simplinho

simplinho is a standalone revised simplex LP solver with a Python extension module and a small modeling API.

The core solver is header-only C++ in include/simplex/, the Python bindings live in bindings/, and the top-level build produces a simplinho module via pybind11.

Highlights

• Primal revised simplex and dual revised simplex in one solver
• Automatic mode selection with Auto, Primal, and Dual modes
• Direct Phase II attempts with Phase I fallback when a feasible basis is not available
• Explicit handling of lower and upper bounds, including automatic reformulation of nonstandard variable bounds
• Dual bound flipping with Beale-style bound-flip ratio logic
• Presolve passes for row reduction, scaling, singleton elimination, bound tightening, dual fixing, and early infeasible/unbounded detection
• Markowitz LU factorization with rook pivoting and iterative refinement
• Forrest-Tomlin basis updates, with eta-stack updates as an alternative
• Multi-attempt crash basis construction with Markowitz-threshold triangularization, rotating hybrid/sprint/crash_ii/crash_iii heuristics, and presolved warm-start repair
• Adaptive pricing, Devex pricing, and most-negative pricing in both primal and dual simplex
• Degeneracy management, anti-cycling support, and Harris-style ratio tests
• Rich solve outputs: basis data, reduced costs, dual values, shadow prices, internal tableau data, traces, and Farkas certificates
• A higher-level Python modeling layer with algebraic expressions, named variables, named constraints, and post-solve dual access

What The Project Exposes

There are two main ways to use the solver:

1. simplinho.RevisedSimplex Use the low-level matrix API directly: solve(A, b, c, l, u) for min c^T x subject to Ax = b and l <= x <= u.

2. simplinho.Model Use the modeling API with variables, expressions, constraints, and minimize(...) / maximize(...).

The Python module also exposes:

• RevisedSimplexOptions
• SimplexMode
• LPStatus
• LPBasis
• LPBasisStatus
• status_to_string(...)

Solver Features

Algorithms

• Revised simplex implementation with both primal and dual pivoting
• Automatic fallback behavior when a direct solve needs Phase I work
• Support for bounded variables, free variables, shifted variables, and internally added slacks
• Bound-flip logic for dual iterations when enabled through dual_allow_bound_flip

Numerics

• Dense Markowitz LU factorization
• Rook pivoting for more robust pivot selection
• Iterative refinement in both forward and transpose solves
• Configurable pivot tolerances, refactor frequency, and compression thresholds
• Forrest-Tomlin updates by default, or eta-stack updates via basis_update = "eta"

Pricing And Stability

• pricing_rule = "adaptive" for steepest-edge on smaller dual bases and Devex on larger ones, with periodic weight resets
• pricing_rule = "devex" for Devex pricing
• pricing_rule = "most_negative" for a simple reduced-cost rule in primal and most-infeasible-row pricing in dual
• crash_attempts, crash_markowitz_tol, crash_strategy, and repair_mapped_basis tune the initial basis search and post-presolve basis repair
• Degeneracy tracking and anti-cycling support
• Harris-style primal and dual ratio tests
• Optional Bland rule toggle

Presolve And Diagnostics

• Presolve with row/column simplifications, scaling, singleton eliminations, and bound tightening
• Early infeasible and unbounded detection in presolve when possible
• Verbose tracing with optional basis and presolve details
• Final internal tableau, reduced costs, dual values, and shadow prices on the solved internal model
• Farkas certificate output when infeasibility is proven through the dual path

Build

The CMake project currently builds a Python extension named simplinho.

Requirements

• CMake 3.16+
• A C++20 compiler
• Python 3.13 development headers

If local copies of Eigen or pybind11 are not present in a nearby _deps directory, CMake will fetch them with FetchContent.

Build Commands

cmake -S . -B build-local
cmake --build build-local -j

That produces a shared object like build-local/simplinho.cpython-313-...so.

