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A Gauss-Newton approach to solving generalized inequalities. (English) Zbl 0623.90072
Consider the generalized inequality $g\left(x\right){\le }_{K}0$, where g is a mapping between normed linar spaces and ${\le }_{K}^{\text{'}\text{'}}$ denotes the partial order induced by a closed convex cone K. The authors turn to study the global minimization of the functional $\rho \left(x\right):=dis\left(g\left(x\right),-k\right):=inf\left\{\parallel g\left(x\right)+k\parallel :$ $k\in K\right\}$ and give a new algorithm based on the Gauss- Newton approach. The algorithm replaces directional derivatives $\rho$ ’(x;d) by ${\Delta }\left({x}_{k}\right):={\rho }^{*}\left({x}_{k}\right)-\rho \left({x}_{k}\right)$, (where ${\rho }^{*}\left({x}_{k}\right):=dist\left(g\left({x}_{k}\right),-\left(k+\left[{g}^{\text{'}}\left({x}_{k}\right)\right]\right)\right)\right)$ 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:
 90C30 Nonlinear programming 65K05 Mathematical programming (numerical methods) 49M37 Methods of nonlinear programming type in calculus of variations