## Quadratic Program (QP) Solver that only depends on NumPy/SciPy?

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I would like students to solve a quadratic program in an assignment without them having to install extra software like cvxopt etc. Is there a python implementation available that only depends on NumPy/SciPy?

I ran across a good solution and wanted to get it out there. There is a python implementation of LOQO in the ELEFANT machine learning toolkit out of NICTA (http://elefant.forge.nicta.com.au as of this posting). Have a look at optimization.intpointsolver. This was coded by Alex Smola, and I've used a C-version of the same code with great success.

**A quadratic program (QP) that only depends on NumPy / SciPy ,** I came across a good solution and wanted to get it there. There is a python LOQO implementation in the ELEFANT machine learning toolkit from NICTA If a generic nonlinear solver is enough, a better answer is that by @ali_m which only depends on Numpy/Scipy – divenex Nov 27 at 16:28 @divenex: the OP didn't ask for a QP solver, they asked for something to solve a QP problem that only depended on numpy / scipy and well, the solvers in mystic essentially only have a numpy dependency

I'm not very familiar with quadratic programming, but I think you can solve this sort of problem just using `scipy.optimize`

's constrained minimization algorithms. Here's an example:

import numpy as np from scipy import optimize from matplotlib import pyplot as plt from mpl_toolkits.mplot3d.axes3d import Axes3D # minimize # F = x[1]^2 + 4x[2]^2 -32x[2] + 64 # subject to: # x[1] + x[2] <= 7 # -x[1] + 2x[2] <= 4 # x[1] >= 0 # x[2] >= 0 # x[2] <= 4 # in matrix notation: # F = (1/2)*x.T*H*x + c*x + c0 # subject to: # Ax <= b # where: # H = [[2, 0], # [0, 8]] # c = [0, -32] # c0 = 64 # A = [[ 1, 1], # [-1, 2], # [-1, 0], # [0, -1], # [0, 1]] # b = [7,4,0,0,4] H = np.array([[2., 0.], [0., 8.]]) c = np.array([0, -32]) c0 = 64 A = np.array([[ 1., 1.], [-1., 2.], [-1., 0.], [0., -1.], [0., 1.]]) b = np.array([7., 4., 0., 0., 4.]) x0 = np.random.randn(2) def loss(x, sign=1.): return sign * (0.5 * np.dot(x.T, np.dot(H, x))+ np.dot(c, x) + c0) def jac(x, sign=1.): return sign * (np.dot(x.T, H) + c) cons = {'type':'ineq', 'fun':lambda x: b - np.dot(A,x), 'jac':lambda x: -A} opt = {'disp':False} def solve(): res_cons = optimize.minimize(loss, x0, jac=jac,constraints=cons, method='SLSQP', options=opt) res_uncons = optimize.minimize(loss, x0, jac=jac, method='SLSQP', options=opt) print '\nConstrained:' print res_cons print '\nUnconstrained:' print res_uncons x1, x2 = res_cons['x'] f = res_cons['fun'] x1_unc, x2_unc = res_uncons['x'] f_unc = res_uncons['fun'] # plotting xgrid = np.mgrid[-2:4:0.1, 1.5:5.5:0.1] xvec = xgrid.reshape(2, -1).T F = np.vstack([loss(xi) for xi in xvec]).reshape(xgrid.shape[1:]) ax = plt.axes(projection='3d') ax.hold(True) ax.plot_surface(xgrid[0], xgrid[1], F, rstride=1, cstride=1, cmap=plt.cm.jet, shade=True, alpha=0.9, linewidth=0) ax.plot3D([x1], [x2], [f], 'og', mec='w', label='Constrained minimum') ax.plot3D([x1_unc], [x2_unc], [f_unc], 'oy', mec='w', label='Unconstrained minimum') ax.legend(fancybox=True, numpoints=1) ax.set_xlabel('x1') ax.set_ylabel('x2') ax.set_zlabel('F')

Output:

Constrained: status: 0 success: True njev: 4 nfev: 4 fun: 7.9999999999997584 x: array([ 2., 3.]) message: 'Optimization terminated successfully.' jac: array([ 4., -8., 0.]) nit: 4 Unconstrained: status: 0 success: True njev: 3 nfev: 5 fun: 0.0 x: array([ -2.66453526e-15, 4.00000000e+00]) message: 'Optimization terminated successfully.' jac: array([ -5.32907052e-15, -3.55271368e-15, 0.00000000e+00]) nit: 3

**Quadprog python,** Quadratic Program (QP) Solver that only depends on NumPy/SciPy? Posted by: admin April 4, 2018 Leave a comment. Questions: I would like students to solve Description This is a simple and light-weight convex quadratic programming (QP) solver. It only uses numpy and it is distributed as a single python file module.

