Exponential convexifying of polynomials (Q2088659)

The paper under review deals with the question of convexifying a polynomial which is positive on a closed convex subset of euclidean \(n\)-space. In [\textit{K. Kurdyka} and \textit{S. Spodzieja}, SIAM J. Optim. 25, No. 4, 2512--2536 (2015; Zbl 1331.11025)] the first and the third author have established the following results for a polynomial \(f\) which is positive on a closed convex subset \(X\) of \(\mathbb{R}^n\): {Special case:} \(X\) is compact. \begin{itemize} \item There exists a positive integer \(N\) such that the function \((1+|x|^2)^Nf(x)\) is strongly convex on \(X\). \item There exists a positive integer \(N\) such that the function \((1+|x-\xi|^2)^Nf(x)\) is strongly convex on \(X\) for every \(\xi\in X\). \end{itemize} {General case:} \begin{itemize} \item Assume that the leading form of \(f\) is positive on \(\mathbb{R}^n\setminus\{0\}.\) For each \(R>0\) there exists \(N_0\) such that for any \(N>N_0\) the function \((1+|x-\xi|^2)^Nf(x)\) is strongly convex on \(X\) for every \(\xi\in X\) with \(|\xi|\leq R\). \end{itemize} In the present paper convexification using exponential and double exponential factors are investigated. Here are the results: {Special case 1:} \(X\) is compact. \begin{itemize} \item There exists \(N_0\) such that for every \(N>N_0\) the function \(e^{N|x-\xi|^2}f(x)\) is strongly convex on \(X\) for every \(\xi\in \mathbb{R}^n\). \end{itemize} {General case:} \begin{itemize} \item Assume that the leading form of \(f\) is positive on \(\mathbb{R}^n\setminus\{0\}.\) There exists \(N_0\) such that the function \(e^{N|x-\xi|^2}f(x)\) is strongly convex on \(X\) for every \(\xi\in \mathbb{R}^n\). \item Assume that there is \(m>0\) such that \(f(x)\geq m\) for all \(x\in X\). For each \(R>0\) there exists \(N_0\) such that for any \(N>N_0\) the function \(e^{e^{N|x-\xi|^2}}f(x)\) is strongly convex on \(X\) for every \(\xi\in X\) with \(|\xi|\leq R\). \item Assume that \(X\) is semialgebraic. For each \(R>0\) there exists \(N_0\) such that for any \(N>N_0\) the function \(e^{e^{N|x-\xi|^2}}f(x)\) is strongly convex on \(X\) for every \(\xi\in X\) with \(|\xi|\leq R\). \end{itemize} The number \(N_0\) can be effectively computed. The results are applied for searching critical points of plynomials on convex closed semialgebriac sets.