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On a subproblem of trust region algorithms for constrained optimization. (English) Zbl 0711.90062
The author studies a subproblem which appears in some trust region algorithms for constrained optimization, where the original problem is to minimize a nonlinear function subject to equality constraints. It is the minimization of a general quadratic function with two special quadratic constraints. Some peculiarities of the problem, mainly with respect to the signs of the eigenvalues of the Hessian of the Lagrangian function at a solution are investigated. The main result is that the Hessian of the Lagrangian has at most one negative eigenvalue if the Lagrangian multipliers are unique. An example is presented to show that the Hessian may have a negative eigenvalue when one constraint is inactive at the solution.
Reviewer: M.Todorov

MSC:
90C20 Quadratic programming
65K05 Numerical mathematical programming methods
90-08 Computational methods for problems pertaining to operations research and mathematical programming
90C30 Nonlinear programming
Software:
GQTPAR
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References:
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[2] R. Fletcher,Practical Methods for Optimization, Vol. 2: Constrained Optimization (Wiley, Chichester, 1981). · Zbl 0474.65043
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[4] J.J. MorĂ© and D.C. Sorensen, ”Computing a trust region step,”SIAM Journal on Scientific and Statistical Computing 4 (1983) 553–572. · Zbl 0551.65042
[5] M.J.D. Powell and Y. Yuan, ”A trust region algorithm for equality constrained optimization,” Report DAMTP 1986/NA2, University of Cambridge (Cambridge, UK). · Zbl 0816.90121
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