Linear and Nonlinear Programming

Linear and Nonlinear Programming, 4th Edition, by David G. Luenberger and Yinyu Ye, introduces fundamental concepts and methods in mathematical optimization. Published by Springer, this textbook examines the theoretical properties and algorithmic strategies used to solve linear models.

The text treats linear programming as both a continuous optimization problem over a convex polyhedron and a combinatorial problem of selecting an optimal extreme point. It explains that optimal solutions occur at polyhedral vertices, while noting that the number of candidate vertices can grow exponentially with problem dimension.

The authors also address algorithmic complexity. The volume discusses the simplex method's traversal along edges between adjacent vertices and considers polynomial-time methods such as Khachiyan's ellipsoid approach.