A geometric programming framework for univariate cubic L1 smoothing splines

Univariate cubic L-1 smoothing splines are capable of providing shape-preserving C-1-smooth approximation of multi-scale data. The minimization principle for univariate cubic L-1 smoothing splines results in a nondifferentiable convex optimization problem that, for theoretical treatment and algorithm design, can be formulated as a generalized geometric program. In this framework, a geometric dual with a linear objective function over a convex feasible domain is derived, and a linear system for dual to primal conversion is established. Numerical examples are given to illustrate this approach. Sensitivity analysis for data with uncertainty is presented.