Metadata-Version: 2.4
Name: cvxium
Version: 0.3.0
Summary: Efficient Interior Point Method solvers for convex optimization
Author-email: Bob Wilson <bob.wilson@convexanalytics.com>
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License-File: LICENSE.txt
Keywords: convex,interior-point,newton,optimization
Classifier: Intended Audience :: Science/Research
Classifier: License :: OSI Approved :: Apache Software License
Classifier: Programming Language :: Python :: 3
Classifier: Topic :: Scientific/Engineering :: Mathematics
Requires-Python: <4,>=3.11
Requires-Dist: matplotlib<4,>=3.10.1
Requires-Dist: numpy<3,>=2.2.5
Requires-Dist: scipy<2,>=1.15.2
Description-Content-Type: text/markdown

# Cvxium

[![CI](https://github.com/rwilson4/Cvxium/actions/workflows/ci.yml/badge.svg)](https://github.com/rwilson4/Cvxium/actions/workflows/ci.yml)

Cvxium (pronounced "Calcium") is a Python framework for building fast
Interior Point Method (IPM) solvers for convex optimization problems of
the form:

```
minimize    f0(x)
subject to  A x = b
            fi(x) <= 0,  i = 1, ..., n
```

The framework's distinguishing feature is a clean interface for
exploiting **Hessian structure** to accelerate Newton steps. A generic
solver inverts an n×n dense matrix at each iteration — O(n³). By
encoding the Hessian's structure (diagonal, low-rank update, arrow
sparsity pattern, etc.), the same iteration can run in O(n). This
allows problems to scale to dimensions in the thousands or even higher.

## Why Cvxium?

Most convex optimization needs are well-served by existing tools. Here
is how Cvxium compares:

| Tool               | When to use it                                               | Why not Cvxium                                                                            |
|--------------------|--------------------------------------------------------------|-------------------------------------------------------------------------------------------|
| **scipy.optimize** | General-purpose unconstrained/constrained optimization       | Handles arbitrary problems with minimal setup; no structural speedups needed              |
| **Cvxpy**          | Rapid prototyping of convex programs; standard problem forms | Modeling-layer convenience; dispatches to mature solvers (OSQP, SCS, ECOS) under the hood |
| **Gurobi / CPLEX** | LP, QP, MIP at industrial scale                              | Commercial license; exceptional performance on problems they support, including integers  |
| **OSQP / SCS**     | Large-scale QPs and conic programs                           | Fast first-order methods; good default choice when the problem fits their form            |

**Use Cvxium when:**

- Your problem has a custom convex structure that does not map cleanly
  onto a standard QP/LP/SOCP form — e.g., KL-divergence objectives,
  Huber loss with non-standard constraints, or specialized barrier
  functions.
- The Hessian has exploitable structure (diagonal, diagonal plus
  low-rank, arrow sparsity) that off-the-shelf solvers cannot leverage.
- You need predictable, low-overhead performance without a commercial
  license or a large solver dependency.

**Do not use Cvxium when:**

- Your problem fits a standard form that Cvxpy or Gurobi handles well.
  Those tools are mature, well-tested, and require far less code.
- You need integer variables. Cvxium is strictly continuous convex
  optimization.
- You want a solver you can just call. Cvxium's value is in the
  framework: you implement the math, it handles the IPM loop. If you
  are not willing to derive gradients and Hessians (or have an AI
  agent do this for you), use Cvxpy.

## Installation

```bash
pip install cvxium
```

Or with [uv](https://docs.astral.sh/uv/):

```bash
uv add cvxium
```

## Quick start: ready-made solvers

For the most common problem types, Cvxium ships concrete solvers that
require no subclassing.

### Find x satisfying Ax = b, x ≥ lb

`EqualityWithBoundsSolver` finds a feasible point and, if requested,
minimizes `‖x‖₂²` subject to the constraints:

```python
import numpy as np
from cvxium import EqualityWithBoundsSolver, OptimizationSettings

A = np.random.randn(20, 100)   # p=20 equality constraints, M=100 variables
w_true = np.random.rand(100) + 0.1
b = A @ w_true
lb = 0.01

solver = EqualityWithBoundsSolver(A=A, b=b, lb=lb)

# Feasibility: find any strictly feasible point
result = solver.solve()
assert np.all(result.solution > lb)
assert np.allclose(A @ result.solution, b)

# Optimize: minimize ‖x‖₂²
result = solver.solve(fully_optimize=True)
```

The solver detects infeasibility via the dual certificate and raises
`ProblemCertifiablyInfeasibleError` when the problem has no solution.

