# A two-sided relaxation scheme for Mathematical Programs with Equilibrium Constraints

@article{DeMiguel2005ATR, title={A two-sided relaxation scheme for Mathematical Programs with Equilibrium Constraints}, author={Victor DeMiguel and Michael P. Friedlander and Francisco J. Nogales and Stefan Scholtes}, journal={SIAM J. Optim.}, year={2005}, volume={16}, pages={587-609} }

We propose a relaxation scheme for mathematical programs with equilibrium constraints (MPECs). In contrast to previous approaches, our relaxation is two-sided: both the complementarity and the nonnegativity constraints are relaxed. The proposed relaxation update rule guarantees (under certain conditions) that the sequence of relaxed subproblems will maintain a strictly feasible interior---even in the limit. We show how the relaxation scheme can be used in combination with a standard interior…

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## References

SHOWING 1-10 OF 25 REFERENCES

### Interior Methods for Mathematical Programs with Complementarity Constraints

- MathematicsSIAM J. Optim.
- 2006

The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions, and the results are then extended to an interior-relaxation approach.

### Exact Penalization of Mathematical Programs with Equilibrium Constraints

- Mathematics
- 1999

We study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints (MPECs). In the first part, we prove that a…

### Interior-Point Algorithms, Penalty Methods and Equilibrium Problems

- Computer ScienceComput. Optim. Appl.
- 2006

This paper considers the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)—as nonlinear programs, using an interior-point approach, and examines the use of penalty methods to get around these difficulties.

### Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints

- MathematicsMath. Program.
- 2004

A primal-dual interior-point method is proposed, which solves a sequence of relaxed barrier problems derived from MPEC, and global convergence results are deduced under fairly general conditions other than strict complementarity or the linear independence constraint qualification for MPEC.

### INTERIOR-POINT METHODS FOR NONCONVEX NONLINEAR PROGRAMMING: COMPLEMENTARITY CONSTRAINTS

- Computer Science
- 2002

In this paper, we present the formulation and solu- tion of optimization problems with complementarity constraints using an interior-point method for nonconvex nonlinear program- ming. We identify…

### Some properties of regularization and penalization schemes for MPECs

- MathematicsOptim. Methods Softw.
- 2004

Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.

### Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints

- MathematicsSIAM J. Optim.
- 2001

It is shown that every local minimizer of the MPEC which satisfies the linear independence, upper level strict complementarity, and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric NLP.

### Solving mathematical programs with complementarity constraints as nonlinear programs

- Computer ScienceOptim. Methods Softw.
- 2004

Experience indicates that sequential quadratic programming (SQP) methods are very well suited for solving MPCCs and at present outperform interior-point solvers both in terms of speed and reliability.

### Mathematical Programs with Equilibrium Constraints

- Computer Science
- 1996

Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.

### Global Convergence of an Elastic Mode Approach for a Class of Mathematical Programs with Complementarity Constraints

- MathematicsSIAM J. Optim.
- 2005

We prove that any accumulation point of an elastic mode approach, that approximately solves the relaxed subproblems, is a C-stationary point of the problem of optimizing a parametric mixed P…