Solver Ladder & Optimization¶

Non-linear optimization subject to non-linear inequality constraints can be sensitive to starting points and conditioning. Risk Bridge implements a hierarchical solver ladder to maximize numerical stability and convergence reliability.

The Solver Ladder Architecture¶

                 Start
                   │
                   ▼
┌──────────────────────────────────────┐
│  Step 1: Unconstrained Warm Start   │
│  - BFGS with analytic gradient       │
│  - Minimizes joint NLL l(θ)          │
└──────────────────┬───────────────────┘
                   │ Yields θ_ML
                   ▼
┌──────────────────────────────────────┐
│  Step 2: Primary Constrained Solver  │
│  - Method: trust-constr              │
│  - Warm-started at θ_ML              │
│  - Uses analytic objective gradient  │
│    and analytic constraint Jacobian  │
└──────────────────┬───────────────────┘
                   │
         Feasible? ├────────────► Success (Done)
                   │ No
                   ▼
┌──────────────────────────────────────┐
│  Step 3: Secondary Fallback Solver   │
│  - Method: SLSQP                     │
│  - Warm-started at θ_ML              │
│  - Uses analytic gradient & Jacobian │
└──────────────────┬───────────────────┘
                   │
         Feasible? ├────────────► Success (Done)
                   │ No
                   ▼
┌──────────────────────────────────────┐
│  Step 4: Diagnostic Validation       │
│  - Compute max_violation = max |g_k| │
│  - If max_violation <= tol: Accept   │
│  - Else: Flag cmle_success = False   │
└──────────────────────────────────────┘

1. Step 1: Unconstrained Warm Start (BFGS)¶

Before attempting constrained estimation, the unconstrained MLE is computed:

\[ \hat{\theta}_{\text{ML}} = \arg\min_{\theta} \ell(\theta) \]

We use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton algorithm with exact analytic gradients. This step converges rapidly because \(\ell(\theta)\) is smooth and well-behaved over source data.

The resulting \(\hat{\theta}_{\text{ML}}\) serves two critical roles: 1. It provides an unconstrained benchmark against which the impact of calibration constraints is measured. 2. It serves as a warm start for constrained optimization.

2. Step 2: Primary Constrained Solver (`trust-constr`)¶

Starting from \(\hat{\theta}_{\text{ML}}\), we invoke SciPy's trust-constr algorithm:

Algorithm: An interior-point trust-region algorithm specifically suited for non-linear equality and inequality constraints.
Analytic Jacobians: Both the objective gradient and the \(K \times p_\theta\) constraint Jacobian \(J(\theta)\) are computed analytically:

\[ J_{kj}(\theta) = \frac{\partial g_k(\theta)}{\partial \theta_j} \]

Numerical Guardrails:
Probability terms are clamped away from exact boundaries: \(\hat{p} \in [10^{-12}, 1 - 10^{-12}]\).
Scale parameter \(\sigma\) is lower-bounded: \(\sigma \ge 10^{-8}\).

3. Step 3: Fallback Solver (`SLSQP`)¶

If trust-constr terminates prematurely or fails to satisfy feasibility, the ladder falls back to Sequential Least Squares Programming (SLSQP):

Solves a sequence of quadratic programming subproblems.
Warm-started at \(\hat{\theta}_{\text{ML}}\).
Uses the same analytic gradients and constraint Jacobians.

4. Step 4: Rigorous Feasibility Acceptance¶

Regardless of whether the underlying solver flags convergence, Risk Bridge independently audits the solution by evaluating the constraint violations at the final parameter estimate \(\hat{\theta}\):

\[ \text{max\_violation} = \max_{k=1,\dots,K} \lvert g_k(\hat{\theta}) \rvert \]

The constrained fit is accepted as valid (cmle_success = True) if and only if:

\[ \text{max\_violation} \le \epsilon \]

where \(\epsilon\) is the configured --calibration-tolerance. Both the solver status string and cmle_max_violation are preserved in fit_diagnostics.csv.