Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.07450 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929535413387264 |
|---|---|
| author | Ricceri, Biagio |
| author_facet | Ricceri, Biagio |
| contents | In this paper, given a topological space $X$, an interval $I\subseteq {\bf R}$ and five continuous functions $φ, ψ, ω:X\to {\bf R}$, $α, β:I\to {\bf R}$, we are interested in the infimum of the function $Φ:X\to ]-\infty,+\infty]$ defined by $$Φ(x)=\sup_{λ\in I}(α(λ)φ(x)+β(λ)ψ(x))+ω(x)\ .$$ Using a recent minimax theorem ([5]), we build a general scheme which provides the exact value of $\inf_XΦ$ for a large class of functions $Φ$. When additional compactness conditions are satisfied, our scheme provides also the existence of (explicitly detected) functions $γ, η:X\to {\bf R}$ such that, for some $\tilde x\in X$, one has $$γ(\tilde x)φ(\tilde x)+η(\tilde x)ψ(\tilde x)+ω(\tilde x)=\inf_{x\in X}(γ(\tilde x)φ(x)+η(\tilde x)ψ(x)+ω(x))\ .$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the infimum of the upper envelope of certain families of functions Ricceri, Biagio Optimization and Control In this paper, given a topological space $X$, an interval $I\subseteq {\bf R}$ and five continuous functions $φ, ψ, ω:X\to {\bf R}$, $α, β:I\to {\bf R}$, we are interested in the infimum of the function $Φ:X\to ]-\infty,+\infty]$ defined by $$Φ(x)=\sup_{λ\in I}(α(λ)φ(x)+β(λ)ψ(x))+ω(x)\ .$$ Using a recent minimax theorem ([5]), we build a general scheme which provides the exact value of $\inf_XΦ$ for a large class of functions $Φ$. When additional compactness conditions are satisfied, our scheme provides also the existence of (explicitly detected) functions $γ, η:X\to {\bf R}$ such that, for some $\tilde x\in X$, one has $$γ(\tilde x)φ(\tilde x)+η(\tilde x)ψ(\tilde x)+ω(\tilde x)=\inf_{x\in X}(γ(\tilde x)φ(x)+η(\tilde x)ψ(x)+ω(x))\ .$$ |
| title | On the infimum of the upper envelope of certain families of functions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2410.07450 |