lp - EE236A (Fall 2007-08) Lecture 4 The linear programming...

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 4 The linear programming problem variants of the linear programming problem LP feasibility problem examples and some general applications linear-fractional programming 41 Variants of the linear programming problem general form minimize c T x subject to a T i x b i , i = 1 , . . . , m g T i x = h i , i = 1 , . . . , p in matrix notation: minimize c T x subject to Ax b Gx = h where A = a T 1 a T 2 . . . a T m R m n , G = g T 1 g T 2 . . . g T p R p n The linear programming problem 42 inequality form LP minimize c T x subject to a T i x b i , i = 1 , . . . , m in matrix notation: minimize c T x subject to Ax b standard form LP minimize c T x subject to g T i x = h i , i = 1 , . . . , m x in matrix notation: minimize c T x subject to Gx = h x The linear programming problem 43 Reduction of general LP to inequality/standard form minimize c T x subject to a T i x b i , i = 1 , . . . , m g T i x = h i , i = 1 , . . . , p reduction to inequality form : minimize c T x subject to a T i x b i , i = 1 , . . . , m g T i x h i , i = 1 , . . . , p g T i x h i , i = 1 , . . . , p in matrix notation (where A has rows a T i , G has rows g T i ) minimize c T x subject to A G G x b h h The linear programming problem 44 reduction to standard form: minimize c T x + c T x subject to a T i x + a T i x + s i = b i , i = 1 , . . . , m g T i x + g T i x = h i , i = 1 , . . . , p x + , x , s variables x + , x , s recover x as x = x + x s R m is called a slack variable in matrix notation: minimize tildewide c T tildewide x subject to tildewide G tildewide x = tildewide h tildewide x where tildewide x = x + x s , tildewide c = c c , tildewide G = bracketleftbigg A A I G G bracketrightbigg , tildewide h = bracketleftbigg b h bracketrightbigg The linear programming problem 45 LP feasibility problem feasibility problem: find x that satisfies a T i x b i , i = 1 , . . . , m solution via LP (with variables t , x ) minimize t subject to a T i x b i + t, i = 1 , . . . , m variables t , x if minimizer x , t satisfies t , then x satisfies the inequalities LP in matrix notation: minimize tildewide c T tildewide x subject to tildewide A tildewide x tildewide b tildewide x = bracketleftbigg x t bracketrightbigg , tildewide c = bracketleftbigg 1 bracketrightbigg , tildewide A = bracketleftbig A 1 bracketrightbig...
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lp - EE236A (Fall 2007-08) Lecture 4 The linear programming...

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