This might be a late answer, but I found `CVXOPT`

- http://cvxopt.org/ - as the commonly used free python library for `Quadratic Programming`

. However, it is not easy to install, as it requires the installation of other dependencies.

**scipy.optimize.minimize,** I would like students to solve a quadratic program in an assignment without them having to install extra software like cvxopt etc. Is there a python implementation Quadratic Program(QP) Solver that only depends on NumPy/SciPy? (6) I would like students to solve a quadratic program in an assignment without them having to install extra software like cvxopt etc. Is there a python implementation available that only depends on NumPy/SciPy?

`mystic`

provides a pure python implementation of nonlinear/non-convex optimization algorithms with advanced constraints functionality that typically is only found in QP solvers. `mystic`

actually provides more robust constraints than most QP solvers. However, if you are looking for optimization algorithmic speed, then the following is not for you. `mystic`

is not slow, but it's pure python as opposed to python bindings to C. If you are looking for flexibility and QP constraints functionality in a nonlinear solver, then you might be interested.

""" Maximize: f = 2*x[0]*x[1] + 2*x[0] - x[0]**2 - 2*x[1]**2 Subject to: -2*x[0] + 2*x[1] <= -2 2*x[0] - 4*x[1] <= 0 x[0]**3 -x[1] == 0 where: 0 <= x[0] <= inf 1 <= x[1] <= inf """ import numpy as np import mystic.symbolic as ms import mystic.solvers as my import mystic.math as mm # generate constraints and penalty for a nonlinear system of equations ieqn = ''' -2*x0 + 2*x1 <= -2 2*x0 - 4*x1 <= 0''' eqn = ''' x0**3 - x1 == 0''' cons = ms.generate_constraint(ms.generate_solvers(ms.simplify(eqn,target='x1'))) pens = ms.generate_penalty(ms.generate_conditions(ieqn), k=1e3) bounds = [(0., None), (1., None)] # get the objective def objective(x, sign=1): x = np.asarray(x) return sign * (2*x[0]*x[1] + 2*x[0] - x[0]**2 - 2*x[1]**2) # solve x0 = np.random.rand(2) sol = my.fmin_powell(objective, x0, constraint=cons, penalty=pens, disp=True, bounds=bounds, gtol=3, ftol=1e-6, full_output=True, args=(-1,)) print 'x* = %s; f(x*) = %s' % (sol[0], -sol[1])

Things to note is that `mystic`

can generically apply LP, QP, and higher order equality and inequality constraints to any given optimizer, not just a special QP solver. Secondly, `mystic`

can digest symbolic math, so the ease of defining/entering the constraints is a bit nicer than working with the matrices and derivatives of functions. `mystic`

depends on `numpy`

, and will use `scipy`

if it is installed (however, `scipy`

is not required). `mystic`

utilizes `sympy`

to handle symbolic constraints, but it's also not required for optimization in general.

Output:

Optimization terminated successfully. Current function value: -2.000000 Iterations: 3 Function evaluations: 103 x* = [ 2. 1.]; f(x*) = 2.0

Get `mystic`

here: https://github.com/uqfoundation

**scipy.optimize.minimize,** Quadratic Programming solvers in Python with a unified API as follows: minimize Z = Quadratic Program(QP) Solver that only depends on NumPy/SciPy? To solve a quadratic program, simply build the matrices that define it and call the solve_qp function: from numpy import array , dot from qpsolvers import solve_qp M = array ([[ 1. , 2. , 0. ], [ - 8.

The qpsolvers package also seems to fit the bill. It only depends on NumPy and can be installed by `pip install qpsolvers`

. Then, you can do:

from numpy import array, dot from qpsolvers import solve_qp M = array([[1., 2., 0.], [-8., 3., 2.], [0., 1., 1.]]) P = dot(M.T, M) # quick way to build a symmetric matrix q = dot(array([3., 2., 3.]), M).reshape((3,)) G = array([[1., 2., 1.], [2., 0., 1.], [-1., 2., -1.]]) h = array([3., 2., -2.]).reshape((3,)) # min. 1/2 x^T P x + q^T x with G x <= h print "QP solution:", solve_qp(P, q, G, h)

You can also try different QP solvers (such as CVXOPT mentioned by Curious) by changing the `solver`

keyword argument, for example `solver='cvxopt'`

or `solver='osqp'`

.

