NLPQL

Non-Linear Programming by Quadratic Lagrangian (NLPQL) for Python

nlpql solves the smooth nonlinear programming problem

min  f(x)
     g_j(x)  = 0 ,  j = 1, ..., me
     g_j(x) >= 0 ,  j = me+1, ..., m
     xl <= x <= xu

by a sequential quadratic programming method. A quadratic subproblem is formulated from a quadratic approximation of the Lagrangian and a linearization of the constraints, and the steplength is determined by a line search with respect to an augmented Lagrangian merit function. Data fitting, min-max and multicriteria problems are transformed into programs of this form and solved by the same algorithm.

Available codes

Nonlinear programming

code	purpose
`NLPQLP`	distributed and non-monotone line search, internal restarts
`NLPQL`	the original SQP method
`NLPQLY`	easy-to-use, all derivatives approximated internally
`NLPQLB`	active set strategy for a very large number of constraints
`NLPQLG`	successive restarts for a stepwise improvement of local minima
`NLPQLF`	model functions evaluable only on a convex subset

Data fitting and multicriteria optimization

Every code of this group transforms its problem into a nonlinear program of the form above and solves it with NLPQLP.

code	purpose
`NLPLSQ`	constrained nonlinear least squares
`NLPLSX`	least squares with a very large number of terms
`NLPL1`	sum of absolute values
`NLPINF`	maximum norm data fitting
`NLPMMX`	min-max optimization
`NLPJOB`	multicriteria optimization, sixteen scalar transformations

Quadratic programming

code	purpose
`QL`	convex quadratic programming, solves the subproblem of every SQP step

Quick example

Rosenbrock's post office problem, test problem TP37 of Hock and Schittkowski:

import numpy as np
from nlpql import minimize

res = minimize(
    fun=lambda x: -x[0] * x[1] * x[2],
    x0=[10.0, 10.0, 10.0],
    jac=lambda x: np.array([-x[1] * x[2], -x[0] * x[2], -x[0] * x[1]]),
    bounds=[(0.0, 42.0)] * 3,
    constraints={
        "type": "ineq",
        "fun": lambda x: np.array([
            x[0] + 2 * x[1] + 2 * x[2],
            72 - x[0] - 2 * x[1] - 2 * x[2],
        ]),
        "jac": lambda x: np.array([[1.0, 2.0, 2.0], [-1.0, -2.0, -2.0]]),
    },
)

print(res.x)  # [24. 12. 12.]
print(res.fun)  # -3456.0

Problems with a very large number of constraints are handled by the active set code, which only needs the gradients of a working set:

m = 200_000
y = np.arange(m) / (m - 1.0)

res = minimize(
    fun=lambda x: float(np.sum(np.exp(x))),
    x0=[1.0, 0.5, 0.0],
    method="NLPQLB",
    jac=np.exp,
    bounds=[(-100.0, 100.0)] * 3,
    constraints={
        "type": "ineq",
        "fun": lambda x: x[0] + x[1] * y + x[2] * y**2 - 1.0 / (1.0 + y**2),
        "jac": lambda x: np.column_stack([np.ones_like(y), y, y**2]),
    },
    options={"mw": 420, "acc": 1e-10, "rho": 0.1},
)
print(res.fun)  # 4.3011838

Features

upper and lower bounds are handled separately and stay satisfied
an additional variable prevents inconsistent linearized constraints
non-monotone line search and automatic restarts make the method very stable for noisy function and derivative values
objective and constraints may be evaluated simultaneously at several test points of the line search
initial multipliers and an initial quasi-Newton matrix may be provided
reverse communication, so the model functions can be supplied by an external simulation
no COMMON blocks, no SAVE, no memory allocation, therefore thread-safe and re-entrant
the detailed iteration protocol of the original codes is available through options={"iprint": 4}

Installation

pip install nlpql

Requires Python 3.10+ and NumPy. No external runtime dependencies. See the full installation guide for uv, poetry and source builds.

