Minimizing the Condition Number of a Gram Matrix

Abstract

Abstract. The condition number of a Gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the least squares polynomial approximation. Given a polynomial basis, we consider the problem of finding a set of points and/or weights which minimizes the condition number of the Gram matrix. The objective function f in the minimization problem is nonconvex and nonsmooth. We present an expression of the Clarke generalized gradient of f and show that f is Clarke regular and strongly semismooth. Moreover, we develop a globally convergent smoothing method to solve the minimization problem by using the exponential smoothing function. To illustrate applications of minimizing the condition number, we report numerical results for the Gram matrix defined by the weighted Vandermonde-like matrix for least squares approximation on an interval and for the Gram matrix defined by an orthonormal set of real spherical harmonics for least squares approximation on the sphere.

Smoothing methods for nonsmooth, nonconvex minimization

“…Such nonsmooth, nonconvex problems arise from optimal control, design of experiments and distributions of points on the sphere [10,29,71,76]. …”

“…is a smoothing function of |t| and [29,76] Consider minimizing the condition number of a symmetric posi-…”

“…Examples of the class of smooth approximations φ of (t) + defined by (17) can be found in a number of articles, we refer to [2,5,14,[16][17][18]22,24,[27][28][29][30][31]33,39,40,64,92].…”

“…Recently, problem (1) has attracted significant attention in engineering and economics. An increasing number of practical problems require minimizing a nonsmooth, nonconvex function on a convex set, including image restoration, signal reconstruction, variable selection, optimal control, stochastic equilibrium problems and spherical approximations [1,8,10,11,13,21,29,30,33,42,44,61,76,85,101,105]. In many cases, the objective function f is not differentiable at the minimizers, but the constraints are simple, such as box constraints, X = { x ∈ R n | ≤ x ≤ u} for two given vectors ∈ {R ∪ {−∞}} n and u ∈ {R ∪ {∞}} n .…”

“…Smoothing methods are not only efficient for problems with nonsmooth objective functions, but also for problems with nonsmooth constraints. See [1,2,[4][5][6][7]14,[16][17][18]20,22,[24][25][26][27][28][29][30][31]33,39,40,[46][47][48][49][50]54,[56][57][58][59][60]63,64,66,70,72,73,81,82,[88][89][90][91][92]94,97,103,104].…”