L32_Quadratic Programming - Modified Simplex algorithm

L32_Quadratic Programming - Modified Simplex algorithm -...

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QUADRATIC  PROGRAMMING
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Quadratic Programming Quadratic Programming A quadratic programming problem is a non-linear programming problem of the form DX X X c z T + = Maximize Subject to 0 , X b X A
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Here [ ] n m n c c c c b b b b x x x X . . . , . . , . . 2 1 2 1 2 1 = = = = mn m m n n a a a a a a a a a A . . . . . . . . 2 1 2 22 21 1 12 11 = nn n n n n d d d d d d d d d D . . . . . . . . 2 1 2 22 21 1 12 11
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We also assume that the n × n matrix D is symmetric and negative-definite. This means that the objective function is strictly concave . Since the constraints are linear, the feasible region is a convex set.
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In scalar notation, the quadratic programming problem reads: Maximize 2 1 1 1 2 n n j j jj j ij i j j j i j n z c x d x d x x = = ≤ < = + + ∑ ∑ Subject to 0 , . . ,. , . . . . . . . . . . . 2 1 2 2 1 1 2 2 2 22 1 21 1 1 2 12 1 11 + + + + + + + + + n m n mn m m n n n n x x x b x a x a x a b x a x a x a b x a x a x a
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The solution to this problem is based on the KKT conditions. Since the objective function is strictly concave and the solution space is convex, the KKT conditions are also sufficient for optimum. Since there are m+n constraints, we have m+n Lagrange multipliers; the first m of them are denoted by λ 1 , λ 2 , …, λ m ; and the last n of them are denoted by μ 1 , μ 2 , …, μ n .
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The KKT (necessary) conditions are: 0 , . . . , , , , . . . , , . 1 2 1 2 1 n m μ μ μ λ λ λ n j a x d c j n i ij i n i i ij j , . . . , 2 , 1 0 2 . 2 1 1 = = + - + = = μ λ (continued …)
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1 3. 0, 1,2,. . ., 0, 1,2,. . ., n i ij j i j j j a x b i m x j n λ μ = - = = = = 1 4. , 1,2,. . ., 0, 1,2,. . ., n ij j i j j a x b i m x j n = = =
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Denoting the (non-ve) slack variable for the ith constraint i n j j ij b x a = 1 by S i , the 3 rd condition(s) can be written in an equivalent form as: n j x m i S j j i i , . . . , 2 , 1 , 0 , . . . , 2 , 1 , 0 . 3 = = = = μ λ Referred to as " Restricted Basis " conditions.
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Also condition(s) (2) can be rewritten as: 1 1 2. 2 1,2,. . ., n n ij i i ij j j i i d x a c j n λ μ = = - + - = = And condition(s) (4) can be rewritten as: 1 4. , 1,2,. . ., 0, 1,2,. . ., n ij j i i j j a x S b i m x j n = + = = =
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Thus we find the optimal solution * * 1 * 1 , . . . , , n x x x is a solution of the system of m+n linear
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