 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2009/t/t2/b/i/solution

Write each unrestricted coordinate as

x_{j} = u_{j} - v_{j}

with

u_{j}, v_{j} \geq 0

. Then the primal has nonnegative variables

(u, v)

, constraint matrix

[A, - A]

, and objective coefficients

(c, - c)

. Its dual constraints are

A^{T} y \geq c

and

- A^{T} y \geq - c

, hence equality. Therefore

D^{*} : min b^{T} y subject to A^{T} y = c, y \geq 0.

The removed sign restriction on the primal variables becomes equality of the dual column constraints.

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