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\centerline{\bf NONLINEAR OPTIMISATION}
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\centerline{\bf AND THE ASTEROID IDENTIFICATION PROBLEM}

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\centerline{\bf M.E. Sansaturio\dag, A. Milani\ddag\  and L. Cattaneo\ddag}

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\centerline{\it \dag E.T.S. Ingenieros Industriales, Valladolid, Spain}
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\centerline{\it\ddag Space Mechanics Group, University of Pisa, Italy}
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\centerline{\bf Prepared for: IAU Symposium 172, Paris, July 1995}
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The classical procedure of differential correction allows to improve
orbits for which new observations are available; however, it works
only provided the original orbit is within the convergence domain of
the pseudo--Newton method. Given the strong nonlinearity of the
problem, this occurs only when the residuals of the new observations
with respect to the old orbit are quite small. On the contrary, if a
single opposition asteroid, observed only on a short arc, is not
recovered for several years, then the position uncertainty of the
asteroid grows to a very large size, until the asteroid is ``lost'',
that is it can be in widely separated regions of the sky. In this
case, if the same asteroid is observed again, the identification with
the already known one can be very difficult, and even when the
identification is performed correctly the differential correction
algorithm can fail. The only lost asteroid in the numbered asteroid
list is now {\it 719 Albert}, lost since 1911. If the proposed {\it
Spaceguard survey} will be realized, the problem of identifying
millions of new discoveries within catalogues of comparable size will
be one of the main challenges.

We have begun experimenting with a new algorithm of orbit
determination, which is capable of performing both positive and
negative identification of asteroids lost since many years. We are
using {\sl LANCELOT}, a software system developed by Conn, Gould and
Toint to handle large and constrained nonlinear optimisation
problems. We have implemented the asteroid orbit determination, in the
case of two widely separated observed arcs, as a nonlinear least
squares problem; so far we are experimenting with a model problem
(i.e. 2--body orbits). The nonlinearity of the problem is mitigated by
cutting the orbit between the two observed arcs into a large number
(e.g. 50) of small arcs; the joining of each small arc with the next
one is given as a nonlinear constraint. We report on the performance
and applicability of this method to obtain both positive
identification (hence orbit improvement) and negative identification
(when the obtained minimum is too high, proving that the two observed
arcs are incompatible).


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