Exact penalties functions in the problem of estimating the coordinates of the dynamical system under uncertainty

V. V. Karelin

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 4 (2011), pp. 40-46

Résumé

The problem of reducing a constrained mathematical programming problem to an unconstrained one has been given a great deal of attention. In most cases such a reduction is performed with the help of socalled penalty functions. At present the theory of Penalization is well developed and widely used. The exact penalization approach is most interesting and elegant but it generally requires solving a nonsmooth problem even if the original one was smooth. However, recent developments in Nondifferentiable Optimization give some hope that these difficulties will be overcome. To be able to reduce a constrained optimization problem to an unconstrained one via exact penalization it is suitable to represent the constraining set in the form of equality, where the function describing the set must satisfy some conditions on its directional derivatives (or, in general, on its generalized directional derivatives). In the present paper we show how to describe the constraints – given in the form of differential equations – by a (nonsmooth) functional whose directional derivatives satisfy the required properties. We treat one parametric optimization problem. This problem is reduced to a nonsmooth unconstrained optimization problem. It makes it possible to construct a numerical algorithm for the unconstrained optimization problem just allowing one to solve the original parametric optimization problem. Then, by making use of necessary optimality conditions (for a nonsmooth problem) it is shown that the conditions we obtain are equivalent to the well-known ones.

Détail

Keywords: observability, the differential equations, penal functions, not differentiated optimization, management.