Application of Genetic Algorithm in Multi-objective Optimization

Convex and Nonconvex MOOP:
A MOOP is convex if all the objective functions and feasible region are convex. For a convex
function: → , any two pair of solutions , ∈ will satisfy the following conditions:
1 1 ………. where 01
Both spaces (decision and objective function spaces) of a MOOP problem should be evaluated to
test their convexity. Even one of them can be non-convex while another one is convex. A MOLP
has been defined as a convex problem [1].
Ideal Objective Vector:
The ideal objective vector consists of an array with the lower bound of all objective functions of a
MOOP resulting in non-conflicting objective functions. It can only be possible for a feasible
solution when the minimum of all objective functions are identical. Otherwise, it does not exist. If
∗ is a solution vector of variables that minimize or maximize the i th objective in a MOOP having
M conflicting objectives,
∃ ∗ ∈, ∗ ∗ , ∗ ,……. ∗ : ∗
Thus the ideal vector is defined as following,
∗ ∗ ∗ , ∗ ,….. ∗
where
∗
is the optimum value of M th objective function and the point in decision variable space
which determines this vector is the ideal solution.
Utopian Objective Vector:
The objective vector having components slightly less than that of an ideal objective vector for the
minimization of a MOOP problem is called the utopian objective vector. It is used for algorithms
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