A Pivotal Method for Affine Variational Inequalities

@article{Cao1996APM,
title={A Pivotal Method for Affine Variational Inequalities},
author={Menglin Cao and Michael C. Ferris},
journal={Math. Oper. Res.},
year={1996},
volume={21},
pages={44-64}
}
• Published 1 February 1996
• Mathematics, Computer Science
• Math. Oper. Res.
We explain and justify a path-following algorithm for solving the equations ACx = a, where A is a linear transformation from Rn to Rn, C is a polyhedral convex subset of Rn, and AC is the associated normal map. When AC is coherently oriented, we are able to prove that the path following method terminates at the unique solution of ACx = a, which is a generalization of the well known fact that Lemke's method terminates at the unique solution of LCPq, M when M is a P = matrix. Otherwise, we…
• Computer Science
Mathematical Programming
• 2017
A structure-preserving pivotal approach that can efficiently process large-scale sparse instances of the problem with theoretical guarantees and at high accuracy, and can be implemented in a large scale setting using existing sparse linear algebra and linear programming techniques without employing a reduction.
• Computer Science
Math. Program.
• 2018
A structure-preserving pivotal approach that can efficiently process large-scale sparse instances of the problem with theoretical guarantees and at high accuracy, and can be implemented in a large scale setting using existing sparse linear algebra and linear programming techniques without employing a reduction.
• Mathematics
Math. Oper. Res.
• 1999
A path-following algorithm is presented that determines zeros of coherently oriented piecewise-affine functions, and this algorithm is used, together with the T-map, to solve the generalized equation for affine, coherentlyoriented functions F, and polyhedral multifunctions T.
In this paper, we present a continuation method for solving normal equations generated byC2 functions and polyhedral convex sets. We embed the normal map into a homotopyH, and study the existence and
• Mathematics
• 1995
The PATH solver is an implementation of a stabilized Newton method for the solution of the Mixed Complementarity Problem. The stabilization scheme employs a path-generation procedure which is used to
• Computer Science
• 2009
An affine variational inequality solver (PathAVI) is developed that exploits the special underlying polyhedral set structure and is able to process a class of models whose equivalent complementarity reformulations cannot be processed by existing complementarity solvers.
• Economics
Mathematical Programming
• 2012
Affine generalized Nash equilibrium problems (AGNEPs) represent a class of non-cooperative games in which players solve convex quadratic programs with a set of (linear) constraints that couple the
• Economics
Math. Program.
• 2013
This paper treats a large subclass of AGNEPs wherein the coupled constraints are shared by, i.e., common to, the players, and presents and analyzes a modification to Lemke’s method that allows us to compute GNE that are not necessarily VE.
• Mathematics
• 1996
This paper describes a globally convergent path-following method for solving nonlinear equations containing particular kinds of nonsmooth functions called normal maps. These normal maps express
• Mathematics
Ann. Oper. Res.
• 2019
This work shows that the parallel two-terminal case with a fixed number of classes is polynomially solvable, and proposes an extension of Lemke’s algorithm able to solve this problem.

References

SHOWING 1-10 OF 21 REFERENCES

• Mathematics
SIAM J. Optim.
• 1991
A path-following algorithm for finding a solution to the nonlinear stationary point problem on an unbounded, convex, and pointed polyhedron and a condition under which the path of zeros converges to a solution is stated.
The algorithm is a variable dimension fIXed point algorithm having as many rays as the vertices of n and terminates as soon as it hits the boundary of n or it fmds a zero of f.
• Mathematics
Math. Oper. Res.
• 1989
A simplicial variable dimension restart algorithm for the stationary point problem or variational inequality problem on a polytope and the vertex w is obtained from the optimum solution of the linear programming problem maximize.
It is shown here that when the linear transformation giving rise to such a normal map has a certain symmetry property, the necessary and sufficient condition for nonsingularity takes a particularly simple and convenient form, being simply a positive definiteness condition on a certain subspace.
The main result is that the normal map induced by a linear transformation is a Lipschitzian homeomorphism if and only if the determinant of the map in each n-cell of the normal manifold has the same nonzero sign.
The notion of a monotone map is generalized to that of a pseudomonotone map. It is shown that a differentiable, pseudoconvex function is characterized by the pseudomonotonicity of its gradient.
• Mathematics
Math. Oper. Res.
• 1993
This paper proposes a complementary pivoting algorithm for finding a stationary point of an affine function on an unbounded polyhedron by exploiting fully the linearity of the problem.
Some simple constructive proofs are given of solutions to the matric system Mz - \omega - q; z \geqq 0; \omega \geqq 0; and z T \omega - 0, for various kinds of data M, q, which embrace the quadratic