price-contour 0.3.0

High-performance insurance price optimisation via Lagrangian dual decomposition

Price Contour

High-performance insurance price optimisation via Lagrangian dual decomposition.

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Price Contour finds optimal price scenario values across a portfolio of insurance risks subject to business constraints. Give it a scored dataset with objective and constraint values at discrete price points, and it returns the scenario value per quote that maximises your objective while respecting every constraint.

The core algorithm is Lagrangian dual decomposition, implemented in Rust for speed and exposed to Python via zero-copy Polars DataFrames. A portfolio of 1M+ risks solves in seconds.

---

Quick start

uv add price-contour

import polars as pl
import price_contour as pc

# Long-format DataFrame: one row per (quote, price_scenario)
# with pre-computed objective and constraint values
df = pl.read_parquet("scored_quotes.parquet")

optimiser = pc.OnlineOptimiser(
    objective="income",
    constraints={"volume": {"min_pct": 0.90}},  # retain at least 90% of baseline volume
    quote_id="quote_id",
    scenario_index="scenario_index",
    scenario_value="scenario_value",
)

result = optimiser.solve(df)

print(result.converged)        # True
print(result.iterations)       # 23
print(result.lambdas)          # {'volume': 0.147}
print(result.total_objective)  # 1_284_302.5

# Per-quote optimal scenario values as a Polars DataFrame
out = result.dataframe
print(out.head())
# ┌──────────┬──────────────┬────────────────────┬─────────────────────┬──────────────────┐
# │ quote_id │ optimal_step │ optimal_scenario_value │ optimal_income      │ optimal_volume   │
# ╞══════════╪══════════════╪════════════════════╪═════════════════════╪══════════════════╡
# │ Q001     │ 14           │ 1.07               │ 42.30               │ 0.82             │
# │ Q002     │ 11           │ 0.98               │ 18.55               │ 0.91             │
# └──────────┴──────────────┴────────────────────┴─────────────────────┴──────────────────┘

---

What it does

Price Contour operates on pre-computed scenario data. It does not fit models or generate demand curves. Upstream, your pricing pipeline scores every quote at a grid of price scenario values (e.g. 0.8, 0.85, 0.9, ..., 1.2) and computes what the expected income, volume, loss ratio, etc. would be at each point. Price Contour then selects the optimal scenario value per quote across the portfolio.

The input is a long-format Polars DataFrame:

quote_id	scenario_index	scenario_value	income	volume	loss_ratio
Q001	0	0.80	85.2	0.95	0.62
Q001	1	0.90	92.1	0.88	0.59
Q001	2	1.00	100.0	0.80	0.60
Q002	0	0.80	42.0	0.97	0.58
...	...	...	...	...	...

The output is one optimal scenario value per quote, chosen to maximise portfolio-level income while keeping portfolio-level volume above 90% of baseline (or whatever constraints you set).

---

Three optimisation modes

Online optimisation

Find the optimal scenario value per individual quote. Each quote independently picks its best price point, coordinated by shared Lagrange multipliers that enforce portfolio-level constraints.

optimiser = pc.OnlineOptimiser(
    objective="income",
    constraints={
        "volume": {"min_pct": 0.90},                # sum constraint
        "loss_ratio": {                              # ratio constraint
            "numerator":   "incurred",
            "denominator": "premium",
            "max":         0.65,
        },
    },
)
result = optimiser.solve(df)
print(result.lambdas)            # {'volume': 0.147, 'loss_ratio': 1.21}
print(result.total_constraints)  # {'volume': 5400.0, 'loss_ratio': 0.6498}

Both sum and ratio constraints work in all three optimisation modes (online, ratebook, apply) and in the efficient-frontier sweep.

Ratebook optimisation

Find optimal rating factors across rating dimensions. Instead of individual scenario values, find the best factor value for each level of each rating factor (e.g. age band, region, vehicle power), applied uniformly to all quotes sharing that level.

optimiser = pc.RatebookOptimiser(
    objective="income",
    constraints={"volume": {"min_pct": 0.90}},
    factor_columns=[["age_band"], ["region"], ["vehicle_power"]],
)

result = optimiser.solve(df, factors=factor_df)

print(result.factor_tables)
# {'age_band': {'18-25': 1.15, '26-35': 1.02, '36-50': 0.95, '51+': 0.98},
#  'region': {'London': 1.08, 'South East': 1.01, 'North': 0.93},
#  'vehicle_power': {'Low': 0.97, 'Medium': 1.0, 'High': 1.06}}

# Save to disk
result.save("parameters/")

# Convert to rating-step DataFrames
tables = result.to_rating_entries()

Live scoring with stored lambdas

Apply pre-computed Lagrange multipliers to new quotes in a single forward pass, with no iteration. Use this in production to score individual quotes using lambdas learned from a batch solve.

# Batch solve (offline)
result = optimiser.solve(df_portfolio)
lambdas = result.lambdas

# Live scoring (per-quote, no iteration)
applier = pc.ApplyOptimiser(
    lambdas=lambdas,
    objective="income",
    constraints={"volume": {"min_pct": 0.90}},
)
applier.save("config/applier.json")

# Later, in production:
applier = pc.ApplyOptimiser.load("config/applier.json")
live_result = applier.apply(df_single_quote)
optimal_scenario_value = live_result.dataframe["optimal_scenario_value"][0]

---

Efficient frontier

Sweep constraint thresholds to generate the Pareto frontier - the trade-off curve between your objective and constraints. Each point on the frontier is a full portfolio solve at a different constraint target.

frontier = optimiser.frontier(
    df,
    threshold_ranges={"volume": (0.85, 1.0)},
    n_points_per_dim=20,
)

# DataFrame with one row per frontier point
print(frontier.points)
# ┌──────────────────┬─────────────────┬──────────────┬───────────────┬────────────┬───────────┬─────────┬─────────────────┐
# │ threshold_volume │ total_objective │ total_volume │ lambda_volume │ iterations │ converged │ sv_mean │ sv_pct_increase │
# ╞══════════════════╪═════════════════╪══════════════╪═══════════════╪════════════╪═══════════╪═════════╪═════════════════╡
# │ 0.85             │ 1_350_102       │ 0.851        │ 0.089         │ 18         │ true      │ 1.04    │ 0.62            │
# │ 0.86             │ 1_342_891       │ 0.861        │ 0.102         │ 21         │ true      │ 1.03    │ 0.58            │
# │ ...              │ ...             │ ...          │ ...           │ ...        │ ...       │ ...     │ ...             │
# └──────────────────┴─────────────────┴──────────────┴───────────────┴────────────┴───────────┴─────────┴─────────────────┘

Adjacent points are warm-started from each other (nearest-neighbour traversal of the threshold grid), so the full frontier solves much faster than running each point independently. Each point also includes scenario value distribution statistics (sv_mean, sv_std, percentiles, sv_pct_increase/sv_pct_decrease).

Sweeping a ratio target — declare the constraint with None so the constructor doesn't fix it, then supply the range to frontier():

optimiser = pc.OnlineOptimiser(
    objective="income",
    constraints={
        "loss_ratio": {
            "numerator":   "incurred",
            "denominator": "premium",
            "max":         None,       # frontier supplies the target
        },
    },
)
frontier = optimiser.frontier(
    df,
    threshold_ranges={"loss_ratio": (0.55, 0.75)},
    n_points_per_dim=10,
)
# points["threshold_loss_ratio"] = [0.55, 0.572, ..., 0.75]  (user units, verbatim)
# points["total_loss_ratio"]     = actual Σ incurred / Σ premium at each optimum

Mixed sweep — sweep multiple constraints at once via the cartesian product:

frontier = optimiser.frontier(
    df,
    threshold_ranges={
        "volume":     (8000, 12000),    # absolute units
        "loss_ratio": (0.55, 0.75),     # absolute ratio targets
    },
    n_points_per_dim=10,
)
# 10 × 10 = 100 frontier points

Constraints with numeric thresholds may be omitted from threshold_ranges — they are held fixed at the constructor value across the sweep. None thresholds must have a range entry.

