  Shape constraints, such as non-negativity, monotonicity, convexity or
supermodularity, play a key role in various applications of machine learning
and statistics. However, incorporating this side information into predictive
models in a hard way (for example at all points of an interval) for rich
function classes is a notoriously challenging problem. We propose a unified and
modular convex optimization framework, relying on second-order cone (SOC)
tightening, to encode hard affine SDP constraints on function derivatives, for
models belonging to vector-valued reproducing kernel Hilbert spaces (vRKHSs).
The modular nature of the proposed approach allows to simultaneously handle
multiple shape constraints, and to tighten an infinite number of constraints
into finitely many. We prove the convergence of the proposed scheme and that of
its adaptive variant, leveraging geometric properties of vRKHSs. Due to the
covering-based construction of the tightening, the method is particularly
well-suited to tasks with small to moderate input dimensions. The efficiency of
the approach is illustrated in the context of shape optimization,
safety-critical control, robotics and econometrics.
