Matrix analysis
Projected gradient methods for linearly constrained problems
Mathematical Programming: Series A and B
Algorithms for generalized fractional programming
Mathematical Programming: Series A and B
On the sum of the largest eigenvalues of a symmetric matrix
SIAM Journal on Matrix Analysis and Applications
Mathematical Programming: Series A and B - Special issue: Festschrift in Honor of Philip Wolfe part II: studies in nonlinear programming
Solution of monotone complementarity problems with locally Lipschitzian functions
Mathematical Programming: Series A and B - Special issue on computational nonsmooth optimization
First and second order analysis of nonlinear semidefinite programs
Mathematical Programming: Series A and B
Tensor product Gauss-Lobatto points are Fekete points for the cube
Mathematics of Computation
SIAM Journal on Numerical Analysis
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Smoothing Technique and its Applications in Semidefinite Optimization
Mathematical Programming: Series A and B
Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs
SIAM Journal on Numerical Analysis
A variational characterisation of spherical designs
Journal of Approximation Theory
SIAM Journal on Optimization
SIAM Journal on Optimization
Error bounds for approximation in Chebyshev points
Numerische Mathematik
Computational existence proofs for spherical t-designs
Numerische Mathematik
SIAM Journal on Imaging Sciences
Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere
SIAM Journal on Numerical Analysis
Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere
SIAM Journal on Numerical Analysis
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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.