---

Constraint format

Constraints are specified as a dictionary. There are two shapes:

Sum constraints apply to a single column. The dict key is the column name in your DataFrame, the value specifies direction and threshold. Use min / max for absolute thresholds and min_pct / max_pct for thresholds expressed as a fraction of baseline (the portfolio totals at scenario_value = 1.0):

constraints = {
    "volume":  {"min_pct": 0.90},     # portfolio volume >= 90% of baseline
    "premium": {"min": 1_000_000},    # absolute: portfolio premium >= 1M
    "claims":  {"max_pct": 1.05},     # portfolio claims <= 105% of baseline
}

Ratio constraints apply to a ratio of two summed columns (e.g. loss ratio = Σ incurred / Σ premium). The dict key is a display label (does NOT need to be a column); numerator and denominator name the columns:

constraints = {
    "loss_ratio": {
        "numerator":   "incurred",
        "denominator": "premium",
        "max":         0.65,           # portfolio loss ratio <= 0.65
    },
    "combined_ratio": {
        "numerator":   "claims_plus_expenses",
        "denominator": "premium",
        "max_pct":     1.10,           # <= 110% of baseline combined ratio
    },
}

Internally, ratio constraints are linearised as Σ (num − L·denom) ≤ 0 and handed to the same Lagrangian solver. Setting Σ_baseline denom == 0 raises ValueError for _pct modes (baseline ratio undefined). If Σ_optimum denom == 0 at the chosen step set, the ratio reported in total_constraints[label] and summary() is nan (sentinel; the divide is undefined, not silently zero).

None thresholds mark frontier-only constraints — the threshold is supplied by the sweep range:

constraints = {
    "loss_ratio": {
        "numerator":   "incurred",
        "denominator": "premium",
        "max":         None,           # frontier supplies the target
    },
}

frontier = optimiser.frontier(
    df,
    threshold_ranges={"loss_ratio": (0.55, 0.75)},
    n_points_per_dim=10,
)

solve() rejects None thresholds; frontier() requires a threshold_ranges entry for every None constraint. Numeric-threshold constraints are optional in threshold_ranges — omitted ones are held fixed at their constructor value across the sweep.

points["threshold_<name>"] reports the user-supplied range value verbatim (absolute units for min/max, fractions of baseline for min_pct/max_pct); points["total_<name>"] reports the actual aggregate at the optimum (the actual ratio for ratio constraints).

---

Direct Parquet loading

For large datasets, build the internal grid directly from a Parquet file without materialising a DataFrame in Python memory:

grid = pc.build_grid_from_parquet(
    "scored_quotes.parquet",
    constraint_columns=["volume", "loss_ratio"],
    objective="income",
)
result = optimiser.solve(grid)

---

Incremental grid building

For large datasets that don't fit in memory at once, build the internal grid incrementally:

builder = pc.QuoteGridBuilder(
    ["volume", "loss_ratio"],
    quote_id="quote_id",
    scenario_index="scenario_index",
    scenario_value_col="scenario_value",
    objective="income",
)

for chunk in data_source.iter_chunks(100_000):
    builder.append(chunk)

grid = builder.build()
result = optimiser.solve(grid)

---

MLflow integration

Both OnlineOptimiser and RatebookOptimiser produce MLflow-ready summaries:

result = optimiser.solve(df)
summary = optimiser.summary(result)

import mlflow
mlflow.log_params(summary["params"])
mlflow.log_metrics(summary["metrics"])
mlflow.log_dict(summary["artifacts"]["lambdas"], "lambdas.json")
mlflow.log_dict(summary["artifacts"]["config"], "config.json")

---

How it works

The algorithm

Price Contour solves the constrained optimisation problem:

Maximise    sum_i  objective(quote_i, scenario_value_i)
Subject to  sum_i  constraint_k(quote_i, scenario_value_i) >= threshold_k   for all k
            scenario_value_i in {discrete grid}

This is a combinatorial problem (each quote picks from M discrete scenario values). Lagrangian dual decomposition relaxes the coupling constraints into the objective using dual variables (lambdas), decomposing it into N independent per-quote subproblems:

For fixed lambdas:
    Each quote picks:  argmax_m [ objective(i, m) + sum_k lambda_k * constraint_k(i, m) ]

These are independent and embarrassingly parallel.

The outer loop updates lambdas via the subgradient method with adaptive step sizes, iterating until all constraints are satisfied and lambdas converge.