Low-Level Example

import sys
from pathlib import Path

import numpy as np

sys.path.insert(0, str(Path("build-local").resolve()))

import simplinho as simplex

A = np.array([
    [1.0, 1.0],
])
b = np.array([4.0])
c = np.array([1.0, 2.0])
l = np.array([0.0, 0.0])
u = np.array([np.inf, np.inf])

options = simplex.RevisedSimplexOptions()
options.mode = simplex.SimplexMode.Auto
options.pricing_rule = "adaptive"

solver = simplex.RevisedSimplex(options)
solution = solver.solve(A, b, c, l, u)

print(simplex.status_to_string(solution.status))
print("objective:", solution.obj)
print("x:", solution.x)
print("iterations:", solution.iters)
print("stats:", solution.stats.as_dict())
print("log:", solution.log)
print("basis:", solution.basis_state)
print("dual values:", solution.dual_values_internal)
print("reduced costs:", solution.reduced_costs_internal)

You can also reuse that basis as a warm start after bound changes:

basis = solution.basis_state

u2 = np.array([1.5, np.inf])
warm = solver.solve(A, b, c, l, u2, basis=basis)
print(warm.stats.basis_start)

If you keep the same RevisedSimplex instance in Dual mode, it will also automatically reuse the last valid basis on later solve(...) calls with the same row/column dimensions. That makes repeated bound-fixing re-solves behave more like a branch-and-bound node LP loop:

options.mode = simplex.SimplexMode.Dual
solver = simplex.RevisedSimplex(options)

root = solver.solve(A, b, c, l, u)
u_node = np.array([1.5, np.inf])
node = solver.solve(A, b, c, l, u_node)
print(node.stats.basis_start)

Modeling API Example

import sys
from pathlib import Path

sys.path.insert(0, str(Path("build-local").resolve()))

import simplinho as simplex

model = simplex.Model()

x = model.addVar("x", lb=0.0)
y = model.addVar("y", lb=0.0)

c1 = model.addConstr(x <= 3, name="cap_x")
c2 = model.addConstr(y <= 2, name="cap_y")
c3 = model.addConstr(x + 2 * y <= 6, name="mix_cap")

model.maximize(x + y)
solution = model.solve()

print("status   :", simplex.status_to_string(solution.status))
print("objective:", solution.obj)
print("x        :", solution.value(x))
print("y        :", solution.value("y"))
print("all vars :", solution.values)
print("stats    :", solution.stats.as_dict())
print("basis    :", solution.basis)
print("log      :", solution.log)
print("dual c1  :", c1.pi)
print("dual c2  :", c2.pi)
print("dual c3  :", c3.pi)

The modeling layer also supports live edits after construction:

x.obj = 3.0
c3.rhs = 5.0
c3.set_coeff(y, 2.0)

solution = model.reoptimize()

model.deleteConstr(c1)
model.deleteVar(x)
solution = model.reoptimize()

When the model structure stays the same, reoptimize() will automatically try to reuse the last valid basis after edits like bound changes, RHS updates, coefficient changes, or objective changes. You can also pass an explicit saved basis for fast dual re-solves:

basis = solution.basis
x.ub = 1.5
model.options.mode = simplex.SimplexMode.Dual
solution = model.reoptimize(basis)

Useful Outputs

The low-level LPSolution object includes more than just the primal vector:

• status, obj, x, iters
• stats with typed solve telemetry and as_dict()
• basis_state / basis for reusable warm starts
• log_lines and log for verbose solver traces
• basis, basis_internal, nonbasis_internal
• tableau, tableau_rhs, has_internal_tableau
• reduced_costs_internal
• dual_values_internal
• shadow_prices_internal
• trace
• info
• farkas_y, farkas_y_internal, farkas_has_cert
• primal_ray, primal_ray_internal, primal_ray_has_cert

The modeling layer wraps that in ModelSolution, while still exposing the raw solve result as solution.raw.

Repository Layout

• include/simplex/: header-only solver core, presolve, LU, pricing, and degeneracy helpers
• bindings/: pybind11 bindings and modeling API
• src/: reserved for future non-header sources
• simplex_modeling_api_demo.ipynb: notebook showing the modeling API in action
• CMakeLists.txt: standalone build for the Python module