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Unformatted text preview: Two Lectures on the Ellipsoid Method for Solving Linear Programs ∗ 1 Lecture 1: A polynomialtime algorithm for LP Consider the general linear programming problem: δ ∗ = max cx S.T. Ax ≤ b, x ∈ R n , (1) where A = ( a ij ) i,j ∈ Z m × n is an m × n integer matrix, and b = ( b i ) i ∈ Z m and c = ( c j ) j ∈ Z n are integer vectors. We denote by a T i the i th row of A . In the unitcost model of computation, the cost of multiplying two integers x and y is cost( x · y ) = 1. However, in the bitmodel , it is cost( x · y ) = ℓ ( x ) + ℓ ( y ), where ℓ ( x ) = log(1 +  x  ) + 1 = number of bits used to represent x in binary . For instance, consider the following code for computing x = a 2 k : 1. x ← a ; 2. for i = 1 ,...,k , set x ← x 2 . Then in the unitcost model, the cost of this code is k , while in the bitmodel, it is log x ≈ 2 k ℓ ( a ). Let ℓ = max { ℓ ( a ij ) ,ℓ ( b i ) ,ℓ ( c j ) } , and let L = summationdisplay i,j ℓ ( a ij ) + summationdisplay i ℓ ( b i ) + summationdisplay j ℓ ( c j ) be the total number of bits needed to represent the input. In the bitmodel of computation, an algorithm is said to run in polynomialtime, if the number of operations ( { + , − , ∗ ,/ } ) is at most poly( n,m,ℓ ) and the bit length of all numbers involved in the computation is at most poly( n,m,ℓ ). As we shall see, linear programming is polynomialtime solvable in the bitmodel. * These notes were taken by Khaled Elbassioni (MaxPlanckInstitut f¨ur Informatik, Saarbr¨ucken, Germany; ([email protected])) while attending a graduate course on Lin ear Programming taught by Leonid Khachiyan in Fall 1996 at Rutgers University. 1 The Ellipsoid method Let ε > 0 be a given constant. We call ˜ x an εapproximate solution of (1) if c ˜ x ≥ δ ∗ − ε and a T i ˜ x ≤ b i + ε , for i = 1 ,...,m . Assumption 1 We know R ∈ R + , such that (1) has an optimal solution x ∗ in the Euclidean ball B R = { x / bardbl x bardbl ≤ R } . Let h = max { a ij  ,  b i  ,  c j } , i.e., h = 2 ℓ − 1. Then under Assumption 1, the Ellipsoid method computes an εapproximate solution of (1) in O (( n + m ) n 3 log ( Rhn ε ) ) arithmetic operations over O (log ( Rhn ε ) )bit numbers. Fact 1 Let V be the unit ball V = { x / bardbl x bardbl ≤ 1 } , and let V − = V ∩{ x / x n ≥ } . Then there’s an ellipsoid E ′ such that • V − ⊆ E ′ • vol E ′ vol V ≤ e − 1 2( n +1) ≈ 1 − 1 2 n . Proof . Let the center of E ′ be (0 , ,..., , 1 n +1 ) = 1 n +1 e n , and α = 1 − 1 n +1 ≈ 1 − 1 n (see Figure 1). The equation of E ′ is parenleftBig x n − 1 n +1 parenrightBig 2 α 2 + x 2 1 β 2 + x 2 2 β 2 + ··· + x 2 n − 1 β 2 ≤ 1 ....
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 Spring '08
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 Operations Research, Linear Programming, Algorithms, Optimization, Ellipsoid method, vol Br

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