The usage of MAX: whats the difference between this two expressions?

[Weiwei Yao]

Dear Prof. Lofberg,

Thank you very much for your previous helps. YALMIP is so great for us!

Now I have a question: what's the difference between the two expressions below? Why do I get different results with the two expressions ? Could you help me please : )

-----------------------------------------------------------------

x = sdpvar(1,24);

y = sdpvar(1,24);

z = max(max(x),max(y));     %expression A

z = max(x+y);                      %expression B

-----------------------------------------------------------------

Full m-code as follows:

%**********************************************************************

close all;clear;clc;

X = sdpvar(1,24);

Y = sdpvar(1,24);

onoffX = binvar(1,24);

onoffY = binvar(1,24);

F = [onoffX+onoffY<=1,

0<=X<=1e5.*onoffX,

0<=Y<=1e5.*onoffY];

%**********************************************************************

% Is the two expressions the same? Why do I get different results?

% expression A:

                max_XY = max(max(X),max(Y));

% expression B:

                max_XY = max(X+Y);

%**********************************************************************

f = sum(X)+max_XY;

ops = sdpsettings('verbose',0,'debug',1,'solver','gurobi');

result = optimize(F,-f,ops);

f = value(f)

% The result with expression A is 4.3e5, and another is 2.5e6.

[Johan Löfberg]

of course you get different answers

max(max([1 2 3]),max([1 2 3])) = 3

max([1 2 3]+[1 2 3]) = 6

...and using big-M modelling with coefficients in the order 10^5 is a recipe for numerical problems and strange results.

[Weiwei Yao]

Because of the binvar onoffX + onoffY <=1, so X and Y are  mutually exclusive.

max(max([1 0 3]),max([0 2 0])) = 3

max([1 0 3]+[0 2 0]) = 3

Why is that? Maybe the most likely cause is the order 10^5 ?

Best regards

[Johan Löfberg]

Are you asking why max([1 0 3]+[0 2 0]) is equal to 3??

[Weiwei Yao]

I mean X and Y are  mutually exclusive.  So the example you give is inappropriate.

[1 0 3] and [0 2 0] is appropriate, not [1 2 3] and [1 2 3].

Thank you so much!

[Weiwei Yao]

X.*Y always equal zero in my model.  so X=[1 2 3] and Y=[1 2 3]  is  inappropriate.

[Johan Löfberg]

So I guess your question this is, should the two objectives

sum(X)  + max(X+Y)

and

sum(X)  + max(max(X,max(Y))

lead to the same maximizing argument (Y=0, X at upper bound)?

Yes they should, but with this horrible big-M scaling you don't get that. You cannot expect things to work with the bounds go above 10^4 or so, since that is an internal assumption in the big-M modelling that you don't have any bounds on variables involved in big-m models larger than that.

[Weiwei Yao]

Now I find that, when I reduce the coefficients from 1e5 to 1e3, all the confused results disappear. That's great! 

But what's the principle with using big-M modelling? Where's the bound? Maybe the best feasible way is to lower the coefficients. 1e5 is too much larger.

Thank you Prof Lofberg! 

[Johan Löfberg]

YALMIP uses bounds 10^4 on some internal big-M constructs which are required to implement your nonconvex use of the max-operator.