A novel approach for solving multi-parametric problems with nonlinear constraints

Parametric optimization problems appear in many areas of applications even though most of the existing solution methods for such problems are limited to problems with polyhedral constraints. In this article, a global solution strategy is proposed for a general convex multi-parametric problems with nonlinear constraints and bounded regions. The basic idea of the proposed approach is to obtain an approximate parametric solution based on the sensitivity analysis theory in the interior of the nonlinear feasible region, and on finding analytic parametric solutions on the boundaries of the nonlinear constraints. The method employs a barrier function reformulation technique to construct a barrier multi-parametric problem with polyhedral constraints. The proposed method also provides exact solutions to convex multi-parametric problems whose objective function and constraints are polynomials of up to third-degree in the optimization variables and quadratic in the parameters vector.