Classification model selection via bilevel programming
Optimization Methods & Software - Mathematical programming in data mining and machine learning
Optimization and dynamical systems algorithms for finding equilibria of stochastic games
Optimization Methods & Software
Computational Optimization and Applications
On the asymmetric eigenvalue complementarity problem
Optimization Methods & Software - GLOBAL OPTIMIZATION
Combining the regularization strategy and the SQP to solve MPCC - A MATLAB implementation
Journal of Computational and Applied Mathematics
The C-Index: A New Stability Concept for Quadratic Programs with Complementarity Constraints
Mathematics of Operations Research
An $\ell_1$ Elastic Interior-Point Method for Mathematical Programs with Complementarity Constraints
SIAM Journal on Optimization
A direct method for trajectory optimization of rigid bodies through contact
International Journal of Robotics Research
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We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using algorithms and procedures of smooth nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sufficient conditions for their Lagrange multiplier set to be nonempty. MPCCs that have nonempty Lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter. In this context, the elastic mode transforms MPCC into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem, which in turn makes it approachable by sequential quadratic programming (SQP) algorithms. One such algorithm is shown to achieve local linear convergence once the problem is relaxed. Under stronger conditions, we also prove superlinear convergence to the solution of an MPCC using an adaptive elastic mode approach for an SQP algorithm recently analyzed in an MPCC context in [R. Fletcher, S. Leyffer, S. Sholtes, and D. Ralph, Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints, Tech. report NA 210, University of Dundee, Dundee, UK, 2002]. Our assumptions are more general since we do not use a critical assumption from that reference. In addition, we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen.