Note on Multi-objective optimization (The duality principle)

The inequality constraints are treated as `greater-than-equal-to' types, although a `less-than-equal-to' type inequality constraint is also taken care of. In the latter case, he constraint must be converted into a `greater-than-equal-to' type constraint by multiplying the constraint function by -1 (Deb, 1995).

The duality principle (deb, 1995; Rao, 1984; Reklaitis et al., 1983), in the context of optimization, suggests that we can convert a maximization problem into a minimization one by multiplying the objective function by -1.

The duality principle has made the task of handling mixed type of objectives much easier. Many optimization algorithms are developed to solve only one type of optimization problems, such as e.g. minimization problems. When an objective is required to be maximized by using such an algorithm, the duality principle can be used to transform the original objective for maximization into an objective for minimization.