 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/t/t2/b/i/solution

Write

A x = b

A x \leq b

and

- A x \leq - b

. The dual has nonnegative multipliers

u, v

for these inequalities, with objective

b^{T} u - b^{T} v

and condition

A^{T} (u - v) \geq c

. Every real vector

y

can be expressed as

u - v

with

u, v \geq 0

. Hence

D^{'} : min b^{T} y subject to A^{T} y \geq c, y \in R^{m} unrestricted .

The equality constraints remove the sign restriction on the dual variables.

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