Projectionally exposed cones in \(R^ 3\)
\textit{J. Borwein} and \textit{H. Wolkowicz} [J. Math. Anal. Appl. 83, 495-530 (1981; Zbl 0467.90076)] introduced the concept of p-exposed (projectionally exposed) cones, which can be used, if the constraint function of an optimization problem takes values in such a cone; then certain multipliers may be chosen from a smaller set. A closed and pointed cone K is called p-exposed, if every of its faces is. A face F of K is a subcone F such that \(x\in K\), \(y\in F\) and y-x\(\in K\) imply \(x\in F\). A face F of K is exposed if there is an \(f\in K^*\) such that \(F=\{x\in K:\) \(f(x)=0\}\), and p-exposed, if there is a projection P such that \(P(K)=F.\) In this paper the p-exposed cones of \(R^ 3\) are completely characterized: K is p-exposed if and only if each face of K is exposed. The authors believe that this result extends to \(R^ n\) for all n, but it seems to them, as if it is difficult to prove it.
scientific article; zbMATH DE number 4083372