# Could non-supported efficient solutions in multi-objective optimization problem be an optimal solution of a parameterized single-objective problem?

Since all supported efficient solutions in a multi-objective optimization problem are actually the optimal solutions for some weighted sum scalarization single-objective optimization problem with the positive multiplier $$\sum_k \lambda_k = 1$$. Does the above theorem proposed by Geoffrion implies that other non-supported efficient solutions can also be the optimal solutions for some parameterized single-objective problem but with a multiplier $$\sum_k \lambda_k \neq 1$$ ?

## 1 Answer

No. If you have strictly positive weights you will get a supported efficient solution by solving the weighted sum scalarization, regardless of whether the weights sum to one or not. Requiring the weights to sum to 1 is the same as multiplying the objective by a constant, which does not change the set of optimal solutions.

Edit: if you allow your weights to be both positive and negative, all sorts of things may happen. If your problem is a pure binary problem, all feasible solutions are extreme points of the convex hull of feasible solutions. Hence, you may be able to construct weighted sum scalarizations of the objectives with some negative weights, that leads to non-supported efficient solutions.