On finding global optima for the hinge fitting problem

Abstract

This paper considers the data fitting of n given points in by a hinge function, as it appears in Breiman (IEEE Trans. Inform. Theory 39(3) (1993) 999) and Pucar and Sjöberg (IEEE Trans. Inform. Theory 44(3) (1998) 1310). This problem can be seen as a mathematical programming problem with a convex objective function and equilibrium constraints. For the euclidean error, an enumerative approach is proposed, which is a polynomial method in the sample size n, for a fixed dimension m. An alternative formulation for the l1 error is also introduced, which is processed by a Sequential Linear Complementarity Problem approach. Some numerical results with both algorithms are included to highlight the efficiency of those procedures.