Solving Critical Point Conditions for the Hamming and Taxicab Distances to Solution Sets of Polynomial Equations

Minimizing the Euclidean distance (ℓ
₂
-norm) from a given point to the solution set of a given system of polynomial equations can be accomplished via critical point techniques. This article extends critical point techniques to minimization with respect to Hamming distance (ℓ
₀
-"norm") and taxicab distance (ℓ
₁
-norm). Numerical algebraic geometric techniques are derived for computing a finite set of real points satisfying the polynomial equations which contains a global minimizer. Several examples are used to demonstrate the new techniques.