Every essay — page 21
The matrix a constraint makes
Every difficult matrix in the other fields was difficult for a reason the arithmetic supplied: a Hilbert matrix arrives ill conditioned, a Wilkinson matrix grows under elimination, a kernel matrix is dense. Ask a problem to minimise something subject to a constraint and the matrix that results is difficult for a reason the algebra supplies. Its zero block is the second derivative of a Lagrangian with respect to its own multipliers, so no pivot order removes it and no precision changes that; its inertia is known before anything runs; and Cholesky does not fail somewhere on it, it fails at the first constraint row, on a number the problem already contained. Then the field's second surprise, which is the opposite one: an interior-point method drives the condition number of this matrix to 10¹⁵ deliberately, and the answer keeps fifteen digits — because the number that describes the error is not the one every library prints.
A test with no tolerance in it
An interior-point method's own stopping test is a tolerance on μ, and at the tightest it can be set to it stops after 15 iterations with 1.6·10⁻¹². A crossover from the iterate at 1.5 returns a point whose error is 1.8·10⁻¹⁴ — ten times sooner and a hundred times better, from an iterate carrying one correct digit. One attempt costs an eighth of a step, and the guess's own margin says which iterate to spend it on.
The shift that stops at the first right count
A nonconvex solver that finds the wrong inertia adds δI to H and tries again, and the δ it settles on is used as though it measured the curvature it corrects. It does not. On a well-conditioned constraint it is the schedule's number — 1.8·10⁴ times the need at a curvature of 5.6·10⁻⁹, between one and 7.3 times above 10⁻² — and on an ill-conditioned one the loop stops wherever the count first reads right: 63 of 152 saddles at κ(A) = 10⁸, the deepest with curvature 56. Refining the shift by bisection removes the first error and adds to the second.
A constraint the count stops seeing
Let one constraint drift towards being a combination of the others and the inertia of the saddle-point matrix keeps its promise only while σ²/|h| can be resolved — σ the constraint's smallest singular value, h the curvature along the direction it barely constrains. At h = −1 the count stops seeing the constraint at σ = 1.4·10⁻⁹, six decades before any rank test would drop it, and below that it reports a genuine minimum as a saddle on three to six draws in eight. No shift of H brings the constraint back: the correction loop shifts a problem that needed nothing by as much as 2,620. A perturbation of the constraint block does not bring it back either — it decides, at σ = √(|h|δ).
A loop that asks the null space why
An inertia-correction loop sees only an integer, and two different faults produce the same wrong one: curvature that needs a shift, and a constraint too weak for the count to see. One QR of the constraint matrix on a wrong count tells them apart — it shifts none of the 22 weak-constraint minima the ordinary loop shifted by up to 2,621 — and its reduced eigenvalue gives the shift a saddle needs in one step, twice the need exactly, where the schedule overshoots by up to 17,783 times. But at κ(A) = 10⁸ the loop still certifies 57 saddles of 152, because a false certificate is a count that read right, and a check made only on wrong counts never sees it. Asking every time leaves five, all shallower than 2·10⁻⁸.
The residual turns before the error doubles
Regularise a saddle-point matrix past the Hessian's most negative eigenvalue and its pivot signs certify every shallow saddle as a minimum; refinement against the unregularised matrix keeps the test, as a rate, and a saddle at a hundredth of the regularisation grows by only 1.0014 a step — 485 steps to double. That was read as hundreds of steps before the history says anything. The residual, which is what a solver actually has, says it at step 62: a fit of its logarithm over the last ten steps turns positive there and stays positive, while the minimum's is negative from step 10. Across five regularisations the verdict comes at an eighth of the doubling time.
An augmentation read in the smallest eigenvalue
An augmented Lagrangian preconditioner's least work sat at γ = 1, 10⁴ and 1 on three systems, and two of the three optima sat where the smallest generalised eigenvalue was about 0.06 — which suggested a rule written in ν rather than γ. On twenty-four systems whose norms are all one, the least-work γ spans seven decades and ν at the optimum spans a factor of thirty, so there is no one ν. There is still a rule: take the smallest γ at which ν reaches 0.03, and the median system pays 7 per cent over its least work and the worst 29, where the best single γ pays 61. Its price is digits — twelve times the best forward error on the median system, 4,561 times on the worst.
The right-hand side that hides the saddle
Refinement against an unregularised saddle-point matrix gives the second-order verdict the pivot signs cannot, from step 62 on the shallowest saddle — for one right-hand side. The verdict depends on how much of the growing direction the right-hand side contains, and it can contain none. Each decade removed delays the verdict by a fixed number of steps, 147 on the shallowest saddle, set by the growth against the minimum's slowest contraction rather than by the growth alone, which predicted 1,611. Rounding stops the hiding at about 2,700 steps — but by then refinement has solved the system to a residual of 2.6 times ten to the minus thirteen, and any stopping test has already accepted it. A random kick to the starting point costs nothing and finds it.