2016_10_06.pdf - Duality Theory And in fact it is a theory...

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Duality Theory And, in fact, it is a theory that extends to all convex optimization problems, not just linear programs. Duality theory can be approached in various ways. It is a theory that is independent of particular algorithms. However, much of duality theory can be built from what you know about the revised simplex method. Thus, in our first take on duality theory, we start from the simplex method. To begin, we focus on LP’s in standard equality form. (In a few days, we will consider all LP’s.) min c T x s.t. Ax = b x 0 ¯ y T ¯ z A - 1 J ¯ b J = { j 1 , j 2 , . . . , j m } optimal tableau Unless indicated otherwise, always assume a tableau is optimal (i.e., a tableau at which the simplex method would stop). min 3 x 1 + 4 x 2 + 3 x 3 s.t. - 4 x 1 + 2 x 2 + x 3 = 14 3 x 1 + x 2 + x 3 = 8 x 0 1 2 30 1 -1 6 -1 2 2 J = { 2 , 3 } Think of the vector b as quantifying tasks to which you have committed – you must accomplish 14 units of the first task, and 8 of the second. You accomplish tasks by choosing levels (values) of three di erent actions x 1 , x 2 , x 3 . The actions, however, are expensive. 1
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min c T x s.t. Ax = b x 0 ¯ y T ¯ z A - 1 J ¯ b J = { j 1 , j 2 , . . . , j m } min 3 x 1 + 4 x 2 + 3 x 3 s.t. - 4 x 1 + 2 x 2 + x 3 = 14 3 x 1 + x 2 + x 3 = 8 x 0 1 2 30 1 -1 6 -1 2 2 J = { 2 , 3 } ¯ z = c T J ¯ b = c T J ( A - 1 J b ) = ( c T J A - 1 J ) b = ¯ y T b Thus, ¯ y T b = ¯ z = c T J ¯ b 1 2 14 8 = 30 = 4 3 6 2 The fact that ¯ z = ¯ y T b provides economic meaning for the vector ¯ y : Is it in your interest to commit?
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