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Unformatted text preview: EE236A (Fall 200708) Lecture 4 The linear programming problem • variants of the linear programming problem • LP feasibility problem • examples and some general applications • linearfractional programming 4–1 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 4–2 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 4–3 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 4–4 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 4–5 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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 Spring '07
 Dr.Vandenber
 Optimization, ax, Ri, linear programming problem

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