Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems

Abstract

In this paper we present novel theoretical and algorithmic developments for the solution of mixed-integer optimization problems involving uncertainty, which can be posed as multiparametric mixed-integer optimization models, where uncertainty is described by a set of parameters bounded between lower and upper bounds. In particular, we address convex nonlinear formulations involving (i) 0−1 integer variables and (ii) uncertain parameters appearing linearly and separately and present on the right-hand side of the constraints. The developments reported in this work are based upon decomposition principles where the problem is decomposed into two iteratively converging subproblems: (i) a primal and (ii) a master subproblem, representing valid parametric upper and lower bounds on the final solution, respectively. The primal subproblem is formulated by fixing the integer variables which results in a multiparametric nonlinear programming (mp-NLP) problem, which is solved by outer-approximating the nonlinear functions at a number of points in the space of uncertain parameters to derive linear profiles. The aim of the master subproblem is then to propose another set of integer solutions which are better than the integer solutions that have already been analyzed in the primal subproblem. This can be achieved by (i) introducing cuts, (ii) employing outer-approximation (OA) principles, or (iii) using generalized benders decomposition (GBD) fundamentals. The algorithm terminates when the master problem cannot identify a better integer solution.