# The notion of conflicting objective functions in multi-objective optimization

In a paper by Carlos A. Coello titled

A Comprehensive Survey of Evolutionary-Based Multiobjective Optimization Techniques

he states:

"Multiobjective optimization (also called multicriteria optimization, multiperformance or vector optimization) can be defined as the problem of finding [65]: a vector of decision variables which satisfies constraints and optimizes a vector function whose elements represent the objective functions. These functions form a mathematical description of performance criteria which are usually in conflict with each other. Hence, the term "optimize" means finding such a solution which would give the values of all the objective functions acceptable to the designer."

Also, in another book THEORY OF MULTIOBJECTIVE OPTIM IZATION by NAKAYAMA, the author states:

"Let us consider the case in which there are a finite number of objective functions each of which is to be minimized. If there exists a feasible solution, action, or alternative that minimizes all of the objective functions simultaneously, we will have no objection to adopting it as the optimal solution. However, we can rarely expect the existence of such an optimal solution, since the objectives usually conflict with one another. The objectives, therefore, must be traded off"

My question is if one wants to define the notion of a "conflict" mathematically would that work: two functions are said to be in conflict if there doesn't exist a feasible solution that "optimizes- individually (in the usual sense of single-objective optimization)" each objective function. Is this valid?

• You wouldn't be able to define "conflict" in terms of the objective functions only, because it also depends on the feasible regions, e.g., the constraints. One could come up with examples of two objective functions that are "in conflict" by your definition under one set of constraints but not "in conflict" under another set. – LarrySnyder610 May 15 at 14:49