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Articles Related To Mathematica


Now a days, the popularity of scientific computing environments such as IDL, Maple, Mathematica, Matlab and R has increased considerably. Engineer simply feel more productive in such environments. One reason is the simple and clean syntax of command languages in these environments. Another factor is tight integration of simulation and visualization in Maple, R and similar environments you can quickly and conveniently visualize what you just have computed. One problem with the mentioned environments is that they do not work, at least not in an easy way, with other types of numerical software and visualization systems. Many of the environment specific programming languages are also quite simple or primitive. At this point scripting in Python comes in.

 

Python offers the clean and simple syntax of the popular scientific computing environments, the language is very powerful, and there are lots of tools for simulation, visualization, and data analysis programs. Python allows you to build your own Matlab like scientific computing environment, tailored to your specific needs and based on your favorite high performance FORTRAN, C, or C++ codes.

 

Scientific Computing Is More Than Number Crunching: Many computational scientists work with their own numerical software development and realize that much of the work is not only writing computationally intensive number-crunching loops. Very often programming is about shuffling data in and out of different tools, converting one data format to another, extracting numerical data from a text, and administering numerical experiments involving a large number of data files and directories. Such tasks are much faster to accomplish in a language like Python than in FORTRAN, C, C++, and C#.

 

Scripting is particularly attractive for building demos related to teaching or project presentations. Such demos benefit greatly from a GUI, which offers input data specification, calls up a simulation code, and visualizes the results. The simple and intuitive syntax of Python encourages users to modify and extend demos on their own, even if you are newcomers to Python.

 

Python has some clear advantageous over Matlab and similar environments:

  • The Python programming language is more powerful.
  • The Python environment is completely open and made for integration with external tools.
  • A complete toolbox/module with lots of functions and classes can be contained in a single file.
  • Transferring functions as arguments to functions is simpler.
  • Nested, heterogeneous data structures are simple to construct and use.
  • Object-oriented programming is more convenient.
  • Interfacing C, C++, and FORTRAN code is better supported and therefore simpler.
  • Scalar functions work with array arguments to a larger extent (without modifications of arithmetic operators).
  • The source is free and runs on more platforms.

 

How to run Python script

One of the most important skills you need to build as a Python developer is to be able to run Python scripts and code. This is going to be the only way for you to know if your code works as you planned. It’s even the only way of knowing if your code works at all!

 

A Python script is a reusable set of code which is essentially a Python program or a sequence of Python instructions contained in a file. You can run the program by specifying the name of the script file to the interpreter. 

 

This step-by-step will guide you through a series of ways to run Python scripts, depending on your environment, platform, needs, and skills as a programmer. When you try to run Python scripts, a multi-step process begins. 

 

  1. Run Python Scripts Using the Command-Line: A Python interactive session will allow you to write a lot of lines of code, but once you close the session, you lose everything you’ve written. That’s why the usual way of writing Python programs is by using plain text files. By convention, those files will use the .py extension. Open a command-line and type in the word ‘python’ followed by the path to script file and press enter. You’ll see output on your screen.
  2. Run Python Scripts Interactively: It is also possible to run Python scripts and modules from an interactive session. This option offers you a variety of possibilities.
    • Taking advantage of import
    • Use importlib and imp
    • Use runpy.run_module()
    • Hacking exec()
    • Use execfile()
  3. Run Python Scripts from an IDE or a Text Editor: IDE offer the possibility of running your scripts from inside the environment itself. It is common for them to include a Run or Build command, which is usually available from the tool bar or from the main menu.
  4. Run Python Scripts From a File Manager: Running a script by double-clicking on its icon in a file manager is another possible way to run your Python scripts. This option may not be widely used in the development stage, but it may be used when you release your code for production.

 

After you play around with Python on your own or in an online tutorial, I highly recommend to you to write small scripts to strengthen your knowledge. To stay motivated, choose a program that is in some way useful to you, so you can gain insight while figuring out Python. Below are a few ways you can begin to build your expert level in Python script:

 

  • Python Documentation
  • Google and stackoverflow
  • Ask an experience person

 

First, create a very basic version end-to-end. It is much less frustrating than trying to build a super-duper version from scratch. A big plus is that you will have something you can use very fast. Then iterate and add more complex functionality one by one.

