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Une extension de la programmation quadratique successive. (French) Zbl 0559.90081

Analysis and optimization of systems, Proc. 6th int. Conf., Nice 1984, Part 2, Lect. Notes Control Inf. Sci. 63, 16-31 (1984).
[For the entire collection see Zbl 0538.00036.]
Nonlinear programming codes based on successive quadatic programming enjoy a large popularity. They may encounter some difficulties, however, if the quadratic subproblem is non-well posed at some iterations. We give a mean to compute a descent direction of a penalized functional (differentiable or not), using a linear quadratic approximation of the criterion and constraints. At each iteration, the subproblem to be solved is well-posed, even if the linearized constraints are not consistent. If the penalization is quadratic, we get the formulae of M. C. Bartholomew-Biggs [J. Inst. Math. Appl. 21, 67-81 (1978; Zbl 0373.90056)]. In the case of the \(L^{\infty}\) (resp. \(L^ 1)\) penalization, we extend the result of B. N. Pshenichnyj and Yu. M. Danilin [”Numerical methods in extremal problems” (1978; Zbl 0384.65028)] (resp. S. P. Han [J. Optimization Theory Appl. 22, 297- 309 (1977; Zbl 0336.90046)]. We also propose a new exact penalty function based on the \(L^ 2\) norm.

MSC:

90C20 Quadratic programming
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
49M30 Other numerical methods in calculus of variations (MSC2010)