Dual characterizations of norm minimization problems
Nguyen Duy Cuong
TL;DR
The paper develops a unified duality-based framework for norm minimization on product spaces $X^n$, deriving necessary and sufficient dual optimality conditions and explicit descriptions of the entire solution set once a primal solution and its dual vectors are known. It builds a general product-norm construction on $X^n$ via simplex-generated functions $\\psi\,\in\,\Psi_n$ and analyzes the induced dual norms, enabling a coherent treatment of sum, max, and $p$-norm cases within the same framework. The authors then specialize to three canonical problems—the Fermat–Torricelli, Chebyshev centre, and $p$-Fermat–Torricelli—providing concrete dual characterizations and constructive descriptions of solution sets, including intersections of affine and geometric regions like Voronoi cells. They also discuss existence, uniqueness, and compactness of solution sets, with several examples in finite- and infinite-dimensional spaces. Overall, the work advances duality-driven analysis and explicit solution-construction methods for a broad class of norm minimization problems with practical implications in localization and approximation tasks.
Abstract
The paper studies a general norm minimization problem on a product of normed vector spaces. We establish dual necessary and sufficient optimality conditions and derive explicit formulas for the corresponding solution sets. These formulas are obtained under the assumption that one optimal solution together with its associated dual vectors arising from the optimality conditions is known. Three important cases of product norms, namely the sum norm, maximum norm and $p$-norm, are also studied. Several examples in finite and infinite dimensional spaces equipped with various types of norms are presented to illustrate the established results.