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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.

Python and Java both lay claims to being among the top five most popular programming languages at any given time, with Java usually just ahead of Python. However, Python’s popularity is growing at a tremendous rate, and Python overtake Java in 2018.

 

Python is a high-level, interpreted, interactive and object-oriented scripting language. Python was designed to be highly readable which uses English keywords frequently whereas other languages use punctuation and it has fewer syntactical constructions than other languages.

 

Java and Python have many similarities. Both languages have strong cross-platform support and extensive standard libraries. They both treat (nearly) everything as objects. Both languages compile to bytecode, but Python is (usually) compiled at runtime.

 

Python versus Java:

  1. Java language is more about syntax, if one can forget to add curly braces or semicolon in the end then this will show error as your output. But there is nothing like that with python there is no need of semicolon and curly braces in the end but python follows indentation process so that it will make your code readable.
  2. Java programming is statically typed means that one has to explicitly mention the data type of variable. If datatype (int, float, double, character) isn’t mentioned then the error can occur in program. Python is dynamically typed means if one has directly assigned a value to a variable at the runtime then it'll assume data type.
  3. Java codes are more complex than python codes. Try to write a hello world program in both then you'll observe the complexness of the code, four lines of code in Java and same hello world program in python are of 1-line code.
  4. Python has a large and robust standard library makes Python score over other programming languages. The standard library allows you to choose from a wide range of modules according to your precise needs. Each module further enables you to add functionality to the Python application without writing additional code.
  5. Python is an open source programing language, Python helps you to curtail software system development cost significantly. You’ll even use many open source Python frameworks, libraries and development tools to curtail development time without increasing development cost. You even have choice to select from a wide range of open source Python frameworks and development tools per your precise needs.
  6. Interpreted, with tools like IDLE, you can also interpret Python instead of compiling it. While this reduces the program length, and boosts productivity, it also results in slower overall execution.
  7. Python is considered to be the most favorable language for Machine Learning, Artificial Intelligence, IoT and much more.
  8. Python has a more unified support model than Java for the first time, and open source developers are focusing their efforts on the latest version of the language.
  9. After working on large projects in both languages, I feel secure saying that Python’s syntax is more concise than Java’s. It’s easier to get up and running quickly with a new project in Python than it is in Java.
  10. The most two popular frameworks for Python are Django and Flask. Flask is a micro web framework, it gives you the basic functionalities you’d need like routing requests without much overhead. Django is a more featured option and can help you build a powerful backend while capitalizing on efficiency and security, Django is equipped with a powerful ORM layer which facilitates dealing databases and performing different operations on the data.

NLP is a branch of data science that consists of systematic processes for analyzing, understanding, and deriving information from the text information in a smart and efficient manner. By utilizing NLP and its parts, one can organize the massive chunks of text information, perform various automated tasks and solve a wide range of issues like – automatic summarization, machine translation, named entity recognition, relationship extraction, sentiment analysis, speech recognition, and topic segmentation etc.

 

NLTK (Natural Language Toolkit) is a leading platform for building Python programs to work with human language data. It provides easy-to-use interfaces to lexical resources like WordNet, along with a collection of text processing libraries for classification, tokenization, stemming, and tagging, parsing, and semantic reasoning, wrappers for industrial-strength NLP libraries.

 

NLTK has been called “a wonderful tool for teaching and working in, computational linguistics using Python,” and “an amazing library to play with natural language.”

 

Downloading and installing NLTK

  1. Install NLTK: run pip install nltk
  2. Test installation: run python then type import nltk and run nltk.download() and download all packages.

 

Pre-Processing with NLTK

The main issue with text data is that it's all in text format. However, the Machine learning algorithms need some variety of numerical feature vector so as to perform the task. Thus before we have a tendency to begin with any NLP project we'd like to pre-process it to form it ideal for working. Basic text pre-processing includes:

 

  • Converting the whole text into uppercase or lowercase, in order that the algorithm doesn't treat the same words completely different in several cases.
  • Tokenization: Process of converting the normal text strings into a list of tokens i.e. words that we actually want. The NLTK data package includes a pre-trained Punkt tokenizer for English.

 

           import nltk

           from nltk.tokenize import word_tokenize

           text = "God is Great! I won a lottery."

           print(word_tokenize(text))

           Output: ['God', 'is', 'Great', '!', 'I', 'won', 'a', 'lottery', '.']

