(529b) Hybrid Strategy for Mathematical Programming with Complementarity Constraints

Mathematical Programming with Complementarity Constraints(MPCC) is a class of optimization problems where two constraints are said to be complement if one of them vanishes depending on the value of the other constraint. They can be used to represent logic type constraints like IF-ELSE statements without using additional integer variables. This makes MPCCs useful in modeling phenomena like flow reversal, valve dynamics, PI controller saturation and phase change for process engineering problems (Baumrucker et al,2008).

It is well known that MPCCs violate standard conditions(LICQ and MFCQ) at each feasible point which makes it hard to solve them using standard solvers. Reformulations in the form of regularizations or augmented penalty term have been used to solve MPCCs as a sequence of nonlinear programming (NLP) problems. Ralph and Wright (2004) show rate of convergence properties for these reformulations within a neighborhood of the local optimal solution. Unfortunately, this does not guarantee converging to spurious stationary points with feasible descent direction. Leyffer and Munson (2007) proposed a SLPEC-EQP filter based approach that could guarantee convergence to local optimum but the approach is prone to slow convergence.

In this study, we will present a mixed hybrid strategy which uses both the regularizated NLP and LPEC formulation to accelerate the convergence to the local optimal solution of MPCCs. We will use active set information to reduce the size of the complementarities in the LPEC. We will also use parametric sensitivity to reduce the computational effort on solving the NLP to the optimal solution. In the end, we will apply the proposed approach to solve different types of MPCC examples.

References:

1. Baumrucker, B. & Renfro, J. & Biegler, Lorenz. (2008). MPEC problem formulations and solution strategies with chemical engineering applications. Computers & Chemical Engineering. 32. 2903-2913. 10.1016.
2. Daniel Ralph & Stephen J. Wright (2004) Some properties of regularization and penalization schemes for MPECs, Optimization Methods and Software, 19:5, 527-556
3. S. Leyffer, T. Munson, A globally convergent filter method for MPECs. Preprint ANL/MCS-P1457-0907, Argonne National Laboratory, Mathematics and Computer Science Division, 2007