What does inclusion-minimal maximizer mean?

Question

In set theory, combinatorial optimization etc, I came across the term called inclusion-minimal maximizer. What is it's definition?

Answer 1

In combinatorial optimization, it is common for solutions to be represented by sets of objects. There may be multiple optimal solutions to a combinatorial optimization problem. Moreover, since these optimal solutions are represented by sets, some optimal solutions may be strictly contained within others. An inclusion-wise minimal maximizer is a set that maximizes your objective function, while simultaneously not being a proper subset of any other maximizer.

For example, consider the following:

*Example: Consider the set X = {-2, -1, 1,2}. Find a subset of A$\subseteq$ X such that

• A has at most 2 elements, and
• the product of the elements in A is maximized.

For example, A = {-2,1} will give a product of -2 while A = {-1} will give a product of -1.*

Answer: {-2,-1}, {1,2}, and {2} are all maximizers for this problem. However, only {-2,-1} and {2} are inclusion-wise minimal maximizers.