Documentation

Theory — the mathematics of the implemented algorithms
Quickstart — runnable examples
API Reference — functions, options and termination flags
References — literature citations

Provenance

The algorithms were developed by Prof. Dr. K. Schittkowski and co-authors. The sources in this repository were written from scratch from the published user's guides and papers; no source code of the original implementations was used. Subroutine and argument names follow the published documentation, so existing calling programs can be adapted easily. Reimplementation from the documentation was explicitly permitted by the copyright holder.

License

GNU General Public License v3 (GPLv3) — see LICENSE.

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MD5	`2961fe44e2145f994a7178f87f9fd12e`
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Algorithm	Hash digest
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MD5	`f642d4d969d31da399a54c49f98b251a`
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MD5	`8f7767d66d4ba80f9ba29509e12cf57f`
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Algorithm	Hash digest
SHA256	`09546072b31fb0faa4c7cf60704375ddb0c8b7659aadb5db26c28e87c7397c20`
MD5	`fe55519a59351d926ebf22f0bc63c4f3`
BLAKE2b-256	`0850f32e44c7229fc5040c185cde751336617599a8b2fce7a5385b60c76e1987`

nlpql 0.1.0

NLPQL

Available codes

Nonlinear programming

Data fitting and multicriteria optimization

Quadratic programming

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Algorithm	Hash digest
SHA256	`68bde2fa19036905810daa2ddb3916b46ab9ad55cff1002795d4d7ef3c5d460d`
MD5	`4ef86e55aa2e6373d7eb606236c315d1`
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Algorithm	Hash digest
SHA256	`c5d3d62eb684e95603a431e478f01027e346e644557df7717b069225444e0947`
MD5	`f25ae549a3a241172bda1cf526141437`
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Algorithm	Hash digest
SHA256	`d2fdd5268aac9cc7f454ef15a938705ad56a1d6ea48ebdded00ee7dfed506379`
MD5	`f0d715ed6f3185eee31a566a858fbdec`
BLAKE2b-256	`fa64d37f023553efec7cef8b4ccdf700fab9630092b4f47de76f0aa5967334d4`

Algorithm	Hash digest
SHA256	`664cd8ee03cb8a35d935060b4f2150e04ddb2ce57b47fac68d0791a99afc7b6c`
MD5	`2246cdd59acf186f8249c03ac61999c7`
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Algorithm	Hash digest
SHA256	`b91cb04007e847f7869bc0d396051f266f2749dcafb9057a93d78df7d54e3fdb`
MD5	`eca90631672535f43f787f5b53e6c4fe`
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Algorithm	Hash digest
SHA256	`03b08d3303b0ff057381c64f47d8542dd61abba521b74904783c054e2e7ed4de`
MD5	`7ddbebcbe8b2c98e9bdfbf6026a1f2c4`
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Algorithm	Hash digest
SHA256	`864bba257092aa6b69d6547c7eff79ad104365ed0e14220123f9f839a9014f6d`
MD5	`7e5363b02d2a2a996401e7034e71da8f`
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Algorithm	Hash digest
SHA256	`b7ea8922e4199caf48b85d0821169498975a27bb5889d0712ceaee241f03e7d4`
MD5	`317888cb25eb7d465ba0c8614b8833e4`
BLAKE2b-256	`fb779b58d2f2c4425ab1f640cc9ccd34479cad475359c469be4c19e4d0a0f493`

Algorithm	Hash digest
SHA256	`26ced2b8682b20e829fa7793c9579a4302f581f18243ed99f6f1936e7eb02ebe`
MD5	`85b417262f28c4f6997d9624efd606cd`
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Algorithm	Hash digest
SHA256	`0fce945c6112c26374420deebf69155c44479aea0241697e9810a11a9d130eb1`
MD5	`04b6d009c1cd8d7d328a1b6434d2e556`
BLAKE2b-256	`b43397e724e135faf2741640b2509003c30004d300a3ff21f8c0265501175104`