 

Second, decompose large problems to smaller ones by introducing functions. Small, cohesive functions are easy to understand, test and debug.

 

Last, but probably the most important thing to keep in mind, is practice makes perfect. Start small, be patient and practice. Happy coding!

Optimization deals with selecting the simplest option among a number of possible choices that are feasible or do not violate constraints. Python is used to optimize parameters in a model to best fit data, increase profitability of a possible engineering style, or meet another form of objective which will be described mathematically with variables and equations.

 

pyOpt is a Python-based package for formulating and solving nonlinear constrained optimization problems in an efficient, reusable and portable manner. Python programming uses object-oriented concepts, such as class inheritance and operator overloading, to maintain a distinct separation between the problem formulation and the optimization approach used to solve the problem.

 

All optimisation downside solvers inherit from the Optimizer abstract category. The category attributes include the solver name (name), an optimizer kind symbol (category), and dictionaries that contain the solver setup parameters (options) and message output settings (informs). The class provides ways to check and alter default solver parameters (getOption, setOption), as well as a method that runs the solver for a given optimisation problem (solve).

 

Optimization solver

A number of constrained optimization solvers are designed to solve the general nonlinear optimization problem.

  1. PSQP: This optimizer is a preconditioned sequential quadratic programming algorithm. This optimizer implements a sequential quadratic programming method with a BFGS variable metric update.
  2. SLSQP: This optimizer is a sequential least squares programming algorithm. SLSQP uses the Han–Powell quasi-Newton method with a BFGS update of the B-matrix and an L1-test function in the step-length algorithm. The optimizer uses a slightly modified version of Lawson and Hanson’s NNLS nonlinear least-squares solver.
  3. CONMIN: This optimizer implements the method of feasible directions. CONMIN solves the nonlinear programming problem by moving from one feasible point to an improved one by choosing at each iteration a feasible direction and step size that improves the objective function.
  4. COBYLA: It is an implementation of Powell’s nonlinear derivative–free constrained optimization that uses a linear approximation approach. The algorithm is a sequential trust–region algorithm that employs linear approximations to the objective and constraint functions.
  5. SOLVOPT: SOLVOPT is a modified version of Shor’s r–algorithm with space dilation to find a local minimum of nonlinear and non–smooth problems.
  6. KSOPT: This code reformulates the constrained problem into an unconstrained one using a composite Kreisselmeier–Steinhauser objective function to create an envelope of the objective function and set of constraints. The envelope function is then optimized using a sequential unconstrained minimization technique.
  7. NSGA2: This optimizer is a non-dominating sorting genetic algorithm that solves non-convex and non-smooth single and multiobjective optimization problems.
  8. ALGENCAN: It solves the general non-linear constrained optimization problem without resorting to the use of matrix manipulations. It uses instead an Augmented Lagrangian approach which is able to solve extremely large problems with moderate computer time.
  9. FILTERSD: It use of a Ritz values approach Linear Constraint Problem solver. Second derivatives and storage of an approximate reduced Hessian matrix is avoided using a limited memory spectral gradient approach based on Ritz values.

 

To solve an optimization problem with pyOpt an optimizer must be initialized. The initialization of one or more optimizers is independent of the initialization of any number of optimization problems. To initialize SLSQP, which is an open-source, sequential least squares programming algorithm that comes as part of the pyOpt package, use:

>>> slsqp = pyOpt.SLSQP()

This initializes an instance of SLSQP with the default options. The setOption method can be used to change any optimizer specific option, for example the internal output flag of SLSQP:

>>> slsqp.setOption('IPRINT', -1)

Now Schittkowski’s constrained problem can be solved using SLSQP and for example, pyOpt’s automatic finite difference for the gradients:

>>> [fstr, xstr, inform] = slsqp(opt_prob,sens_type='FD')

By default, the solution information of an optimizer is also stored in the specific optimization problem. To output solution to the screen one can use:

>>> print opt_prob.solution(0)

 

Example:

The problem is taken from the set of nonlinear programming examples by Hock and Schittkowski and it is defined as

=======================================================================

      min            − x1x2x3

     x1,x2,x3

 

subjected to     x1 + 2x2 + 2x3 − 72 ≤ 0

                        − x1 − 2x2 − 2x3 ≤ 0

 

                        0 ≤ x1 ≤ 42

                        0 ≤ x2 ≤ 42

                        0 ≤ x3 ≤ 42

 

The optimum of this problem is at (x1∗ , x2∗ , x3* ) = (24, 12, 12), with an objective function value of f ∗ = −3456, and constraint values g (x∗ ) = (0, −72).

