Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Active set — where it appears

The constraints that hold with equality at the solution of an inequality-constrained problem. Which they are is the combinatorial part of the problem, and an interior-point method finds out by driving a diagonal apart until the two groups are sixteen orders of magnitude away from each other.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Also named here as barrier parameter — the same set of essays touches all of them, so they are one junction rather than several.

-14-12-10-8-6-4-2010⁻¹⁷10⁻¹³10⁻⁹10⁻⁵10⁻¹10³10⁷10¹¹10¹⁵log₁₀ μ — the barrier parametercondition number, and relative errorκ₂, condensedκ₂, augmentederror, condensederror, augmentedagainst a BigInt answerκ₂ augmented, μ = 10⁻¹⁴3·10¹⁵its relative error10⁻¹⁵κ₂ condensed2.4·10¹⁶its relative error0.31the same step, written two waysand only one of them is solvable

A condition number sent to infinity

An interior-point method manufactures an ill-conditioned matrix on every iteration, deliberately, because the separating of a diagonal is how it discovers which constraints are active. Written one way the answer keeps fifteen digits at a condition number of 3·10¹⁵. Written the other way — the way almost every code writes it — it has none left.

constraint · Interior-point conditioning -1-0.75-0.5-0.25010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹10⁴log₁₀ σ — the barrier's reduction factorresidual after one reused stepconvergedthe pattern free, the factors notentries moved6off-diagonal0survived at σ = 0.996survived at σ = 0.106 steps0 stepsthe few entries that movedare the ones that dominate

What survives one step of the barrier

An interior-point method solves the same system dozens of times with the same pattern and different numbers, and exactly p entries change between one step and the next. The pattern is reusable for ever. The factorisation is reusable for none of them, and the threshold that says so is a reduction factor of about a per cent against schedules that use ten.

sequence · Reuse 0246810121410⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹interior-point iterationrelative error, and μcertified from iterate 1μthe iterate's errorthe crossover's errorone solve, checkediterations15first certified iterate1iterate error there0.22crossover error there4·10⁻¹⁴the active set arrives long before the digitsand a crossover collects them at once

The active set before the digits

An interior-point method takes fifteen iterations on a quadratic programme with forty constraints, and its iterate has eight correct digits at the eleventh. Take the constraints its diagonal calls active at the first iterate, solve the equality problem they define once, and check the answer against the conditions for optimality. It passes, to thirteen digits. The step's matrix had a condition number of 43 at that iterate, and 7·10¹⁵ at the last.

constraint · Interior-point conditioning the damagewell scaled, nothing done0.9110^±3, nothing done0.095never certified, of 62at 10^±3rows to unit norm0.64rows by right-hand side0.6start at the rows0.43012300.250.50.751rows rescaled by 10ᵏshare that certifiesnothing donestart at the rowsrows to unit normrows by their right-hand sidethe grey line is a well-scaled programme with nothing doneboth equilibrations are flat, and below it

Two repairs for one symptom

Rescale a quadratic programme's constraint rows over six decades and the crossover that certified its answer at iterate 1.5 first certifies at 69.8, with two of six programmes never certifying at all. Normalising the rows removes the spread completely — the same numbers at 10¹, 10² and 10³ either way. Starting the method at the magnitudes the rows imply repairs the iteration count completely and the identification only halfway. They are two repairs and they fix different halves.

constraint · Interior-point conditioning the certificatecertificate, iterate1.5its error1.8·10⁻¹⁴the iterate's own error0.14the tightest μ testμ < 10⁻¹³, iterate15its error1.6·10⁻¹²one attempt ÷ one step0.130246810121410⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹interior-point iterationrelative error returnedμ < 10⁻⁴μ < 10⁻⁶μ < 10⁻⁸μ < 10⁻¹⁰μ < 10⁻¹³the certificatethe iterate it is built fromthe grey line is the certificate's own errorno tolerance on μ reaches it, at any iterate

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.

constraint · Interior-point conditioning

Named alongside it

The objects these essays reach for when they reach for this one.

Barrier parameterCondition numberInterior-point methodCertificateEquilibrationExact ground truthForward errorSaddle-point systemsStopping criterionNewton iterationBackward errorComponentwise condition number

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