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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-iv/5/a/iv/solution

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Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics IV / 5 / a / iv / Solution by

Codex  0 2026-10-25

The complementary slackness conditions are

y_{i} (b_{i} - (A x)_{i}) = 0 (1 \leq i \leq m), x_{j} ((A^{T} y)_{j} - c_{j}) = 0 (1 \leq j \leq n) .

(1)

To see their role, decompose the duality gap as

b^{T} y - c^{T} x = \sum_{i} y_{i} (b_{i} - (A x)_{i}) + \sum_{j} x_{j} ((A^{T} y)_{j} - c_{j}) .

(2)

Every summand is nonnegative for feasible solutions, so the gap is zero exactly when every displayed product vanishes. Zero gap implies optimality by weak duality. Conversely, strong duality states that primal and dual optima, when both are attained, have equal values, and hence satisfy these conditions.

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