(74b) Quadratic Approximation-Based Algorithm for the Solution of Convex Multi-Parametric Non-Linear Programming Problems

Multi-parametric programming and expicit/multi-parametric model predictive control (mp-MPC) have received a lot of attention in the open literature over the past ten years. Despite major advances achieved so far (Pistikopoulos et al., 2004, 2007a, 2008), specially for linear and mixed-integer linear (hybrid) linear systems, the general solution of nonlinear multi-parametric programming problems, a pre-requisite for explicit nonlinear MPC (mp-NMPC), still remains a challenging and difficult task. Previous attempts in the area include the outer-approximation based approach (Dua and Pistikopoulos, 1999; Acevedo and Salgueiro, 2003; Pistikopoulos et al., 2007b), the approximate based approach (Bemporad and Filippi, 2006) and the geometric based approach (Narciso, 2009) with all approaches offering advantages and disadvantages.

In this work, we describe a novel quadratic-based approximation algorithm embedded in a branch and bound framework for convex multi-parametric nonlinear programming problems. The algorithm employs second order information to quadratically approximate the non-linear objective function and first order information to construct outer approximations of the non-linear constraints. A new systematic procedure to characterize the entire parametric space is also presented. The algorithm has been implemented in MATLAB and makes use of the POP (Parametric Optimizer) solver (Bozinis et al., 1999) developed in our research group at Imperial College.

Example problems will be introduced to demonstrate the advantages of the proposed approach - comparisons with current available algorithms will also be presented.

References

Acevedo, J. and M. Salgueiro (2003). An efficient algorithm for convex multiparametric nonlinear programming problems. Ind. Eng. Chem. Res. 42(23), 5883-5890.

Bemporad, Alberto and Carlo Filippi (2006). An algorithm for approximate multiparametric convex programming. Computational Optimization and Applications 35(1), 87-108.

Bozinis, N. A., V. Dua and E. N. Pistikopoulos (1999). A MATLAB implementation of the multiparametric quadratic algorithm. Centre for Process Systems Engineering. Imperial College. London, UK.

Dua, V. and E. N. Pistikopoulos (1999). Algorithms for the solution of multiparametric mixed-integer nonlinear optimization problems. Ind. Eng. Chem. Res. 38(10), 3976-3987.

Narciso, Diogo (2009). Developments in Nonlinear Multiparametric Programming and Control. PhD thesis. London, U. K.

Pistikopoulos, E. N., M. C. Georgiadis and V. Dua (2007a). Multi-Parametric Model-Based Control. Vol. 2. WILEY-VCH. Weinheim.

Pistikopoulos, E. N., M. C. Georgiadis and V. Dua (2007b). Multi-Parametric Programming. Vol. 1. WILEY-VCH. Weinheim.

Pistikopoulos, E. N., N. Bozinis, V. Dua, J. Perkins and V. Sakizlis (2004). Improved process control. European Patent EP1399784.

Pistikopoulos, E.N., N. Bozinis, V. Dua, J. Perkins and V. Sakizlis (2008). Process control using co-ordinate space. United States Patent and Trademark Office Granted Patent No US7433743.