I am studying the max-sum algorithm to solve Distributed Constraint Optimization Problem. I have a very basic doubt about the maximization of a function w.r.t. a single variable. Consider the constraints F(x1,x2) with x1={a,b} and x2={a,b} represented by the following table:
We want to maximize the table w.r.t to x2. I have an exercise provided me by a friend of mine. The solution is:
Is this solution right? My doubt concerns the fact that we have chosen the value x1=a to maximize x2=a (first line on the table) whereas we have chosen x1=b to maximize x2=b. Is it "legal" (mathematically correct) to "assign" two different values to x1 in order to maximize the table?
Your initial function is $F(x_1,x_2)$. The function in the second table is: $$G(x_2) = \max_{x_1} F(x_1,x_2).$$ The function $G$ depends only on $x_2$. A version of your table with a correct header is: