section4 - 4(P Duality Theory maximize subject to cT x Ax b...

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4 Duality Theory Recall from Section 1 that the dual to an LP in standard form ( P ) maximize c T x subject to Ax b, 0 x is the LP ( D ) minimize b T y subject to A T y c, 0 y. Since the problem D is a linear program, it too has a dual. The duality terminology suggests that the problems P and D come as a pair implying that the dual to D should be P . This is indeed the case as we now show: minimize b T y subject to A T y c, 0 y = - maximize ( - b ) T y subject to ( - A T ) y ( - c ) , 0 y. The problem on the right is in standard form so we can take its dual to get the LP minimize ( - c ) T x subject to ( - A T ) T x ( - b ) , 0 x = maximize c T x subject to Ax b, 0 x . The primal-dual pair of LPs P-D are related via the Weak Duality Theorem. Theorem 4.1 (Weak Duality Theorem) If x R n is feasible for P and y R m is feasible for D , then c T x y T Ax b T y. Thus, if P is unbounded, then D is necessarily infeasible, and if D is unbounded, then P is necessarily infeasible. Moreover, if c T ¯ x = b T ¯ y with ¯ x feasible for P and ¯ y feasible for D , then ¯ x must solve P and ¯ y must solve D . We now use The Weak Duality Theorem in conjunction with The Fundamental Theorem of Linear Programming to prove the Strong Duality Theorem . The key ingredient in this proof is the general form for simplex tableaus derived at the end of Section 2 in (2.5). Theorem 4.2 (The Strong Duality Theorem) If either P or D has a finite optimal value, then so does the other, the optimal values coincide, and optimal solutions to both P and D exist. Remark: This result states that the finiteness of the optimal value implies the existence of a solution. This is not always the case for nonlinear optimization problems. Indeed, consider the problem min x R e x . 45
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This problem has a finite optimal value, namely zero; however, this value is not attained by any point x IR . That is, it has a finite optimal value, but a solution does not exist. The existence of solutions when the optimal value is finite is one of the many special properties of linear programs. Proof: Since the dual of the dual is the primal, we may as well assume that the primal has a finite optimal value. In this case, the Fundamental Theorem of Linear Programming says that an optimal basic feasible solution exists. By our formula for the general form of simplex tableaus (2.5), we know that there exists a nonsingular record matrix R IR n × n and a vector y IR m such that the optimal tableau has the form bracketleftbigg R 0 - y T 1 bracketrightbigg bracketleftbigg A I b c T 0 0 bracketrightbigg = bracketleftbigg RA R Rb c T - y T A - y T - y T b bracketrightbigg . Since this is an optimal tableau we know that c - A T y 0 , - y T 0 with y T b equal to optimal value in the primal problem. But then A T y c and 0 y so that y is feasible for the dual problem D . In addition, the Weak Duality Theorem implies that b T y = maximize c T x b T hatwide y subject to Ax b, 0 x for every vector hatwide y that is feasible for D . Therefore, y solves D !!!!
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