A smoothing stochastic simulated annealing method for localized shapes approximation

In B-splines approximation setting, it is known that monotony and convexity (or concavity) shapes can easily be controlled by the spline coefficients. In this paper we deal with the general context of combinations of localized shape constraints. We prove that unimodality constraint is fulfilled simply by an increasing and decreasing sequence of the spline coefficients by using the Descartes' sign rule. Then, the local support property of B-splines is used to locate each constraint on a given interval. We formulate a smoothing spline approximation under inequality constraints in function of the spline coefficients. We also give a simulated annealing algorithm to solve the optimization problem and we establish the almost sure convergence of the efficient solution.