N-variable function maximisation

[hml]

Hello,

I have a function, well not really a function but a computer code that takes N numerical variables and returns a set of independent values.

Some of the N variables are integers, others are floating point. All are >=0 and are bound by maxima.
A subset of these N variables is a set of weights. There are M of them , M < N. And sum over M of weights is constrained to be 1. (ie there are M-1 degrees of freedom and the Mth weight is deduced from the others)

The integer variables are discrete, well, by nature. The floating point variables are continuous in theory, but discrete in practise due to the fact that a computer is used (the machine accuracy is the space between 2 possible values)

The function is not a black box but is not describable as a mathematical function.
It is a run of an alogrithm on a set of data.
1 run of the function is somehow expensive.

There are 3 objectives:
1. maximize 1 of the function return values over the set of all allowed values of the N variables independently of all other return values.
2. maximize 1 of the return values while maintaining another return value < limit.
3. the hardest objective is a dual possibly conflicting objectives
3.1. objective 1
3.2. at the same time, I would want the function around the optimal N-point to be as smooth as possible,
ie: I may find a global optimum at the N-point p1. But then, there may be a close optimum at N-point p2
where the function value is not as high as at p1, but the behavior of it "around" p2 is much smoother than "around" p1, I would then choose p2 instead of p1.

3.2 requires 3 clarifications on how to quantify everything:
3.2.1 How do I define the "neighborhood" of the optimal N point? A distance for each of the N dimensions.
3.2.2 How do I define smoothness? Is there a general quantifiable definition for N-variable functions?
3.2.3 Perhaps I need a weight to balance between 3.1 and 3.2 where if the weight is 0%, I just take the absolute global optimum, while if it's 100%, I would take the smoothest neighborhood optimum, and 50% somewhere between


Which chapters in NR3 are most applicable?

Regards,