Convex Quadratic Approximation

  • Authors:

  • J. Ben Rosen;Roummel F. Marcia

  • Affiliations:

  • Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA, USA;San Diego Supercomputer Center, University of California, San Diego, La Jolla, CA, USA

  • Venue:
  • Computational Optimization and Applications
  • Year:
  • 2004

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Abstract

For some applications it is desired to approximate a set of m data points in \Bbb Rn with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n + 1)(n + 2)/2 parameters, s ≤ m, so as to minimize the approximation error in the L1 norm. The approximating function is q(p,x), where p ∈ \Bbb Rs is the vector of parameters, and x ∈ \Bbb Rn. The Hessian of q(p,x) with respect to x (for fixed p) is positive semi-definite, and its Hessian with respect to p (for fixed x) is shown to be positive semi-definite and of rank ≤n. An algorithm is described for computing an optimal p* for any specified set of m data points, and computational results (for n = 4,6,10,15) are presented showing that the optimal q(p*,x) can be obtained efficiently. It is shown that the approximation will usually interpolate s of the m data points.