samsara 1.0.1
Reverse communication nonlinear optimization solver for smooth unconstrained objectives
Requires Python: >=3.6

# Samsara

1.  [Documentation](#orgedf932c)
2.  [Tutorial](#orge926c85)
3.  [Credits](#org87931d4)

Samsara is a reverse communication nonlinear optimization solver for smooth unconstrained objectives. Samsara is just an oracle that suggests a step (direction and length) using previous information provided to it by the calling routine. It does not execute function evaluations or gradient calculations, but it does build a model of the function being optimized, based on the steps, gradients and function values (if available) passed to it by the user. This repository contains the Python version of Samsara.

This package is available on PyPI and can be installed with the command

```sh
pip install samsara
```


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## Documentation


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## Tutorial

The file `examples/example.py` contains a running example of how to use Samsara. This document tries to explain how all the moving pieces in the example fit together.

The first step is importing samsara

```python
import matplotlib.pyplot as plt
from numpy import abs, amax, array, load, zeros
from numpy.linalg import norm

from samsara import Samsara, BFGS1_product, Broyden1_product,\
  Broyden2_product, Dogleg_QN, Explicit_TR, cdSY_mat, fdSY_mat
```

In order to use Samsara, we need an objective function to minimize and its gradient. For this example we will use Rosenbrocks's function

```math
f(x, y) = 100(y-x^2)^2+(1-x)^2
```

```python
def Objective(x):
    f = 100*(x[1]-x[0]**2)**2 + (1-x[0])**2
    Df = zeros(2)
    Df[0] = -400*(x[1]-x[0]**2)*x[0] - 2*(1-x[0])
    Df[1] = 200*(x[1]-x[0]**2)
    return f, Df
```

The next step is to create the options dictionary that will be passed to the solver.

```python
ngrad_TOL = 6e-6   # eps**(1/3)
step_TOL = 4e-11   # eps**(2/3)
Maxit = 500
options = {}
options['verbose'] = True
options['alpha_0'] = 5e-12
options['gamma'] = .5
options['c'] = .01
options['beta'] = .9999999
options['eta1'] = .995
options['eta2'] = .8
options['eta3'] = .25
options['maxmem'] = 8
options['tr'] = 1e+15
options['ngrad_TOL'] = ngrad_TOL
options['step_TOL'] = step_TOL
options['Maxit'] = Maxit
options['QN_method'] = BFGS1_product
options['Step_finder'] = Explicit_TR
options['History'] = cdSY_mat
options['update_conditions'] = 'Trust Region'
options['initial_step'] = 1e+5*options['step_TOL']
```

We need to unpack this dictionary as parameters in the constructor of the `Samsara` class.

```python
samsara = Samsara(**options)
```

Now that the `Samsara` object was created we need to provide the initial configuration of the main loop.

```python
xold_vec = array([.5, 2.])
fold, gradfold_vec = Objective(xold_vec)
xnew_vec = None
fnew = None
gradfnew_vec = None
stepsize = 999.
ngradfnew = 999.
it = 0
```

