52、利用有限精度预言机的线性规划

利用有限精度预言机的线性规划

1. 迭代细化算法

迭代细化算法用于求解原始 - 对偶可行的线性规划问题。以下是该算法的详细步骤：

Algorithm 1: Iterative Refinement for a Primal and Dual Feasible LP
input: rational LP data A, b, ℓ, c, termination tolerance τ ⩾0
parameters: incremental scaling limit α ∈N, α ⩾2
output: primal–dual solution x∗∈Qn, y∗∈Qm within tolerance τ
1 begin
2    Δ1 ←1
/* initial solve */
3    get ( ¯A,¯b, ¯ℓ, ¯c) ≈(A, b, ℓ, c) in working precision of the oracle
4    call oracle for min{¯cT x : ¯Ax = ¯b, x ⩾¯ℓ}, abort if failure
5    (x1, y1) ←approximate primal–dual solution returned
6    for k ←1, 2, . . . do
/* refinement loop */
7        ˆb ←b −Axk, ˆℓ←ℓ−xk, ˆc ←c −AT yk /* compute residual error */
8        δk ←max{maxj |ˆbj|, maxi ˆℓi, maxi −ˆci, |Σi −ˆℓiˆci|}