at 12:24
  • As written, that's neither a linear programming model nor something that you can solve very quickly. Max of min is a two-player game in normal form, and your constraint is on the Nash equilibrium value. If the entries of the game matrix change with x, this is really going to hurt unless F has special structure. – tmyklebu Nov 20 '13 at 14:27
  • thanks, so basically is should look into the structure of F and discuss the change of saddle point w.r.t X? Emmm this should keep me thinking for a while... – user3013013 Nov 21 '13 at 04:47

1 Answers1

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Preliminary remark: the problem you describe is not a "linear programming model", and there is no way to transform it into a linear model directly (which doesn't mean it can't be solved).

First, note that the Max in the constraint is not necessary, i.e. your problem can be reformulated as:

 Max X
 subject to: Min_b F(a, b, X) <= K  forall a

Now, since you are speaking of 'linear model', I assume that at least F is linear, i.e.:

 F(a, b, X) = Fa.a + Fb.b + FX.X

And the constraint can obviously be written:

 Fa.a + Min_b Fb.b + FX.X <= k forall a

The interesting point is that the minimum on b does not depend on the value of a and X. Hence, it can be solved beforehand: first find u = Min_b Fb.b, and then solve

 Max X
 subject to Fa.a + FX.X <= k - u   forall a

This assume, of course, that the domain of a and b are independant (of the form AxB): if there are other constraints coupling a and b, it is a different problem (in that case please write the complete problem in the question).

Nicolas Grebille
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  • I guess the problem all lies within F(a,b,X) the original formula is: F(a,b,X)=∑X+∑ab, and that's problematic in the sense that the function itself is nonlinear....for a full disclosure, "b" is like a decision demand (and it is constrained by some capacity issues, availability issues, etc) and "a" is like a price. The rationale of this constrain is to say, what is the highest "a" I can get given that the optimal purchase strategy does not exceed a budget. how should I tackle then?? thx!! – user3013013 Dec 02 '13 at 07:06
  • One possible way would be to describe the fact that `U = Min_b F(a,b,X)` through the KKT optimality condition, and add these (as constraints) to the model formulation. However, it is not possible to say if it can work without the full model formulation... I would suggest you ask a new question at math.stackexchange.com with a full description of your model with the economical interpretation above. Also consider including the bound constraints on `a` and `b`, and the meaning of the variable `X` (e.g. I assume `a` should be positive, which is certainly important for convexity and coercivity). – Nicolas Grebille Dec 02 '13 at 22:09