The main loop of the program has to call the oracle in order to obtain a new point candidate. In this example we will accept the candidate unconditionally.

```python
while it < Maxit and ngradfnew > ngrad_TOL and stepsize > step_TOL:
    xnew_vec, xold_vec, fold, gradfold_vec, stepsize =\
	samsara.run(xold_vec, xnew_vec, fold, fnew, gradfold_vec,
		    gradfnew_vec)
    it += 1
    fnew, gradfnew_vec = Objective(xnew_vec)
    ngradfnew = norm(gradfnew_vec)
```

```
First iteration: initialising...
iteration: 0 ; trust region: 4e-06 ; memory: -1
iteration: 1 ; trust region: 40.0 ; memory: 0
iteration: 2 ; trust region: 40.0 ; memory: 1
iteration: 2 ; trust region: 4e-07 ; memory: 0
iteration: 2 ; trust region: 19.193508015823596 ; memory: 0
iteration: 2 ; trust region: 9.596754007911798 ; memory: 0
iteration: 2 ; trust region: 4.798377003955899 ; memory: 0
iteration: 2 ; trust region: 2.3991885019779495 ; memory: 0
iteration: 3 ; trust region: 2.3991885019779495 ; memory: 1
iteration: 4 ; trust region: 0.4894454502747553 ; memory: 2
iteration: 5 ; trust region: 2.4472272513737763 ; memory: 3
iteration: 6 ; trust region: 12.236136256868882 ; memory: 4
iteration: 7 ; trust region: 61.18068128434441 ; memory: 5
iteration: 8 ; trust region: 305.903406421722 ; memory: 6
iteration: 9 ; trust region: 1529.5170321086102 ; memory: 7
iteration: 10 ; trust region: 7647.585160543051 ; memory: 7
iteration: 11 ; trust region: 38237.925802715254 ; memory: 7
iteration: 12 ; trust region: 191189.62901357625 ; memory: 7
iteration: 13 ; trust region: 955948.1450678813 ; memory: 7
iteration: 14 ; trust region: 4779740.725339407 ; memory: 7
iteration: 15 ; trust region: 23898703.626697034 ; memory: 7
iteration: 16 ; trust region: 119493518.13348517 ; memory: 7
iteration: 17 ; trust region: 597467590.6674259 ; memory: 7
in trust region subproblem
iteration: 17 ; trust region: 0.004495650683068379 ; memory: 6
in trust region subproblem
iteration: 17 ; trust region: 0.0004495650683068379 ; memory: 5
in trust region subproblem
iteration: 18 ; trust region: 0.0022478253415341896 ; memory: 6
in trust region subproblem
iteration: 19 ; trust region: 0.011239126707670948 ; memory: 7
iteration: 20 ; trust region: 0.056195633538354745 ; memory: 7
iteration: 21 ; trust region: 0.11239126707670949 ; memory: 7
iteration: 22 ; trust region: 0.22478253415341898 ; memory: 7
```

The last step in using Samsara is to output the results

```python
if stepsize <= options['step_TOL']:
    print('Algorithm stagnated:  stepsize tolerance violated.')
if it >= options['Maxit']:
    print('Algorithm exceeded:  maximum step count violated.')

print('iterations:', it, '; optimal value:', fnew)
```

    iterations: 30 ; optimal value: 1.8348194092063077e-17

and to plot the behavior of the function values.

```python
samsara.plot(fopt=fnew)
plt.savefig('./samsara_results.png')
```

![Plot Results](python/doc/samsara_results.png "Plots")

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## Credits

Development on SAMSARA began in 2007 with funding from the National Science Foundation of the USA, DMS-0712796.

Contributors include:

-   Russell Luke (main author), Institute for Numerical and Applied Mathematics, University of Göttingen

-   Student helpers:
    -   Rachael Bailine (Matlab and Fortran version), University of Delaware
    -   Patrick Rowe (Fortran version), University of Delaware
    -   Brian Rife (Fortran version), University of Delaware
    -   Marco Bedolla (Fortran version), University of Delaware
    -   Benedikt Rascher-Friesenhausen (Python version), University of Göttingen
    -   Titus Pinta, University of Göttingen

Special thanks to Laurence Marks at Northwestern University and Peter Blaha at the Technical University of Vienna who provided much of the inspiration for SAMSARA.

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propagate the contents of its contributor version.

  In the following three paragraphs, a "patent license" is any express
agreement or commitment, however denominated, not to enforce a patent
(such as an express permission to practice a patent or covenant not to
sue for patent infringement).  To "grant" such a patent license to a
party means to make such an agreement or commitment not to enforce a
patent against the party.

  If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone
to copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
available, or (2) arrange to deprive yourself of the benefit of the
patent license for this particular work, or (3) arrange, in a manner
consistent with the requirements of this License, to extend the patent
license to downstream recipients.  "Knowingly relying" means you have
actual knowledge that, but for the patent license, your conveying the
covered work in a country, or your recipient's use of the covered work
in a country, would infringe one or more identifiable patents in that
country that you have reason to believe are valid.

