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A Gauss-Newton approach to solving generalized inequalities. (English) Zbl 0623.90072
Consider the generalized inequality g(x) K 0, where g is a mapping between normed linar spaces and `` K '' denotes the partial order induced by a closed convex cone K. The authors turn to study the global minimization of the functional ρ(x):=dis(g(x),-k):=inf{g(x)+k: kK} and give a new algorithm based on the Gauss- Newton approach. The algorithm replaces directional derivatives ρ ’(x;d) by Δ(x k ):=ρ * (x k )-ρ(x k ), (where ρ * (x k ):=dist(g(x k ),-(k+[g ' (x k )]))) and avoids the difficulties of the subgradient approach. The authors show also the convergence of this algorithm. Their perspective is a more geometric one, thereby eliminating the dependence on polyhedrality and finite dimensionality.
Reviewer: Y.Ling

MSC:
90C30Nonlinear programming
65K05Mathematical programming (numerical methods)
49M37Methods of nonlinear programming type in calculus of variations