# How to solve non-linear optimization problems in Python 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*x*x`

`    g = [0.0]*2`

`    g = x + 2.*x + 2.*x - 72.0`

`    g = -x - 2.*x - 2.*x`

`   `

`    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)`

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