lagrange_projection

std.optim.lagrange_projection · Level L0

The Lagrange condition for minimizing ‖x − c‖² subject to one linear constraint g·x = h (g ≠ 0): the minimizer x* = c − ((g·c − h)/‖g‖²)·g, the multiplier λ with ∇f(x*) = λ·∇(g·x) (this sign convention), and the residuals of stationarity, 2(x* − c) − λg, and of feasibility, g·x* − h, both 0. The condition is necessary, not sufficient, in general; here the problem is convex, so x* is the global minimizer. It is not the KKT condition for inequalities.

x* = c − ((g·c − h)/‖g‖²) g, 2(x* − c) = λ g

Signature

lagrange_projection(c: f64[n], g: f64[n], h: f64[]) → f64[n], f64[], f64[n], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference the projection formula in exact arithmetic in exact rational arithmetic, on all 40 test cases.
All 388 float64 results inside the running error bound; the closest uses 87% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 69% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 64%
bit-equal to the NumPy formula in float64: 84%
largest error, in units in the last place: 7.7e+3

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls

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Called by

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sha256:27c492c10d08dd10f07f3eed016a9eb5683fd59456a2b7ea882d7412d94faf67

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.optim.lagrange_projection · Ω:lagrange_projection
Pins: sha256:d13b78fcb1038fb4406d56bc4c7817659f193e09be01ad5c2723ff4340e42a38this graph and the 1 it reaches through calls
Evidence: sha256:bc6dd0fc42742b43da16dad33a647105dd66dcafb867894767edb7ccdff382b6the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.