# Multi Objective Optimization Using Only Single Objective Optimization Algorithms

Let's say there is a single objective optimization problem SP (e.g. a linear program) and an optimization algorithm SA for it (e.g. simplex). Then applying SA to SP yields an optimal solution s=SA(SP).

Now let's assume there is a multi objective optimization problem MP (e.g. a linear program with two linear objectives) but there is no access to a multi objective optimization algorithm to compute the (Pareto-optimal) solutions (front).

One option is of course to linearise the MP to turn it into a SP, basically the result is a weighted sum (in the example of two terms) but this obviously does not result in all Pareto-optimal solutions but it totally depends on the chosen weights.

Is there any way to iteratively apply the single objective algorithm SA to the multi objective MP to mimic a multi objective optimization algorithm MA?

Let your two objective functions be $$f_1,f_2$$. Define a single objective function
$$f(x) = \alpha f_1(x) + (1-\alpha) f_2(x),$$
where $$\alpha$$ is a constant. If you sweep $$\alpha$$ over all possible values in $$[0,1]$$ and, for each such $$\alpha$$, solve the single-objective optimization problem (minimizing $$f$$), then the set of solutions you obtain includes all Pareto-optimal solutions to the original multi-objective optimization problem.