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weak duality

Every dual-feasible value is a lower bound on every primal-feasible value, with no optimality required on either side.

For a primal \min\ c^\top x s.t. Ax\ge b,\ x\ge0 and its dual \max\ b^\top y s.t. A^\top y\le c,\ y\ge0, weak duality states b^\top y\le c^\top x for every primal-feasible x and every dual-feasible y, proved by chaining the two feasibility inequalities through a nonnegative dot product. It holds even far from either optimum, which is what makes a single dual-feasible point already useful as a certified lower bound (weak-duality-lp).

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