ScalingStacks

Proposition 3.64 . [02MD]

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Proposition 3.64.

Let Λ\Lambda be a polyhedron in NℝN_{\mathbb{R}} and ff a piecewise affine concave function with dom⁡(f)=Λ{\operatorname{dom}}(f)=\Lambda given as

Λ\displaystyle\Lambda =⋂1≤j≤k{u∈Nℝ∣⟨aj,u⟩+αj≥0},\displaystyle=\bigcap_{1\leq j\leq k}\{u\in N_{\mathbb{R}}\mid\langle a_{j},u\rangle+\alpha_{j}\geq 0\},
f⁡(u)\displaystyle f(u) =mink+1≤j≤l⁡(⟨aj,u⟩+αj)for ​u∈Λ\displaystyle=\min_{k+1\leq j\leq l}(\langle a_{j},u\rangle+\alpha_{j})\quad\text{for }u\in\Lambda

with aj∈Mℝa_{j}\in M_{\mathbb{R}} and αj∈ℝ\alpha_{j}\in\mathbb{R}. Then

stab⁡(f)\displaystyle\operatorname{stab}(f) =cone⁡(a1,…,ak)+conv⁡(ak+1,…,al),\displaystyle=\operatorname{cone}(a_{1},\dots,a_{k})+\operatorname{conv}(a_{k+1},\dots,a_{l}),
f∨​(x)\displaystyle f^{\vee}(x) =sup{∑j=1l−λjαj|λj≥0,∑j=k+1lλj=1,∑j=1lλjaj=x} for x∈stab(f).\displaystyle=\sup\bigg\{\sum_{j=1}^{l}-\lambda_{j}\alpha_{j}\bigg|\ \lambda_{j}\geq 0,\sum_{j=k+1}^{l}\lambda_{j}=1,\ \sum_{j=1}^{l}\lambda_{j}a_{j}=x\bigg\}\ \text{ for }x\in\operatorname{stab}(f).

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