Advanced Methods Guide

Advanced Methods Guide

The mathematical and algorithmic underpinnings: CAG, the LCPG algorithm, Pyomo code, notation tables, and certifications.

What is the Advanced Methods page?

Advanced Methods is a research reference organized into five tabs: the CAG descent method, the LCPG algorithm pseudocode, the Pyomo reference code, the notation/symbol table, and mathematical correctness certifications. It documents exactly how the LCO engine solves the problem and proves its guarantees.

Componentwise Approximated Gradient (CAG) constructs descent directions when the true gradient is unstable or noisy — common on irregular hotel revenue surfaces with discontinuities at inventory thresholds.

Classical Newton direction

d^Newton = -H^-1(x) ∇f(x)

CAG-Modified Descent Direction (Eq. 37)

d^CAG = -H^-1(x) q(x)

Floor preservation (Eq. 39)

∇L_j(x)^T d_CAG(x) ≥ 0 for all j < k

Operational advantages: resilience to demand shocks, handling of non-convex surfaces, guaranteed feasibility under noisy gradients, better multi-property convergence, and RL fine-tuning compatibility.

How to use it

  1. Start with CAG to understand how descent directions stay valid on noisy surfaces.
  2. Read LCPG to see how intra-tier parameters are tuned online.
  3. Copy the Pyomo code to reproduce the 10×5 instance locally.
  4. Keep the Notation table open when reading the formulas elsewhere in the app.
  5. Review Certifications for the formal correctness guarantees behind each method.
Advanced Methods • CAG, LCPG, code, notation & certifications • Valamontes, 2026