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A Gauss-Newton approach to solving generalized inequalities. (English) Zbl 0623.90072

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

90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
49M37 Numerical methods based on nonlinear programming
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