On leveraging constrained smooth additive regression models for global optimization
Many real-world decision-making processes rely on solving mixed-integer nonlinear programs (MINLPs). However, finding high-quality solutions to MINLPs is often computationally demanding, motivating the development of specialized algorithms to improve their tractability. In this work, we propose Mixed-Integer Smoothing Surrogate Optimization with Constraints (MISSOC), a novel optimization algorithm that builds and solves approximations of challenging MINLPs. MISSOC approximates complicating functions in an MINLP using smooth additive regression models with \unboldmath{$B-$}splines. Expert knowledge can be incorporated into the approximating functions through shape constraints related to bounds, monotonicity and curvature over the observed domain. A surrogate of the original problem is then obtained by replacing the original complicating functions with their approximations, making it more tractable in practice. MISSOC presents an innovative integration of statistical modeling into mathematical optimization and fills a gap in the literature by building surrogates that are both data-driven and knowledge-driven. The proposed algorithm is illustrated on the real-world Water Distribution Network problem and evaluated through a set of experiments that include benchmark instances and the real-world Hydro Unit Commitment problem. Together, they demonstrate that MISSOC handles MINLPs with integer variables and complicating functions appearing in the objective or in the constraints. MISSOC is evaluated with different state-of-the-art solvers and with the Sequential Convex MINLP (SC-MINLP) algorithm. The latter exploits the separable structure of the approximating functions, which are sums of piecewise univariate polynomials. The experiments show that MISSOC can obtain high-quality solutions for challenging MINLPs, particularly when used in combination with the SC-MINLP algorithm.