An extreme point result for convexity, concavity and monotonicity of parameterized linear equation solutions Original Research Article

In many applications, it is useful to know how the solution to a set of simultaneous linear equations depends on parameters θ entering into the coefficients. To this end, this paper addresses the classical equation Ax=b with n×n matrix A=A(θ) and n×1 vector b=b(θ) depending on an m-tuple of parameters θ with components θi entering in a rank-one manner. Given such a system, the following problems are considered: For solution component xi(θ) and parameter θj, determine if the first and second order partial derivatives of xi with respect to θj are of one sign for all θ in a prescribed hypercube Θr of radius rgreater-or-equal, slanted0; i.e., we determine which components enter the solution either monotonically, convexly or concavely. In this paper, we provide extreme point results for these problems. Namely, we need only check the sign of three specially constructed multilinear functions at the extreme points (vertices) of Θr in order to ascertain whether the desired one-sign condition is satisfied over the entire hypercube. Central to the proof of extremality is a special “multilinear factorization” of the partial derivatives of xi(θ). This leads to a simple method to compute the so-called radii of convexity, concavity and monotonicity.