The complexity of interpolating given data in three space with a convex function of two variables

Abstract Three questions concerning the interpolation of a set ( x i , y i , z i ), i = 1,…, N , in R 3 by a convex function of two variables ( z = f ( x , y )) are examined. A given set of N points can be interpolated by a convex function if and only if it can be interpolated by a convex piecewise planar function. Every possible piecewise planar interpolation of the data is determined by a triangulation of the ( x , y ) points in the plane. An algorithm is presented which builds up an acceptable triangulation by sequentially adding points. The algorithm either terminates as soon as it discovers that the interpolation is impossible or terminates with the desired triangulation. Numerical experiments are presented which indicate the effectiveness of the algorithm. Suppose that the points in the plane, ( x i , y i ), i = 1,…, N , are fixed and it is desired to minimize a function f ( z 1 , z 2 ,…, z N ) subject to the constraint that the triples ( x i , y i , z i ) can be interpolated by a convex function. Convexity can be represented by a set of linear inequality constraints among the z 's but many of these constraints may be redundant. For efficiency it is important to reduce the list to the independant constraints and some minimization algorithms actually require independent constraints. An efficient algorithm for generating the set of independent linear inequalities is given. Finally it is shown that the number of independent constraints depends on the location of the ( x , y ) points and varies from zero to O ( N 3 ). It is conjectured that the expected number of independent constraints for N points chosen randomly from a uniform distribution on a square is O ( N 2 log N ). Both theoretical and numerical justification for the conjecture are given. Finally it is shown that there are O ( N 2 ) independent constraints when the points are arranged in a square (or triangular) lattice.