simplex_max

std.optim.simplex_max · Level L4

Maximize cᵀx subject to A·x ≤ b and x ≥ 0, for integer A, b ≥ 0 and c, by the fraction-free simplex method with Bland's rule: a While over simplex_step on the integer tableau [A | I | b; −c | 0 | 0]. Every entry stays an integer, so every decision is exact. status is 1 at an optimum and 0 when the objective is unbounded. The optimum is z_num/z_den in lowest terms; x and the duals y are x_num/den and y_num/den, and bᵀy = cᵀx by strong duality. The reference is the ordinary simplex method on Fractions.

max cᵀx s.t. A·x ≤ b, x ≥ 0

Signature

simplex_max(A: i64[m, n], b: i64[m], c: i64[n]) → i64[], i64[], i64[], i64[n], i64[m], i64[]

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 ordinary simplex method on Fractions, Bland's rule; den = |det B| in exact rational arithmetic, on all 40 test cases.
All 384 int64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the int64 nearest the exact value): 100%
bit-equal to the NumPy formula in int64: 100%
largest error, in units in the last place: 0

Note

b ≥ 0 makes the origin feasible, so no first phase is needed. Exact while the tableau's integers, which are minors of [A | I | b], stay below 2⁵³; the verification uses m, n ≤ 5 and entries up to 20.

Identity

Calls

gcd simplex_running simplex_step

Called by

—

sha256:7681986778429b28a54d2d410b6ddb4792c80f849706ad475a7a39df41f0342c

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.simplex_max · Ω:simplex_max
Pins: sha256:13d7ebfb3752006e6192bc70e7712534474fc3222e80835ec481526e84f06c8fthis graph and the 5 it reaches through calls
Evidence: sha256:a667c30a48586ed3ebf83078cac76ed7189642c492b18f3956a23f670e275fe2the 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.