MATH.20:11 - SoTA-Echoing

The working question is how to obtain a useful justified comparison without first solving the whole problem. The selected line combines feasible construction, relaxation or a comparison identity with explicit order propagation and an examination of the remaining gap.

Vandenberghe’s Duality, EE236A lecture 6, especially 6-4 derives the relation between feasible primal and dual values. Its contribution to :4.2/:5.1 is that a feasible construction and a general inequality can bound the same optimum from opposite sides. A complete optimal solution is unnecessary for each bound. Direct solution is cheaper for an easy instance; bounding becomes useful when a partial construction already settles the receiving question. Equality and attainment require the particular argument, not an unrestricted assumption of strong duality.

Higham’s What Is Backward Error? and What Is a Condition Number? explain why a discrepancy must be combined with the relevant sensitivity and why norm, perturbation class and requested output matter. The adopted consequence is :4.3/:5.2: derive a bound for the requested error and inspect the inverse action. A bare residual is a cheaper observable but can answer a different question. A proved componentwise or structure-specific estimate can be more informative than a coarse whole-vector estimate; no one condition number is imposed on every problem.

The current Mathematics in Lean treatment of monotonicity and set inclusion makes the quantified preservation steps explicit. It supports :4.3-.4 and the set branch of :5.3. These mechanisms are usable in ordinary proofs or formal checking; choosing Lean is optional.

The synthesis retains the cheapest comparison that supports the requested result and exposes the part of its proof that could be tightened. Reopen it when the domain, objective, available premises or required operation changes, or when a different bound would resolve a still-open use with less work.