Linear systems

Part of the Chapman and Hall Mathematics Series book series (CHMS)


Consider the linear programming problem
$${\rm{Minimize }}\left\{ {\mathop c\nolimits^T \upsilon :A\upsilon \geqslant b,\upsilon \geqslant 0} \right\}.$$
where ν ∈ ℝ n , c ∈ ℝ n , b ∈ ℝ m , A ∈ ℝ m×n , and ⩾ is taken componentwise. The constraint set Q is a closed polyhedron (the intersection of a finite number of closed halfspaces); since linear implies convex, any local minimum is a global minimum; if Q is nonempty and bounded, thus compact, then a minimum of (LP) is attained, at one of the finite number of extreme points of Q (see 2.1). Note that (LP) may be re- written with constraints \(C\upsilon \ge k\left( {{\rm{with }}C = \left[ {\matrix{ A \cr I \cr } } \right],k = \left[ {\matrix{ b \cr 0 \cr } } \right]} \right)\).


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  1. Beal, E.M.L. (1970), Advanced features for general mathematical programming, Chapter 4 of Integer and Nonlinear Programming, J. Abadie (Ed.), North-Holland.Google Scholar
  2. Hadley, G. (1962), Linear Programming, Addison-Wesley, Reading. (For a general and detailed account of linear programming and the simplex algorithm.)Google Scholar

Copyright information

© B. D. Craven 1978

Authors and Affiliations

  1. 1.University of MelbourneAustralia

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