Scalarization

To solve multicriterion optimization problems, i.e. problems that have an objective function with a vector output such as $$f_0:\mathbb R^n \to \mathbb R^m$$, we can convert the output of $$f_0$$ into a scalar by multiplying with a vector $$\lambda$$ -- i.e., we minimize $$\lambda^T f_0(x)$$.

Importantly, $$\lambda$$ must be positive w.r.t. the generalized inequality over the cone that we are minimizing, i.e., $$\lambda \ge_{K^*} 0$$ (see Figure 4.9 ).