Performance

The Rust core uses:

• Quote-major memory layout - each quote's M scenario values are contiguous, optimising the per-quote argmax inner loop for cache locality
• Rayon parallelism - the argmax across quotes is parallelised in grain sizes of 4096 quotes
• Adaptive step scaling - per-constraint scale factors normalise for differing magnitudes, so the algorithm works equally well for constraints ranging from 0.1 to 1,000,000
• Lambda averaging - smooths the oscillations inherent in discrete Lagrangian relaxation where all quotes can flip simultaneously

Ratebook mode

For ratebook optimisation, coordinate descent iterates over rating factors. For each factor, a grouped Lagrangian solve finds the best discrete factor value per group (e.g. per age band), with the individual quote scenario value computed as the product of all factor values times a per-quote residual. The inner grouped solve uses the same Lagrangian machinery with remapping to the nearest grid point.

---

Architecture

price-contour/
├── crates/
│   ├── price-contour-core/        # Pure Rust: algorithms, data structures, solver
│   │   └── src/
│   │       ├── data.rs            # QuoteGrid, SolverConfig, SolveResult, GroupMapping
│   │       ├── solver/
│   │       │   ├── online.rs      # Lagrangian dual decomposition
│   │       │   ├── grouped.rs     # Grouped solve (ratebook inner loop)
│   │       │   ├── argmax.rs      # Per-quote Lagrangian argmax (parallel)
│   │       │   ├── lambda.rs      # Subgradient lambda updates
│   │       │   └── apply.rs       # Fixed-lambda forward pass
│   │       ├── frontier.rs        # Efficient frontier sweeping
│   │       ├── constants.rs       # Solver defaults
│   │       └── error.rs           # Error types
│   └── price-contour/             # PyO3 bindings (thin wrappers)
│       └── src/
│           ├── solver_py.rs       # DataFrame ingestion + solve
│           ├── grouped_py.rs      # Grouped solve bindings
│           ├── apply_py.rs        # Apply bindings
│           ├── frontier_py.rs     # Frontier bindings
│           ├── builder_py.rs      # QuoteGridBuilder bindings
│           ├── grid_py.rs         # QuoteGrid bindings
│           └── parquet_grid_py.rs # Parquet → QuoteGrid loader
├── python/
│   └── price_contour/
│       ├── solver.py              # OnlineOptimiser, ratio linearisation, validation
│       ├── ratebook.py            # RatebookOptimiser + RatebookResult
│       ├── apply.py               # ApplyOptimiser + apply_from_grid
│       ├── frontier.py            # FrontierResult helpers + frontier_summary
│       ├── builder.py             # QuoteGridBuilder wrapper
│       ├── _ratio_results.py      # Shared ratio reporting (actual ratios + column stitching)
│       └── _frontier_helpers.py   # Shared frontier orchestrator (used by online + ratebook)
├── tests/
│   └── python/                    # Integration tests
├── notebooks/                     # Demo notebooks
├── docs/                          # Design documentation
└── scripts/                       # Utility scripts

The pure-Rust core (price-contour-core) has no Python dependencies and can be tested independently with cargo test. The PyO3 crate (price-contour) is a thin binding layer that converts between Polars DataFrames and the internal QuoteGrid representation with zero-copy where possible.

---

Development

# Clone
git clone https://github.com/PricingFrontier/price-contour.git
cd price-contour

# Install in development mode (compiles Rust, links Python)
uv sync --all-groups
maturin develop

# Run Rust tests
cargo test

# Run Python tests
pytest

# Rebuild after Rust changes
maturin develop

Requirements: Rust toolchain (stable), Python 3.10+, maturin.

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API reference

OnlineOptimiser

Method	Description
solve(df_or_grid, *, lambdas=None)	Run full optimisation. Returns SolveResult. Ratio constraints require a DataFrame (the linearisation needs raw numerator/denominator columns); a pre-built QuoteGrid with ratio constraints raises ValueError.
frontier(df_or_grid, *, threshold_ranges, n_points_per_dim=10, initial_lambdas=None)	Sweep the efficient frontier. Returns FrontierResult. Numeric thresholds are optional in threshold_ranges (held fixed if omitted); None thresholds require a range.
summary(result)	Package result into MLflow-ready params, metrics, artifacts dicts.
config_dict()	Serialisable solver configuration.

