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A Gauss-Newton approach to solving generalized inequalities. (English) Zbl 0623.90072
Consider the generalized inequality $g(x)\le\sb K0$, where g is a mapping between normed linar spaces and $``\le\sb K''$ denotes the partial order induced by a closed convex cone K. The authors turn to study the global minimization of the functional $\rho (x):=dis(g(x),-k):=\inf \{\Vert g(x)+k\Vert:$ $k\in K\}$ and give a new algorithm based on the Gauss- Newton approach. The algorithm replaces directional derivatives $\rho$ ’(x;d) by $\Delta (x\sb k):=\rho\sp*(x\sb k)-\rho (x\sb k)$, (where $\rho\sp*(x\sb k):=dist(g(x\sb k),-(k+[g'(x\sb k)])))$ 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

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