433 research outputs found
Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra
The set of Nash equilibria of a finite game is the set of nonnegative
solutions to a system of polynomial equations. In this survey article we
describe how to construct certain special games and explain how to find all the
complex roots of the corresponding polynomial systems, including all the Nash
equilibria. We then explain how to find all the complex roots of the polynomial
systems for arbitrary generic games, by polyhedral homotopy continuation
starting from the solutions to the specially constructed games. We describe the
use of Groebner bases to solve these polynomial systems and to learn geometric
information about how the solution set varies with the payoff functions.
Finally, we review the use of the Gambit software package to find all Nash
equilibria of a finite game.Comment: Invited contribution to Journal of Economic Theory; includes color
figure
A Special Homotopy Continuation Method For A Class of Polynomial Systems
A special homotopy continuation method, as a combination of the polyhedral
homotopy and the linear product homotopy, is proposed for computing all the
isolated solutions to a special class of polynomial systems. The root number
bound of this method is between the total degree bound and the mixed volume
bound and can be easily computed. The new algorithm has been implemented as a
program called LPH using C++. Our experiments show its efficiency compared to
the polyhedral or other homotopies on such systems. As an application, the
algorithm can be used to find witness points on each connected component of a
real variety
Numerical homotopies to compute generic points on positive dimensional algebraic sets
Many applications modeled by polynomial systems have positive dimensional
solution components (e.g., the path synthesis problems for four-bar mechanisms)
that are challenging to compute numerically by homotopy continuation methods. A
procedure of A. Sommese and C. Wampler consists in slicing the components with
linear subspaces in general position to obtain generic points of the components
as the isolated solutions of an auxiliary system. Since this requires the
solution of a number of larger overdetermined systems, the procedure is
computationally expensive and also wasteful because many solution paths
diverge. In this article an embedding of the original polynomial system is
presented, which leads to a sequence of homotopies, with solution paths leading
to generic points of all components as the isolated solutions of an auxiliary
system. The new procedure significantly reduces the number of paths to
solutions that need to be followed. This approach has been implemented and
applied to various polynomial systems, such as the cyclic n-roots problem
A Collection of Challenging Optimization Problems in Science, Engineering and Economics
Function optimization and finding simultaneous solutions of a system of
nonlinear equations (SNE) are two closely related and important optimization
problems. However, unlike in the case of function optimization in which one is
required to find the global minimum and sometimes local minima, a database of
challenging SNEs where one is required to find stationary points (extrama and
saddle points) is not readily available. In this article, we initiate building
such a database of important SNE (which also includes related function
optimization problems), arising from Science, Engineering and Economics. After
providing a short review of the most commonly used mathematical and
computational approaches to find solutions of such systems, we provide a
preliminary list of challenging problems by writing the Mathematical
formulation down, briefly explaning the origin and importance of the problem
and giving a short account on the currently known results, for each of the
problems. We anticipate that this database will not only help benchmarking
novel numerical methods for solving SNEs and function optimization problems but
also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on
Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE
Congress on Evolutionary Computation (at Sendai International Center, Sendai,
Japan, from 25th to 28th May, 2015.
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