  If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license to some of the parties
receiving the covered work authorizing them to use, propagate, modify
or convey a specific copy of the covered work, then the patent license
you grant is automatically extended to all recipients of the covered
work and works based on it.

  A patent license is "discriminatory" if it does not include within
the scope of its coverage, prohibits the exercise of, or is
conditioned on the non-exercise of one or more of the rights that are
specifically granted under this License.  You may not convey a covered
work if you are a party to an arrangement with a third party that is
in the business of distributing software, under which you make payment
to the third party based on the extent of your activity of conveying
the work, and under which the third party grants, to any of the
parties who would receive the covered work from you, a discriminatory
patent license (a) in connection with copies of the covered work
conveyed by you (or copies made from those copies), or (b) primarily
for and in connection with specific products or compilations that
contain the covered work, unless you entered into that arrangement,
or that patent license was granted, prior to 28 March 2007.

  Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may
otherwise be available to you under applicable patent law.

  12. No Surrender of Others' Freedom.

  If conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License.  If you cannot convey a
covered work so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you may
not convey it at all.  For example, if you agree to terms that obligate you
to collect a royalty for further conveying from those to whom you convey
the Program, the only way you could satisfy both those terms and this
License would be to refrain entirely from conveying the Program.

  13. Use with the GNU Affero General Public License.

  Notwithstanding any other provision of this License, you have
permission to link or combine any covered work with a work licensed
under version 3 of the GNU Affero General Public License into a single
combined work, and to convey the resulting work.  The terms of this
License will continue to apply to the part which is the covered work,
but the special requirements of the GNU Affero General Public License,
section 13, concerning interaction through a network will apply to the
combination as such.

  14. Revised Versions of this License.

  The Free Software Foundation may publish revised and/or new versions of
the GNU General Public License from time to time.  Such new versions will
be similar in spirit to the present version, but may differ in detail to
address new problems or concerns.

  Each version is given a distinguishing version number.  If the
Program specifies that a certain numbered version of the GNU General
Public License "or any later version" applies to it, you have the
option of following the terms and conditions either of that numbered
version or of any later version published by the Free Software
Foundation.  If the Program does not specify a version number of the
GNU General Public License, you may choose any version ever published
by the Free Software Foundation.

  If the Program specifies that a proxy can decide which future
versions of the GNU General Public License can be used, that proxy's
public statement of acceptance of a version permanently authorizes you
to choose that version for the Program.

  Later license versions may give you additional or different
permissions.  However, no additional obligations are imposed on any
author or copyright holder as a result of your choosing to follow a
later version.

  15. Disclaimer of Warranty.

  THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
APPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
IS WITH YOU.  SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

  16. Limitation of Liability.

  IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
SUCH DAMAGES.

  17. Interpretation of Sections 15 and 16.

  If the disclaimer of warranty and limitation of liability provided
above cannot be given local legal effect according to their terms,
reviewing courts shall apply local law that most closely approximates
an absolute waiver of all civil liability in connection with the
Program, unless a warranty or assumption of liability accompanies a
copy of the Program in return for a fee.

                     END OF TERMS AND CONDITIONS

            How to Apply These Terms to Your New Programs

  If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.

  To do so, attach the following notices to the program.  It is safest
to attach them to the start of each source file to most effectively
state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.

    samsara
    Copyright (C) 2017  scientific_practical

    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with this program.  If not, see <http://www.gnu.org/licenses/>.

Also add information on how to contact you by electronic and paper mail.

  If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:

    samsara  Copyright (C) 2017  scientific_practical
    This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
    This is free software, and you are welcome to redistribute it
    under certain conditions; type `show c' for details.

The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License.  Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".

  You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<http://www.gnu.org/licenses/>.

  The GNU General Public License does not permit incorporating your program
into proprietary programs.  If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library.  If this is what you want to do, use the GNU Lesser General
Public License instead of this License.  But first, please read
<http://www.gnu.org/philosophy/why-not-lgpl.html>.