Shape-constrained spline estimation of multivariate functions using semidefinite programming

Function estimation problems can often be formulated as optimization problems where the approximating function must satisfy certain shape constraints, such as nonnegativity, monotonicity, unimodality, or convexity. Such constraints reduce to the nonnegativity of linear functionals of the approximating function. A frequently used approach to function estimation problems is approximation by splines, where shape constraints take the form of conic inequalities with respect to cones of nonnegative polynomials. In the multivariate setting these constraints are intractable; hence we consider tractable restrictions involving weighted-sum-of-squares cones. We present both theoretical justifications of the proposed approach and computational results.