We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 18 December 2009
Chapter 10 gives an introduction to linear programming on oriented matroids. It does not presuppose any experience with linear programming – but for the operations research practitioner it might offer an alternative view on linear programming from a matroid theory point of view. Our aim is to give the non-expert a smooth, geometric access to the fundamental ideas of oriented matroid programming, as developed in Bland (1974, 1977a). This necessitates extra care at the points where the terminologies from linear programming and from combinatorial geometry clash.
This chapter is intended to demonstrate that the oriented matroid framework can add to the understanding of the combinatorics and of the geometry of the simplex method for linear programming. In fact, the oriented matroid approach gives a geometric language for pivot algorithms, interpreting linear programs as oriented matroid search problems. We find that locally consistent information (orientation of the edges at a vertex) imply the existence of global extrema in pseudoarrangements. We believe that these techniques and results are applicable also to problems in other areas of mathematics.
The oriented matroid framework deals with (pseudo)linear programs, where
– positive cocircuits correspond to feasible vertices, and
– positive circuits correspond to bounding cones.
Both of these are described by oriented matroid bases, corresponding to temporary coordinate systems. Pivot algorithms are now modeled by basis exchanges, and the duality of linear programming becomes a manifestation of oriented matroid duality.
The benefits of geometric understanding are of course not one-sided: the linear programming frame work offers insight into the structure of oriented matroids, and the pivot algorithms of linear programming provide important search techniques for oriented matroids.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
