## New method for minimizing regular functions with constraints on parameter region

Paper: 180
Session: A (talk)
Speaker: Yaschenko, Sergey, Joint Institute for Nuclear Research, Dubna
Keywords: algorithms, analysis, simulation

New method for minimizing regular functions
with constraints on parameter region
I.N.Silin, V.S.Kurbatov, S.V.Yaschenko

Joint Institute for Nuclear Research, Head Post Office,
P.O.Box 79, 101000 Moscow, Russia
(e-mail yaschenk@nusun.jinr.dubna.su)

Approximately thirty years ago a linearization method for
minimizing chisquare - like functionals was proposed [1], the
subroutine FUMILI was developed by one of the authors
(I.N.Silin) [2] and became available for users. The simplest
case of constrained fit for constraints of type a >= X >= b was
implemented in it.
In FUMILI as the second derivatives of minimized function their
approximate values are used with the neglection of members
containing second derivatives of the functional argument, and
special technics is used for stabilizing iteration process. In
most cases it proved to be very effective. For chisquare-like
functionals approximate matrix of second derivatives is
nonnegatively defined. The problems appear when such matrix has
eigenvalues close or equal to zero. In addition to this FUMILI
cannot minimize functions of arbitrary structure.
For many years I.N.Silin worked on a new algorithm with the
aim to overcome these shortcomings. The work on the algorithm
was intensified when the other author (V.S.Kurbatov) has built a
practically acceptable algorithm for reducing the order of the
problem by taking into account nonlinear constraints ([3] and
[4]). The more constraints exist, the more stable the solution
search must be, if we do it accurately. The idea is: let us assume that we have a quadratic
approximation for function of np variables X and nc (nc < np)
constraints in form of equalities. If the constraints are
regular functions of parameters X we may linearize them in the
vicinity of any point X0 and express nc parameters over other
np-nc ones. The problem is, how to choose this nc parameters,
and we solved this problem [5]. Substituting such expression
into quadratic function we obtain it in another form, this time
depending on np-nc parameters (in fact we are reducing the
dimensionality of the problem). Using this technique we can
take into account the constraints of general type, both
inequalities and equalities [5].
Finally a new code [5] was created and called FUMIVI :
FUnction MInimization by Vallies Investigation. The main
features of the new code are :
- Regular functions of arbitrary structure can be minimized.
- When minimizing chisquare - like functions or functionals of
more general case one can use the linearization method, and
in case of degeneration FUMIVI may automatically switch to
accurate calculation of second derivatives matrix.
- Nonlinear regular constraints of arbitrary type can be used.
- Both analytical and numerical calculation of derivatives is
possible.

The new code was extensively tested both with model and real
events of the experiment on rare K-meson decays. Unlike the
other methods used new one converges (and fast) for any
nonabsurd data. Nonuniquennes of solutions for some events is
observed what means that such cases have to be analyzed by the
physicists.
References

1. S.N.Sokolov, I.N.Silin, Preprint JINR D-810, Dubna 1961.
2. CERN Program Library, D510, FUMILI.
3. A.J.Ketikian, E.V.Komissarov, V.S.Kurbatov, I.N.Silin,
Nucl. Instr. and Meth. A314(1992)578.
4. A.J.Ketikian, V.S.Kurbatov, I.N.Silin,
Proceedings of CHEP-92, CERN 92-07, 1992,p.833
5. V.S.Kurbatov, I.N.Silin,
Nucl. Instr. and Meth. A345(1994)346.