A Gauss-Newton Approach to Solving Generalized Inequalities

Generalized inequalities are systems of the form gx ≤K 0, where g maps between normed linear spaces and “≤K” denotes the partial order induced by the closed convex cone K e.g., K = R+m1 × {0}Rm2. In this paper a Gauss-Newton type algorithm is presented for minimizing the distance function $$\rhox := \mbox{dist}gx,-K := \inf\{\Vert gx + k\Vert{:}\ k\in K\}.$$ The technique globalizes the well-known Newton methods for solving generalized inequalities, and overcomes the difficulties associated with subgradient methods for the global minimization of ρ.