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Numerical continuation and singularity detection methods for parametric nonlinear programming. (English) Zbl 0811.90103

The numerical continuation and singularity detection methods investigated in this paper are based on a transformation of the nonlinear programming problem into a nonlinear system. This is done using the Fritz-John necessary optimality conditions and the active set. It also involves a nonstandard normalization of the Lagrange multipliers. After the discussion of this transformation, the paper continues with a presentation of numerical methods that can be used to solve the linear systems arising in a predictor corrector continuation. These methods are based on the Schur complement and the bordering algorithm and include symmetric factorization, the null space, and the range space method. This presentation is followed by a discussion of critical point types and by a comprehensive study of singularity detection. It is shown how the information computed previously can be used to perform these tasks efficiently.
The paper concludes with an illustration of these techniques using a model problem from design optimization.

MSC:

90C31 Sensitivity, stability, parametric optimization
93B05 Controllability
35L05 Wave equation
65H10 Numerical computation of solutions to systems of equations
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90-08 Computational methods for problems pertaining to operations research and mathematical programming
90C90 Applications of mathematical programming

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