Consider a linear program with bounded variables: where the upper bounds u j are positive…

Consider a linear program with bounded variables:

where the upper bounds u j are positive constants. Let yi for i = 1, 2, . . . , m and wj for j = 1, 2, . . . , m be variables in the dual problem.

a) Formulate the dual to this bounded-variable problem.

b) State the optimality conditions (that is, primal feasibility, dual feasibility, and complementary slackness) for the problem.

c)  denote the reduced costs for variable x j determined by pricing out the ai j constraints and not the upper-bounding constraints. Show that the optimality conditions are equivalent to the bounded-variable optimality conditions.

given in Section 2.6 of the text, for any feasible solution x j (j = 1, 2, . . . , n) of the primal problem.

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