**Cvxopt vs scipy,** Options '3-point' and 'cs' are available only to 'trust-constr'. If jac is a For detailed control, use solver-specific options. optionsdict Depending on the method each iteration may use several function evaluations. dispbool Kraft, D. A software package for sequential quadratic programming. 1988. Tech. The above objective function is convex if and only if P is positive-semide nite, which is the realm we are concerned with. 1 The CVXOPT QP framework expects a problem of the above form, de ned by the pa-

**Quadratic Programming in Python,** If not given, chosen to be one of BFGS, L-BFGS-B, SLSQP, depending if the problem Bounds for variables (only for L-BFGS-B, TNC and SLSQP). This section describes the available solvers that can be selected by the 'method' parameter. [12], (1, 2) Kraft, D. A software package for sequential quadratic programming. Me resulta extraño que una pregunta pidiendo un python implementado QP solver que sólo depende numpy/scipy y no requiere software adicional como cvxopt… tiene una respuesta que recomienda cvxopt y otro (el aceptado respuesta) que recomienda lo que en esencia es mantenido enlaces python para otro idioma (es decir, una implementación de python).

**Docs,** Many binaries depend on numpy-1. non-convex optimization • Unconstrained or CVXOPT is the only other popular, open source linear programming solver with a Quadratic Programming with Python and CVXOPT This guide assumes that The Nelder-Mead solver doesn't support constrained optimization, but there are several others that do. TNC and L-BFGS-B both support only bound constraints (e.g. x[0] >= 0), which should be fine for your case. COBYLA and SLSQP are more flexible, supporting any combination of bounds, equality and inequality-based constraints.

We will now see how to solve quadratic programs in Python using a number of Standard form of quadratic programs A quadratic program (QP) is written in The quadprog module works directly on NumPy arrays so there is no need for type The Toeplitz matrix used to generate inequalities is just an upper-tridiagonal The state of open-source quadratic programming convex optimizers I explore here a few open-source optimizers on a relatively simple problem of finding a good convex subset, but with many constraints: 30104 constraints for essentially 174 variables. My particular problem can be easily expressed in the form of a quadratic programming problem.

##### Comments

- If you could provide some links on what you mean by a quadratic program and maybe an example or two, it would allow more people to answer this question. Please update your question, because I am not too sure what you mean by QP and I might know how to write your program, although I don't know what it requires. Thank you!
- Sorry for not clarifying. QP is a special linear algebra problem, see Wikipedia (en.wikipedia.org/wiki/Quadratic_programming).
- I find it odd that a question asking for a
**python**implemented QP solver that**only**depends on`numpy`

/`scipy`

and**doesn't**require additional software**like cvxopt**… has one answer that recommends`cvxopt`

and another (the accepted answer) that recommends what's essentially unmaintained python bindings to another language (i.e. a non-python implementation). - I don't believe the project is active. The download link is broken, but this link works: elefant.forge.nicta.com.au/download/release/0.4/index.html There's a C++ fork of the project at users.cecs.anu.edu.au/~chteo/BMRM.html, but I don't believe it is active either.
- The links in this answer are broken and the suggested software is not pure Python (+Numpy/Scipy)
- I doubt that this is very efficient. I think an implementation of LOQO: An Interior Point Code for Quadratic Programming (citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.2191) will be faster.
- How hard are the problems you need your students to solve? SLSQP solves my (admittedly rather simple) example in about 1.33msec. It can also handle any combination of bounds, inequality and equality constraints. If your heart is set upon using a particular solver that is optimised for QP then you will probably have to (A) have your students install extra dependencies, or (B) write it yourself.
- Thanks for your follow up. The students should use it to solve an Support Vector Machine problem to compare it to a more efficient algorithm they should implement. It's a convex problem in about 100 variables. I might implement the LOQO, just thought I can't be the first.
- It's worth adding 'jac':(lambda x:-A) to the constraint definition, to make the solver more robust.
- I was trying to implement some basic machine learning algorithms from scratch. SVM was on the todo list but I had no confident to pull it out. After reading your answer, I managed to write a svm of my own (github.com/Sacry/mla_sani/blob/master/mla_sani/supervised/…) and it works pretty as expected. I'm really really appreciated for your answer, thank you very much.
- Well, as you described, it's not easy to install :-) Upvote as my thanks for the suggestion but I think I'll try another options first.
- @JimRaynor I have no problem installing
`cvxopt`

directly with`pip install cvxopt`

in OS X. That's it.`pip`

takes care of everything. And I have installed`cvxopt`

in several machines already. Surely you need to have compilers installed, but that's also straightforward and if you are using`scipy`

you most likely have them already. In case it helps, I use Anaconda as a Python distribution (which is fully free) and installing Anaconda is also straightforward. You don't need admin privileges and there isn't anything you need to config. Just download it, install it, and it's ready to go. - This library was one of the reasons I switched to Anaconda for the ease of managing the dependencies. I just couldn't install it with pip. If you already have Anaconda, use
`conda install -c https://conda.anaconda.org/omnia cvxopt`

and it's done. I'm on Windows 10 and Python 2.7.