Warm-starting outer approximation for parameterized convex MINLP

We address the challenge of efficiently solving parameterized sequences of convex Mixed-Integer Nonlinear Programming (MINLP) problems through warm-starting techniques. We focus on an outer approximation (OA) approach, for which we develop the theoretical foundation and present two warm-starting techniques for solving sequences of convex MINLPs. These types of problem sequences arise in several important applications, such as, multiobjective MINLPs using scalarization techniques, sparse linear regression, hybrid model predictive control, or simply in analyzing the impact of certain problem parameters.
The main contribution of this paper is the mathematical analysis of the proposed warm-starting framework for OA-based algorithms, which shows that a simple adaptation of the linear relaxation from one problem to the next can greatly improve computational performance. In the case that the parameters depend linearly on the parameter, we prove under some assumptions that one of the proposed warm-starting techniques results in only one OA iteration to find an optimal solution and verify optimality. Numerical results demonstrate noticeable performance improvements compared to two common initialization approaches, and show that the warm-starting can also in practice result in a single iteration to converge for several problems in the sequences. Our methods are especially effective for problems where consecutive problems in the sequence are similar, and where the integer part of the optimal solutions remains constant for several problems in the sequence. The results show that it is possible, both in theory and practice, to perform warm-starting to significantly enhance the computational efficiency of solving parameterized convex MINLPs.