Bellman Functions: A Toy Problem

Abstract: Suppose we have two functions $f_1,f_2$ defined on an interval $I$ that have their respective bounds $a_i \leq f_i \leq b_i$ and averages $\langle f_i \rangle_I = x_i$, and we want to see how small or large their scalar product can be. This means we are interested in the following optimization problems: \[ \mathbb{B}^{\max} \overset{\mathrm{def}}{=} \sup\left\{ \langle f_1 f_2 \rangle_I : a_i \leq f_i \leq b_i,\; \langle f_i \rangle_I = x_i \right\}, \] \[ \mathbb{B}^{\min} \overset{\mathrm{def}}{=} \inf\left\{ \langle f_1 f_2 \rangle_I : a_i \leq f_i \leq b_i,\; \langle f_i \rangle_I = x_i \right\}. \]

This is one of the simplest problems regarding Bellman functions that nevertheless helps to get some understanding of their theory.

Mathematics

Audience: learners

Fraser Valley Math & Stats Seminar

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