Advanced Methods

CAG, LCPG, Pyomo code, certifications, and notation tables

Componentwise Approximated Gradient (CAG)

CAG provides a principled mechanism for constructing descent directions even when the true gradient exhibits instability, noise, or undefined behavior in individual components. Hotel optimization surfaces are inherently irregular — revenue functions may experience sharp discontinuities when room inventory thresholds are crossed.

Classical Newton direction

d^Newton = -H^-1(x) ∇f(x)

CAG-Modified Descent Direction (Eq. 37)

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

Descent guarantee (Theorem 2.1, Eq. 38)

∇f(x*) = 1, and d ≥ 0 ⟹ ∇f(x*)^T d = d ≥ 0

Floor preservation (Eq. 39)

∇L_j(x)^T d_CAG(x) ≥ 0 for all j < k

Operational Advantages

1. Resilience to demand shocks
2. Improved handling of non-convex revenue surfaces
3. Guaranteed feasibility under noisy gradients
4. Enhanced convergence in multi-property orchestration
5. Compatibility with RL fine-tuning