Lagrange undetermined multipliers

Prince, E.; Finger, L. W.; Konnert, J. H.

doi:10.1107/97809553602060000611

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 8.3, p. 693

Section 8.3.1.1. Lagrange undetermined multipliers

E. Prince,^a L. W. Finger^b and J. H. Konnert^c

^a NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA,^bGeophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 20015-1305, USA, and ^cLaboratory for the Structure of Matter, Code 6030, Naval Research Laboratory, Washington, DC 20375-5000, USA

8.3.1.1. Lagrange undetermined multipliers

| top | pdf |

The classical technique for application of constraints is the use of Lagrange undetermined multipliers, in which the set of p parameters, [x_j] , is augmented by [p-q] $[(q\lt p)]$ additional unknowns, λ_k, one for each constraint relationship desired. The problem may be stated in the form: find the minimum of $[S=\textstyle\sum\limits_{i=1}^nw_i[y_i-M_i({\bf x})]^2,\eqno (8.3.1.1a)]$ subject to the condition $[f_k({\bf x})=0\quad (k=1,2,\ldots,p-q).\eqno (8.3.1.1b)]$ This may be shown (Gill, Murray & Wright, 1981) to be equivalent to the problem: find a point at which the gradient of $[S^{\prime }=\textstyle\sum\limits_{i=1}^nw_i[y_i-M_i({\bf x})]^2+\textstyle\sum\limits_{k=1}^{p-q}\lambda _k\;f_k({\bf x})\eqno (8.3.1.2)]$ vanishes. Solving for the stationary point leads to a set of simultaneous equations of the form $[\partial S^{\prime }/\partial x_j=\partial S/\partial x_j+\textstyle\sum\limits _{k=1}^{p-q}\lambda _k\partial f_k({\bf x})/\partial x_j=0\eqno (8.3.1.3a)]$ and $[\partial S^{\prime }/\partial \lambda _k=f_k({\bf x})=0.\eqno (8.3.1.3b)]$ Thus, the number of equations, and the number of unknowns, is increased from p to 2p − q. In cases where the number of constraint relations is small, and where it may be difficult to solve the relations for some of the parameters in terms of the rest, this method yields the desired results without too much additional computation (Ralph & Finger, 1982). With the large numbers of parameters, and large numbers of constraints, that arise in many crystallographic problems, however, the use of Lagrange multipliers is computationally inefficient and cumbersome.

References

Gill, P. E., Murray, W. & Wright, M. M. (1981). Practical optimization. New York: Academic Press.Google Scholar

Ralph, R. L. & Finger, L. W. (1982). A computer program for refinement of crystal orientation matrix and lattice constants from diffractometer data with lattice symmetry constraints. J. Appl. Cryst. 15, 537–539.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.3, p. 693