 

#======================================================================

# Standard Python modules

#======================================================================

import os, sys, time

import pdb

#======================================================================

# Extension modules

#======================================================================

#from pyOpt import *

from pyOpt import Optimization

from pyOpt import PSQP

from pyOpt import SLSQP

from pyOpt import CONMIN

from pyOpt import COBYLA

from pyOpt import SOLVOPT

from pyOpt import KSOPT

from pyOpt import NSGA2

from pyOpt import ALGENCAN

from pyOpt import FILTERSD

 

#======================================================================

def objfunc(x):

   

    f = -x[0]*x[1]*x[2]

    g = [0.0]*2

    g[0] = x[0] + 2.*x[1] + 2.*x[2] - 72.0

    g[1] = -x[0] - 2.*x[1] - 2.*x[2]

   

    fail = 0

    return f,g, fail  

 

#======================================================================

# Instantiate Optimization Problem

opt_prob = Optimization('Hock and Schittkowski Constrained Problem',objfunc)

opt_prob.addVar('x1','c',lower=0.0,upper=42.0,value=10.0)

opt_prob.addVar('x2','c',lower=0.0,upper=42.0,value=10.0)

opt_prob.addVar('x3','c',lower=0.0,upper=42.0,value=10.0)

opt_prob.addObj('f')

opt_prob.addCon('g1','i')

opt_prob.addCon('g2','i')

print opt_prob

 

# Instantiate Optimizer (PSQP) & Solve Problem

psqp = PSQP()

psqp.setOption('IPRINT',0)

psqp(opt_prob,sens_type='FD')

print opt_prob.solution(0)

 

# Instantiate Optimizer (SLSQP) & Solve Problem

slsqp = SLSQP()

slsqp.setOption('IPRINT',-1)

slsqp(opt_prob,sens_type='FD')

print opt_prob.solution(1)

 

# Instantiate Optimizer (CONMIN) & Solve Problem

conmin = CONMIN()

conmin.setOption('IPRINT',0)

conmin(opt_prob,sens_type='CS')

print opt_prob.solution(2)

 

# Instantiate Optimizer (COBYLA) & Solve Problem

cobyla = COBYLA()

cobyla.setOption('IPRINT',0)

cobyla(opt_prob)

print opt_prob.solution(3)

 

# Instantiate Optimizer (SOLVOPT) & Solve Problem

solvopt = SOLVOPT()

solvopt.setOption('iprint',-1)

solvopt(opt_prob,sens_type='FD')

print opt_prob.solution(4)

 

# Instantiate Optimizer (KSOPT) & Solve Problem

ksopt = KSOPT()

ksopt.setOption('IPRINT',0)

ksopt(opt_prob,sens_type='FD')

print opt_prob.solution(5)

 

# Instantiate Optimizer (NSGA2) & Solve Problem

nsga2 = NSGA2()

nsga2.setOption('PrintOut',0)

nsga2(opt_prob)

print opt_prob.solution(6)

 

# Instantiate Optimizer (ALGENCAN) & Solve Problem

algencan = ALGENCAN()

algencan.setOption('iprint',0)

algencan(opt_prob)

print opt_prob.solution(7)

 

# Instantiate Optimizer (FILTERSD) & Solve Problem

filtersd = FILTERSD()

filtersd.setOption('iprint',0)

filtersd(opt_prob)

print opt_prob.solution(8)

 

Solving non-linear global optimization problems could be tedious task sometimes. If the problem is not that complex then general purpose solvers could work. However, as the complexity of problem increases, general purpose global optimizers start to take time. That is when need to create your problem specific fast and direct global optimizer’s need arises.

 

We have an specialized team with PHD holders and coders to design and develop customized global optimizers. If you need help with one, please feel free to send your queries to us.

 

We first understand the problem and data by visualizing it. After that we create a solution to your needs.

 

Please do read to understand what a solver is and how it works - If you want to create your own simple solver. This is not exactly how every solver works, however, this will give you a pretty solid idea of what is a solver and how it is supposed to work.

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