Next to the most known methods: Ellipsoid Method, which has theoretical the best property but has big numerical problems. Barrier Method, which also runs in polytime, but lacks warmstart for MIPS. Primal/Dual Simplex Method, which is theoretical worse, but in practice good. Are there ideas for different methods for solving LPs/literature for other methods? What are their weak points? asked 19 Feb '16, 19:08 opti100 
You can solve LP's using a variety of general purpose convex optimization methods (projected (sub)gradient descent, bundle methods, etc.) However, these kinds of approaches aren't going to be faster in theory or in practice than an interior point method or simplex. You can also take a look at FourierMotzkin elimination, but it's an exponential time algorithm and hopelessly slow in practice. answered 21 Feb '16, 12:14 Brian Borchers I believe for some largescale LPs these classic methods may not always be the best, so there is a lot of research on using firstorder methods to solve these problems (one example is http://www.optimizationonline.org/DB_FILE/2011/11/3233.pdf)
(22 Feb '16, 10:34)
Andreas

If your goal in looking at alternative methods is developing intuition and deepening your understanding, I'd recommend Kipp Martin's Large Scale Linear and Integer Optimization: A Unified Approach, which discusses FourierMotzkin and others in terms of projections and immersions, as well as Simplex and Interior Point. answered 23 Feb '16, 19:53 Leo 
There is some research in other ways to solve LPs were "other" does not mean fundamentally different but building on what is already known to find a better way to do things. Two areas come to mind:
answered 23 Feb '16, 09:43 Philipp Chri... A fairly comprehensive survey of algorithm engineering for the simplex method is this book by Maros: http://www.amazon.com/ComputationalTechniquesInternationalOperationsManagement/dp/1402073321/ref=sr_1_1?ie=UTF8&qid=1456522791&sr=81&keywords=maros+simplex
(26 Feb '16, 16:41)
Matthew Salt... ♦
