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Chapter 3 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 3...

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MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 3: The Simplex Method 1 / 81
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Consider the standard form of a LP min c 0 x s.t. Ax = b x 0 . Assume A is an m × n matrix and rank ( A ) = m with m n . Let P = { x R n | Ax = b , x 0 } . If P 6 = φ , then P has an extreme point since it does not contain a line. Therefore, either the optimal value is unbounded or there exists an optimal solution which can be found among the finite set of extreme points. 2 / 81
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We now develop a procedure to identify the optimal solution. We derive conditions to check for optimality. Based on a procedure for obtaining a new basic feasible solution from an old basic feasible solution and the optimality conditions, the simplex method will be developed. 3 / 81
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Optimality Conditions In this section, we derive optimality conditions to check whether a basic feasible solution is optimal. This is useful in the development of simplex method. The optimality conditions also provide a clue for identifying a direction to improve the objective value in a neighbourhood of a basic feasible solution. 4 / 81
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Let x be a basic feasible solution with the set { B (1) , · · · , B ( m ) } of basic indices, so that B = ( A B (1) A B (2) · · · A B ( m ) ) and x B = x B (1) · · · x B ( m ) = B - 1 b 0 . 5 / 81
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Definition For a polyhedron P , and a point x P , a vector d is a feasible direction if x + θ d P for some θ > 0. Consider a BFS x = ( x B , x N ) 0 and a feasible direction d = ( d B , d N ) such that the direction brings non-basic variable j into the basis, leaving all other non-basic variables at zero. 6 / 81
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Then the j-th basic direction is defined as: d j = 1 and d i = 0 for non-basic variables i 6 = j d B = - B - 1 A j To guarantee the direction is feasible, we need B - 1 b - θ B - 1 A j 0 . 7 / 81
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How far we can move in this direction d , depends on the BFS x ? 1 If B - 1 A j 0 , then the polyhedron is unbounded in the x j -direction. Eg: P = { ( x 1 , x 2 ) R 2 | x 1 = 1 , x 1 , x 2 0 } . 8 / 81
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2 If x is nondegenerate, i.e. x B > 0 . If ( B - 1 A j ) k > 0 for some component k , then θ ( B - 1 b ) k ( B - 1 A j ) k . Ratio Test Let θ * = min ( ( B - 1 b ) k ( B - 1 A j ) k ( B - 1 A j ) k > 0 ) = ( B - 1 b ) l ( B - 1 A j ) l . The feasible solution ˆ x = x + θ * d is a basic feasible solution which is adjacent to x , with associated basic variables { x B (1) , · · · , x B ( l - 1) , x B ( l +1) , · · · , x B ( m ) , x j } . Eg: P = { ( x 1 , x 2 ) R 2 | x 1 + x 2 = 1 , x 1 , x 2 0 } . 9 / 81
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3 If x is degenerate, i.e. there is some zero basic variable x B ( k ) , then the nonzero vector d may not be a feasible direction. This happens if a basic variable x B ( k ) = 0 and d B ( k ) < 0, so that x + θ d 6∈ P for all positive scalars θ . Hence θ * = 0. Eg: P = { ( x 1 , x 2 , x 2 , x 3 ) R 4 | x 1 + x 2 + x 3 = 1 , x 1 + x 4 = 1 , x 1 , x 2 , x 3 , x 4 0 } . 10 / 81
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Moving from x to ˆ x = x + θ d , the change in objective function value is c 0 ˆ x - c 0 x = θ c 0 d . Suppose d , with d B = - B - 1 A j , is the feasible direction obtained as before. Then the change in the objective value for a unit increase in x j is c 0 d = c j - c 0 B B - 1 A j , where c B = ( c B (1) , c B (2) , · · · , c B ( m ) ) 0 .
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