 

  • Noise removal: Process of removing everything that isn’t in a standard number or letter.
  • Stop word removal: A stop word is a commonly used word (such as “the”, “a”, “an”, “in”). We would not want these words or taking up valuable processing time. For this, we can remove them easily, by storing a list of words that you consider to be stop words. NLTK (Natural Language Toolkit) in python has a list of stopwords stored in sixteen different languages. You can find them in the nltk_data directory.  home/Saad/nltk_data/corpora/stopwords is the directory address.

           import nltk

           from nltk.corpus import stopwords

           set(stopwords.words('english'))

 

  • Stemming: Stemming is the process of reducing the words to its root form. Example if we were to stem the following words: “Connects”, “Connecting”, “Connected”, “and Connection”, the result would be a single word “Connect”.

           # import these modules

           from nltk.stem import PorterStemmer

           from nltk.tokenize import word_tokenize   

           ps = PorterStemmer()  

           # choose some words to be stemmed

           words = ["Connect", "Connects", “Connected”, "Connecting", "Connection", "Connections"]

 

           for w in words:

           print(w, " : ", ps.stem(w)) 

 

  • Lemmatization: Lemmatization is the process of grouping along the various inflected forms of a word in order that they may be analyzed as a single item. Lemmatization is similar to stemming but it brings context to the words. Therefore it links words with similar meaning to one word.

           # import these modules

           from nltk.stem import WordNetLemmatizer  

           lemmatizer = WordNetLemmatizer()  

           print("rocks :", lemmatizer.lemmatize("rocks"))

           print("corpora :", lemmatizer.lemmatize("corpora"))  

           # a denotes adjective in "pos"

          print("better :", lemmatizer.lemmatize("better", pos ="a"))

 

          -> rocks : rock

          -> corpora : corpus

          -> better : good

 

Now we need to transform text into a meaningful vector array. This vector array is a representation of text that describes the occurrence of words within a document. For example, if our dictionary contains the words {Learning, is, the, not, great}, and we want to vectorize the text “Learning is great”, we would have the following vector: (1, 1, 0, 0, 1). A problem is that extremely frequent words begin to dominate within the document (e.g. larger score), however might not contain as much informational content. Also, it will offer additional weight to longer documents than shorter documents.

 

One approach is to rescale the frequency of words or the scores for frequent words called Term Frequency-Inverse Document Frequency.

 

  • Term Frequency: is a scoring of the frequency of the word in the current document.

           TF = (Number of times term t appears in a document)/ (Number of terms in the document)

 

  • Inverse Document Frequency: It is a scoring of how rare the word is across documents.

           IDF = 1+log(N/n), where, N is the number of documents and n is the number of documents a term t has appeared in.

 

           Tf-idf weight is a weight often used in information retrieval and text mining.

           Tf-IDF can be implemented in scikit learn as:

 

           from sklearn.feature_extraction.text import TfidfVectorizer

           corpus = [

           ...     'This is the first document.’

           ...     'This document is the second document.’

           ...     'And this is the third one.’

           ...     'Is this the first document?',]

           >>> vectorizer = TfidfVectorizer()

           >>> X = vectorizer.fit_transform(corpus)

           >>> print(vectorizer.get_feature_names())

           ['and', 'document', 'first', 'is', 'one', 'second', 'the', 'third', 'this']

           >>> print(X.shape)

           (4, 9)

 

  • Cosine similarity: TF-IDF is a transformation applied to texts to get two real-valued vectors in vector space. We can then obtain the Cosine similarity of any pair of vectors by taking their dot product and dividing that by the product of their norms. That yields the cosine of the angle between the vectors. Cosine similarity is a measure of similarity between two non-zero vectors.

           Cosine Similarity (d1, d2) =  Dot product(d1, d2) / ||d1|| * ||d2||

 

          import numpy as np

          from sklearn.metrics.pairwise import cosine_similarity

          # vectors

          a = np.array([1,2,3])

          b = np.array([1,1,4])

          # manually compute cosine similarity

          dot = np.dot(a, b)

          norma = np.linalg.norm(a)

          normb = np.linalg.norm(b)

          cos = dot / (norma * normb)

 

After completion of cosine similarity matric we perform algorithmic operation on it for Document similarity calculation, sentiment analysis, topic segmentation etc.

 

I have done my best to make the article simple and interesting for you, hope you found it useful and interesting too.

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