Gradient methods and conic least-squares problems

This paper presents two reformulations of the dual of the constrained least squares problem over convex cones. In addition, it extends Nesterov’s excessive gap method 1 [21] to more general problems. The conic least squares problem is then solved by applying the resulting modified method, or Nesterov’s smooth method [22], or Nesterov’s excessive gap method 2 [21], to the dual reformulations. Numerical experiments show that this approach obtains relatively accurate solutions for large-scale problems using less CPU time than interior-point method based state-of-art software do and is more accurate than these software on problem instances possibly with dual degeneracy [2]. The approach extends easily to related conic convex quadratic programs.