Dual characterizations of norm minimization problems

TL;DR

This paper develops dual characterizations and explicit formulas for solutions to general norm minimization problems on product spaces, including special cases like sum, maximum, and p-norms, with applications in finite and infinite dimensions.

Contribution

It introduces dual necessary and sufficient conditions and explicit solution formulas for norm minimization problems, extending understanding of optimality in product norm spaces.

Findings

Derived dual optimality conditions for general norm minimization.

Explicit formulas for solution sets under known primal-dual solutions.

Analyzed specific cases: sum norm, maximum norm, and p-norm.

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.