Solving minmax location problems via epigraphical projection

Abstract

For solving minmax location problems in Hilbert spaces H_i, i = 1, …, n, numerically by proximal methods we present in this chapter first a general formula of the projection onto the epigraph of the function h : H₁ × … × H_n → ℝ, defined by $h\left( {x_{1} , \ldots ,x_{n} } \right)\text{: = }\sum\limits_{i = 1}^{n} {\left. {w_{i} } \right\|} \left. {x_{i} } \right\|_{{H_{i} }}^{{\beta_{i} }}$. We consider the situations when β_i = 1, i = 1, …, n, and w_i = 1, β_i = 2, i = 1, …, n, where the formulae given for instance in [3, 33, 34] turn out to be special cases for n = 1 of our considerations.