Do computational geometers use Lagrange multipliers? Can anyone point me to an example of a problem that (more or less) originated in computational geometry whose solution requires the use of Lagrange multipliers (or Kuhn-Tucker conditions, or dual variables in a linear program, etc.)?  I have not been able to find such an example in any of the literature that I own.
 A: Two examples, neither a direct hit on what you seek, I think.
But maybe they will trigger connections for others to answer better.

(1) Moody T. Chu and Matthew M. Lin.
  "Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization."
  SIAM J. Sci. Comput., 30(3), 1131–1155.
  (Journal link.)

This paper's focus is approximating a polytope with another
with fewer facets.
At the heart of their work is a convex-hull fitting problem, for which
they use Lagrange multipliers.

          


          

(Image from Chu PDF presentation.)




(2) Kojima, Masakazu, Nimrod Megiddo, and Shinji Mizuno. "A Lagrangian Relaxation Method for Approximating the Analytic Center of a Polytope." IBM Thomas J. Watson Research Division, 1992.

Whether calculating the analytic center of a polytope "originated 
in computational geometry" is debatable.
Finally, Lagrange multipliers find use in robotics, again not really
originating in computational goemetry.
A: A convex optimization method for constructing a set of points in the plane with prescribed (combinatorial) Delaunay triangulation is given in 
Euclidean structures on simplicial surfaces and hyperbolic volume
I Rivin - Annals of Mathematics, 1994