### Find x satisfying Ax = b, x ≥ lb, ‖Bx − c‖∞ ≤ ψ

`EqualityWithBoundsAndImbalanceConstraintSolver` adds an L∞ imbalance
constraint, useful when exact balance on a subset of covariates is
required alongside a bound on a larger set:

```python
from cvxium import EqualityWithBoundsAndImbalanceConstraintSolver

solver = EqualityWithBoundsAndImbalanceConstraintSolver(
    A=A, b=b, lb=lb,
    B=B, c=c, psi=0.05,  # ‖Bx − c‖∞ ≤ 0.05
)
result = solver.solve()
```

### Solve a quadratic program with equality and bound constraints

`QuadraticProgramEqualityBoundsSolver` solves:

```
minimize    x^T Q x + c^T x
subject to  A x = b
            x >= xl
```

It accepts optional `Q_vector_multiply` and `Q_solve` callables to
exploit structure in Q:

```python
from cvxium import QuadraticProgramEqualityBoundsSolver

solver = QuadraticProgramEqualityBoundsSolver(Q=Q, c=c, A=A, b=b, xl=xl)
result = solver.solve()

print(result.solution)        # optimal x
print(result.objective_value) # primal objective
print(result.dual_value)      # dual lower bound (duality gap = objective - dual)
print(result.nits)            # outer IPM iterations
print(result.inner_nits)      # Newton iterations per centering step
```

When Q has structure (e.g., diagonal plus rank-one), passing
`Q_vector_multiply` and `Q_solve` callables can yield a further ~12×
speedup over the dense path. See USAGE.md for the full pattern.

## Building a custom solver

Cvxium's real power is its framework for new problem types. You
subclass one of the base classes, implement a handful of methods
(objective, gradient, Hessian multiply, Newton step, dual), and the
IPM loop is handled for you. A library of composable structured linear
system solvers (`solve_diagonal`, `solve_rank_one_update`,
`solve_rank_p_update`, `solve_kkt_system`, etc.) makes it
straightforward to go from a mathematical description of the Hessian
to a fast Newton step.

Full guidance — including worked examples, the class hierarchy, and
the numerical helpers reference — is in USAGE.md and can be retrieved
at runtime:

```python
import cvxium
cvxium.usage()
```

An AI agent can implement a custom solver from an existing codebase
with a prompt like:

> Look at the optimization problem being solved in `<function>`. Learn
> how to use Cvxium by running `python -c 'import cvxium;
> print(cvxium.usage())'`. Make a plan to refactor `<function>` using
> Cvxium.

## Exception hierarchy

```
BacktrackingLineSearchError
├── ConstraintBoundaryError       — step would violate a constraint
├── InvalidDescentDirectionError  — Newton step is not a descent direction
└── SevereCurvatureError          — backtracking condition never satisfied

OptimizationError
├── CenteringStepError            — inner Newton loop failed
└── InteriorPointMethodError      — outer IPM loop failed

ProblemInfeasibleError            — no feasible point exists
ProblemCertifiablyInfeasibleError — dual certificate proves infeasibility
ProblemMarginallyFeasibleError    — feasible set is non-empty but has no interior
```

## Optimization settings

`OptimizationSettings` controls the IPM:

```python
from cvxium import OptimizationSettings

settings = OptimizationSettings(
    barrier_multiplier=10.0,    # factor by which t increases each outer iteration
    outer_tolerance=1e-8,       # duality gap threshold for convergence
    outer_tolerance_soft=None,  # looser threshold for feasibility-only problems
    max_outer_iterations=100,
    max_inner_iterations=100,
    verbose=False,
)
```

## References

- Boyd, Stephen and Vandenberghe, Lieven. *Convex Optimization*.
  Cambridge University Press, 2004.
