# On the Global Solution of Linear Programs with Linear Complementarity Constraints

@article{Hu2008OnTG, title={On the Global Solution of Linear Programs with Linear Complementarity Constraints}, author={Jing Hu and John E. Mitchell and J. S. Pang and Kristin P. Bennett and Gautam Kunapuli}, journal={SIAM J. Optim.}, year={2008}, volume={19}, pages={445-471} }

This paper presents a parameter-free integer-programming-based algorithm for the global resolution of a linear program with linear complementarity constraints (LPCCs). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvability—of an LPCC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the…

## Figures and Tables from this paper

## 106 Citations

### On convex quadratic programs with linear complementarity constraints

- Computer ScienceComput. Optim. Appl.
- 2013

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The…

### An algorithm for global solution to bi-parametric linear complementarity constrained linear programs

- Computer ScienceJ. Glob. Optim.
- 2015

This work develops a domain-partitioning algorithm that solves a series of the linear subproblems and/or convex quadratically constrained subprograms obtained by the relaxations of the complementarity constraint.

### Solving linear programs with complementarity constraints using branch-and-cut

- Computer ScienceMath. Program. Comput.
- 2019

A branch-and-cut algorithm to find a global optimum for this class of optimization problems, where the branches branch directly on complementarities, and the computational results show that the approach is a strong alternative to constructing an integer programming formulation using big-M terms to represent bounds for variables.

### A branch and cut approach to linear programs with linear complementarity constraints

- Computer Science
- 2011

A branch and cut algorithm is proposed to globally solve the LPCC problem, where branching is imposed directly on complementarity constraints, and the computational result shows that this approach has the capability to solve this type of problem with finite optimal value.

### A branch-and-bound method for discretely-constrained mathematical programs with equilibrium constraints

- Computer ScienceAnn. Oper. Res.
- 2013

A branch-and-bound algorithm for discretely-constrained mathematical programs with equilibrium constraints (DC-MPEC), which uses Benders decomposition to form a master problem and a subproblem and a new dynamic partition scheme that ensures that the algorithm converges to the global optimum.

### Obtaining Tighter Relaxations of Mathematical Programs with Complementarity Constraints

- Computer Science
- 2012

Computational results for linear programs with complementarity constraints (LPCCs) are included, comparing the benefit of the various constraints on the value of the relaxation, and showing that the constraints can dramatically speed up the solution of the LPCC.

### Branch-and-bound algorithms for the partial inverse mixed integer linear programming problem

- MathematicsJ. Glob. Optim.
- 2013

In the presented algorithms, linear programs with complementarity constraints (LPCCs) need to be solved repeatedly as a subroutine, which is analogous to repeatedly solving linear programs for MILPs, so the computational complexity can be expected to be much worse than that of MILP or LPCC.

### Conic approximation to quadratic optimization with linear complementarity constraints

- Computer Science, MathematicsComput. Optim. Appl.
- 2017

A conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints by adaptively refining the outer approximation of the feasible set to identify an optimal solution or an $$ϵ-optimal solution of the original problem.

### OPTIMIZATION WITH LINEAR COMPLEMENTARITY CONSTRAINTS

- Computer Science
- 2014

In this paper some of the most relevant applications of the MPLCC and formulations of nonconvex optimization problems as MPL CCs are presented and the most important nonlinear programming methods, complementarity algorithms, enumerative techniques and 0 - 1 integer programming approaches for theMPLCC are reviewed.

### Sequential Linearization Method for Bound-Constrained Mathematical Programs with Complementarity Constraints

- Mathematics, Computer ScienceSIAM J. Optim.
- 2022

An algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables that enforces descent on the objective function to promote global convergence to B-stationary points is proposed.

## References

SHOWING 1-10 OF 53 REFERENCES

### AN ACTIVE SET METHOD FOR MATHEMATICAL PROGRAMS WITH LINEAR COMPLEMENTARITY CONSTRAINTS

- Computer Science, Mathematics
- 2007

This work proposes a primal-dual active set projected Newton method for MPLCCs, that maintains the feasibility of all iterates and has strong convergence properties.

### Combinatorial Benders' Cuts for Mixed-Integer Linear Programming

- Computer ScienceOper. Res.
- 2006

Computational results on two specific classes of hard-to-solve MIPs indicate that the new method produces a reformulation which can be solved some orders of magnitude faster than the original MIP model.

### On Using the Elastic Mode in Nonlinear Programming Approaches to Mathematical Programs with Complementarity Constraints

- MathematicsSIAM J. Optim.
- 2005

It is shown that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen, and the assumptions are more general since they do not use a critical assumption from that reference.

### 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.

### A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—I. Theory

- Computer Science, Mathematics
- 1990

### An Implementable Active-Set Algorithm for Computing a B-Stationary Point of a Mathematical Program with Linear Complementarity Constraints

- Computer ScienceSIAM J. Optim.
- 2002

This work shows that B-stationarity can be achieved if the algorithm is modified and an additional error bound condition holds and asserted that, under a uniform LICQ on the ∈-feasible set, this algorithm generates iterates whose cluster points are B- stationary points of the problem.

### Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints

- Mathematics, Computer ScienceSIAM J. Optim.
- 2000

A class of functions, parameterized by a real scalar, is introduced to approximate these nonsmooth problems by smooth nonlinear programs to improve the prospect of feasibility and stability of the constraints of the associated non linear programs and their quadratic approximations.

### State of the art in global optimization: computational methods and applications

- Computer Science
- 1996

A Finite Algorithm for Global Minimization of Separable Concave Programs and Accelerating Convergence of Branch-and-Bound Algorithms for Quadratically Constrained Optimization Problems.

### 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…