A Note on Scalarization Error in Lexicographic Optimization

By Samuel Belko, published on July 25, 2025.

In this post, I describe an insight I realized in my master's thesis on design optimization of hybrid railway vehicles.

For brevity, we consider a one-dimensional, unconstrained setting. Let $f_1, f_2 : \mathbb{R} \to \mathbb{R}$ and suppose that for some constants $L,U$ and all $x$ holds $L \leq f_2(x) \leq U$.

Consider a lexicographic problem

For some $K>0$, consider a scalarized version of (1)

Furthermore, consider an optimizer $x^*_{\text{lexi}}$ of (1) and $x^*_{\text{scal}}$ of (2). By the lexicographic constraint in problem (1), we get

Optimality of $x^*_{\text{scal}}$ implies

Hence, we obtain that

holds. In particular

follows.

In conclusion, by choosing $K$ sufficiently small, $f_1 (x^*_{\text{scal}})$ and $f_1 ( x^*_{\text{lexi}})$ can become arbitrarily close. Irrespective of the choice of $K$, $f_2(x^*_{\text{scal}})$ is always at least as good as $f_2( x^*_{\text{lexi}})$.

The above insight is simple, yet it proved to be useful for greatly speeding up solving (1) by merging objectives and solving (2) instead. Though brief, I hope this this post sheds some light on scalarization in the lexicographic setting.