Dual-price stabilization

Documentation index · Tree lifecycle

Column generation can repeatedly produce similar master duals and columns. BALDES optionally smooths the dual point sent to pricing, using a stored center and bounded retries when smoothed pricing finds no fresh columns. It is enabled by default in cg::Options.

An explicit cg::Options.proximal configuration replaces smoothing with box-step, three-piece or five-piece dual regularization on a private LP copy. Its optional multi-point epigraph also stays private; ordinary master duals are retried before accepting a no-column result. Auxiliary objectives never become original bounds or incumbents. See network proximal controls. Automatic three/five-piece curvature is available through ProximalParameters::automatic_three_piece() / automatic_five_piece() or JSON/Python proximal={"curvature": {}}. The solve loop supplies initialization, physical-row participation, raw-dual measurements, iteration updates, activity retries and compatible warm/child inheritance. See automatic curvature controls.

The implementation is DualPriceSmoothing, integrated by the column-generation loop. The sections below describe the implemented update and retry rules.

Smoothing weight

For raw master duals raw, stored center, and an outward point outward:

pricing[i] = (1 - effective_alpha) * outward[i]
           + effective_alpha * center[i]

By default outward = raw. Larger alpha means more weight on the center. At alpha zero pricing uses the outward point; at alpha one it uses the center. Reducing alpha moves pricing back toward the current master duals.

After interpolation, a positive dual on a LessEqual row or a negative dual on a GreaterEqual row is set to zero when its magnitude exceeds the configured tolerance. Equality rows have no such sign restriction.

Parameters

These are Rust master::stabilization::Parameters, configured through RootOptions::cg.smoothing or directly in cg::Options. They are separate from the named BALDES pricing options.

The reference stores alpha_factor, beta_factor, min_gap and max_gap as single-precision controls. The constructor rounds these fields through f32 before using them as f64. A factor that rounds to zero disables stabilization.

The column-generation caller passes the current pricing stage. Its gap compares the inherited node lower bound with the best artificial-free restricted-LP objective accepted before this solve, using the reference comparison tolerance. It tracks these separately from the global integer incumbent. A new cut or enumeration round carries the certified lower bound and resets the LP bound. Unknown bounds use finite -1e12/+1e12 sentinels; a known lower bound with an unknown LP bound therefore gives a gap slightly below one. Warm starts retain the parent certificate.

Lifecycle

The first master iteration uses raw duals. Cut/enumeration transitions restart the schedule; row cleanup applies the row map. Removing columns alone does not change row coordinates and does not require a center remap.

Pricing retries and alpha updates

Feedback includes the fresh negative-column count, a subgradient vector, Lagrangian value, and pricing phase. In the column-generation driver, the subgradient starts at the row RHS and subtracts multiplicity times the best returned acceptable column's coefficients. A near-zero or rounded-zero reduced cost does not suppress this contribution.

On the first pricing pass in angle-adaptive mode (alpha_factor == 1):

if no fresh negative columns or angle > 1e-12:
    alpha -= 0.1
else if angle < -1e-12 and alpha is below the near-one gate:
    alpha += 0.1 * (1 - alpha)

The angle threshold is 1e-12, independently of Parameters::tolerance. The near-one gate includes the general tolerance and a relative margin. Updating base alpha affects subsequent master iterations; retries update the current effective alpha separately. The angle is calculated only on that first adaptive pricing pass and retained during subsequent mispricing retries.

If fresh negative columns exist, or smoothing is inactive, after_pricing returns false and stops retrying. Otherwise it increments misprices and uses:

dynamic mode: effective_alpha = 1 - 2 * (1 - effective_alpha)
fixed-base mode: effective_alpha = 1 - misprices * (1 - alpha)

It republishes a smoothed point only while the retry count is at most max_updates and effective alpha exceeds the absolute-plus-relative zero gate (tolerance + abs(effective_alpha) * 1e-10). When either gate fails, it deactivates smoothing and requests a final retry at raw duals. “Fixed” describes the base alpha, not an unchanging effective retry weight. The dynamic formula preserves the compiled reference's floating-point operation order; rewriting it as 2 * effective_alpha - 1 can change the result.

Center updates

A nonfinite Lagrangian value is ignored. If the current center phase is less than or equal to the feedback phase, a finite center bound is replaced only when the new value improves it beyond tolerance. A lower-numbered feedback phase can reset the center phase and accept a different bound without that same improvement requirement.

The accepted pricing point and feedback subgradient become a pending center, installed by the next after_master call. Consequently the stored center bound is not unconditionally monotone across phase changes.

The angle helper computes:

toward = pricing - center
angle = dot(toward, feedback_subgradient)
      / (norm(toward) * norm(feedback_subgradient))

The division is unguarded. A zero denominator may yield NaN; comparisons with it then fail. This behavior differs from forcing an angle of zero.

Optional directional construction

With positive beta_factor, let d = raw - center, l = norm(d), and g be the stored normalized center subgradient. A negligible l returns raw duals. For beta_factor == 1, beta is the cosine between d and g; a negligible subgradient also returns raw duals. Otherwise the configured beta is used. Beta is clamped to [0, 1] before constructing:

outward = center + (1 - beta) * d + beta * l * g

A negligible subgradient contributes zero to its directional term. When the constructed step has norm above 1e-12, it is rescaled to length l. This changes the outward point before the alpha interpolation. Mispricing retries use the raw outward point. Restart invalidates the automatic direction until a subsequent accepted center provides a new normalized subgradient. Retaining the old vector alone does not make it valid. The 54-case parameter corpus covers this capacity-row transition.

Configure or disable

A complete Rust example, requiring clp:

use baldes::{master::root::RootOptions, inputs::{Instance, Rounding}};

fn main() -> baldes::Result<()> {
    let mut instance = Instance::read("instances/Solomon/R101.txt")?;
    instance.truncate(10)?;
    let mut options = RootOptions::default();
    options.cg.smoothing.initial_alpha = 0.5;
    options.cg.smoothing.beta_factor = 0.0;
    options.cg.smoothing.max_updates = 10;
    let result = instance.solve_root(8, Rounding::None, options)?;
    println!("{:?}: root LP={}", result.master.status, result.master.lp);
    Ok(())
}

Set options.cg.stabilize = false to disable it in Rust. The CLI exposes:

cargo run --release --features clp --bin baldes -- \
  vrptw instances/Solomon/R101.txt --customers 10 --no-stabilization

Python exposes the enable/disable switch through the public root API:

import pybaldes

result = pybaldes.solve_root(
    "instances/Solomon/R101.txt", customers=10, stabilize=False,
)
print(result.status, result.lp)

The Python root API does not expose the individual smoothing parameters.

An unavailable best pricing solution differs from a zero subgradient. Feedback carries that availability explicitly; the adaptive angle retains its preceding value when a best solution is unavailable, matching the native early return. Completed CG rounds use the LP objective accepted within convergence tolerance for subsequent stabilization gap bookkeeping, while the published pricing bound remains conservative.

When an unavailable pricing solution accompanies an accepted pending center, the center uses raw LP duals. The saved normalized subgradient remains intact; the fallback RHS vector does not replace the direction or make it valid. Customer-only elementarity IDs and canonical retained-path export order preserve ordering through directional enumeration.