On systems of linear equations with nonnegative coefficients: log-convexity of the Perron root and the \(l^2\)-norm of the positive solution with applications
Let \(\mathbb{R}_+^K\) denote the \(K\)-dimensional vector space over the nonnegative real numbers. Let \(\xi \in \mathbb{R}^K\) be a given parameter vector and \(\nu _{k,l}:\mathbb{R}^K\rightarrow \mathbb{R}_+\), \(1\leq k\), \(l\leq K\) be given nonnegative continuous functions. The set of such functions is denoted by \(\vartheta.\) For given \(\lambda >0\) and \(b\in \mathbb{R}_+^K\), \(b\neq 0,\) consider the system of linear equations \((\lambda I-V(\xi))p=b\) where \(V(\xi)=(V_{k,l}(\xi))\in \mathbb{R}_+^{K\times K}.\) Given a parameter vector \(\xi \in \mathbb{R}^K,\) if \(V(\xi)\) is irreducible then a positive solution to the system of linear equations exists if and only if \(\rho (V(\xi)) <\lambda\) where \(\rho (V(\xi))\) denotes the spectral radius of \(V(\xi).\) The set of all vectors \(\xi \in \mathbb{R}^K\) for which \(\rho (V(\xi)) <\lambda\) holds is called the feasibility set and is denoted by \(\mathbb{F}.\) If there is an additional constraint on \(p\in \mathbb{R}_+^K\) such that \(\| p\| _1=\sum_{k} p_k \leq P< \infty\) for some given \(P>0,\) then the parameter \(\xi \in \mathbb{R}^K\) is said to be feasible if \(p(\xi)>0\) and \(\| p(\xi)\| _1 \leq P.\) The feasibility set also depends upon \(P\) and is denoted by \(\mathbb{F}\)\((P).\) The sets \(\mathbb{F}\) and \(\mathbb{F}\)\((P)\) are not always convex. In this paper the authors prove that if all positive functions in \(\vartheta \) are log-convex then both the functions \(\rho (V(\xi))\) and \(\| p(\xi)\| _1\) are log-convex on \(\mathbb{R}^K\) and \(\mathbb{F}\), respectively. As corollaries they deduce that both the sets \(\mathbb{F}\) and \(\mathbb{F}\)\((P)\) are convex. The strict log-convexity for some special functions are discussed. The authors show an application of these results to wireless communication networks.
scientific article; zbMATH DE number 2102335
scientific article