RatebookOptimiser

Method	Description
solve(df_or_grid, factors, *, factor_columns=None, lambdas=None)	Run ratebook optimisation via coordinate descent. Returns RatebookResult.
frontier(df_or_grid, factors, *, threshold_ranges, n_points_per_dim=5, factor_columns=None, initial_lambdas=None)	Sweep the efficient frontier via coordinate descent at each threshold. Returns FrontierResult.
summary(result)	Package result into MLflow-ready dicts.

ApplyOptimiser

Method	Description
apply(df)	Single-pass scoring with fixed lambdas. Returns ApplyResult. For ratio constraints, min_pct/max_pct resolve L = pct × baseline_LR from the apply-time DataFrame (live-scoring contract), not the solve-time baseline.
save(path)	Save config + lambdas to JSON. Ratio specs round-trip verbatim.
ApplyOptimiser.load(path)	Load from saved JSON. Rejects unknown keys.

QuoteGridBuilder

Method	Description
append(df)	Add a chunk of quotes.
build()	Finalise and return a QuoteGrid.

SolveResult

Property	Type	Description
converged	bool	Whether the solver converged.
iterations	int	Number of iterations taken.
lambdas	dict[str, float]	Final Lagrange multipliers (shadow prices) per constraint.
total_objective	float	Portfolio-level objective at optimal solution.
total_constraints	dict[str, float]	Portfolio-level constraint totals.
baseline_objective	float	Objective at scenario_value = 1.0.
baseline_constraints	dict[str, float]	Constraints at scenario_value = 1.0.
dataframe	pl.DataFrame	Per-quote results with optimal scenario values.
history	list[dict] | None	Per-iteration convergence records (if record_history=True).
n_quotes	int	Number of quotes in the grid.
n_steps	int	Number of scenario value steps.
scenario_values	list[float]	The scenario value grid.
grid	QuoteGrid	The internal grid (reusable for subsequent solves or apply).

ApplyResult

Property	Type	Description
total_objective	float	Portfolio-level objective.
total_constraints	dict[str, float]	Portfolio-level constraint totals.
baseline_objective	float	Objective at scenario_value = 1.0.
baseline_constraints	dict[str, float]	Constraints at scenario_value = 1.0.
lambdas	dict[str, float]	Applied Lagrange multipliers.
dataframe	pl.DataFrame	Per-quote results with optimal scenario values.

FrontierResult

Property	Type	Description
points	pl.DataFrame	One row per frontier point with threshold_*, total_objective, total_*, lambda_*, iterations, converged, and scenario value statistics (sv_mean, sv_std, sv_min, sv_p5–sv_p95, sv_max, sv_pct_increase, sv_pct_decrease).
n_points	int	Number of frontier points.

RatebookResult

Property	Type	Description
factor_tables	dict[str, dict[str, float]]	Factor name to level-value mapping.
lambdas	dict[str, float]	Final Lagrange multipliers.
total_objective	float	Portfolio-level objective at optimal solution.
total_constraints	dict[str, float]	Portfolio-level constraint totals.
baseline_objective	float	Objective at scenario_value = 1.0.
baseline_constraints	dict[str, float]	Constraints at scenario_value = 1.0.
converged	bool	Whether coordinate descent converged.
cd_iterations	int	Coordinate descent iterations.
clamp_rate	float	Fraction of remappings that hit a grid boundary.
per_factor_results	list[GroupedSolveResult]	Per-factor inner solve results.
save(path)		Save factor tables to a directory (one JSON per factor).
to_rating_entries()	dict[str, pl.DataFrame]	Convert to rating-step DataFrames.

Utility functions

Function	Description
build_grid_from_parquet(path, *, constraint_columns, ...)	Build a QuoteGrid directly from a Parquet file without materialising a DataFrame in Python. Sum constraints only — ratio constraints require a DataFrame.
apply_from_grid(grid, lambdas, constraints)	Single-pass Lagrangian apply on an existing QuoteGrid. Returns ApplyResult. Sum constraints only; ratio constraints raise ValueError (use ApplyOptimiser.apply(df) on a DataFrame instead — the grid path can't carry numerator/denominator columns for linearisation).
frontier_summary(frontier_result, selected_index)	Package a frontier result into MLflow-ready params, metrics, artifacts dicts.

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License

Price Contour is licensed under the GNU Affero General Public License v3.0.