Classification of optimization Techniques

Classification of optimization
Technique/methods/
Activity 2
Sheleme Mosisa Feyisa (PhD Candidate, MEng)
shelememosisa@gmail.com
Email:
Phone:+251922484024

Content
 Optimization problem
 Optimization process
 Solution methods of optimization problems
 Optimization software

 An optimization problem seeks to find the largest (the smallest)
value of a quantity (such as maximum revenue or minimum surface
area) given certain limits to a problem.
 An optimization problem can usually be expressed as “find the
maximum (or minimum) value of some quantity Q under a certain
set of given conditions”.
Definition of Optimization problems

Problems that can be modelled and solved by optimization
techniques
 Scheduling Problems (production, airline, etc.)
 Network Design Problems
 Facility Location Problems
 Inventory management
 Transportation Problems
 Minimum spanning tree problem
 Shortest path problem
 Maximum flow problem
 Min-cost flow problem

1. Classical Optimization
 Useful in finding the optimum solution or unconstrained maxima
or minima of continuous and differentiable functions.
 Analytical methods make use of differential calculus in locating
the optimum solution

cont.…
 Have limited scope in practical applications as some of them
involve objective functions which are not continuous and/or
differentiable.
 Basis for developing most of numerical techniques that involved
into advanced techniques more suitable to today’s practical
problem

Three main types of problems can be handled:
 Single Variable functions
 Multivariable functions with no constraints,
 Multiple functions with both equality and inequality
constraints
In problems with equality constraints the LaGrange multiplier method
can be used
If the problem has inequality constraints, the Kuhn-Tucker conditions can
be used to identify the optimum solution

Linear Program (LP)
 studies the case in which the objective function (f ) is linear and the set design
variable space (A) is specified Using only linear equalities and inequalities.
 (P) Easy, fast to solve, convex
2. Numerical Methods
https://stanford.edu/class/ee364a/

Optimization Problem Types
Non-Linear Program (NLP)
 studies the general case in which the objective function or the constraints or
both contain nonlinear parts.
 (P) Convex problems easy to solve
 Non-convex problems harder, not guaranteed to find global optimum

Integer Programs (IP)
 studies linear programs in which some or all variables are constrained to
take on integer values
Quadratic programming
 allows the objective functions to have quadratic terms, while the set (A) must be
specified with linear equalities and inequalities

Stochastic Programming
 studies the case in which some of the constraints depend on random variables
Dynamic programming
 studies the case in which the optimization strategy is based on splitting the
problem into smaller sub-problems.

3. Advanced Methods
 Swarm Intelligence Based Algorithms
 Bio-inspired (not SI-based) algorithms
 Physical and chemistry based algorithms
 others

Solution Methods for Discrete Optimization Problems
 Integer Programming
 Network Algorithms
 Dynamic Programming
 Approximation Algorithms

Flow chart of algorithm Optimization Problems
 An algorithm is a step-by step procedure
to solve a given problem
 A flowchart is a type of diagram that represents
an algorithm or process
 A pseudo code is a compact and informal high-level
description of a program.
https://www.youtube.com/watch?v=vOEN65nm4YU&list=PLG6eP
ePp5vvYVEjRanyndt7ZSqTzillom&index=1

Software for Optimization problems
 EXCEL
 PYTHON
 MATLAB
 .
 .
 https://www.youtube.com/watch?v=ATd0MZQGN7I&list=PPSV

EXCEL
 Microsoft Excel solver is a powerful add-on tool to solve and analyze
optimization problems.
 Solver can be used to adjust parameters in a model to best fit data, increase
profitability of a potential engineering design, or meet some other type of
objective that can be described mathematically in a spread sheet.

Example: problem optimization by EXCEL
This problem has a nonlinear objective that the optimizer attempts to minimize. The variable
values at the optimal solution are subject to (s.t.) both equality (=40) and inequality (>25)
constraints. The product of the four variables must be greater than 25 while the sum of
squares of the variables must also equal 40. In addition, all variables must be between 1 and 5
and the initial guess is x1 = 1, x2 = 5, x3 = 5, and x4 = 1.
𝑶𝒃𝒋𝒆𝒄𝒕𝒊𝒗𝒆: 𝒎𝒊𝒏 𝒙𝟏𝒙𝟒 𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 + 𝒙𝟑
𝐂𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐭𝐬: 𝒙𝟏𝒙𝟐𝒙𝟑𝒙𝟒 ≥ 𝟐𝟓
𝒙𝟏
𝟐 + 𝒙𝟐
𝟐 + 𝒙𝟑
𝟐 + 𝒙𝟒
𝟐=40
𝟏 ≤ 𝒙𝟏,𝒙𝟐, 𝒙𝟑 , 𝒙𝟒 ≤ 𝟓
𝒙𝟎 = (𝟏, 𝟓, 𝟓, 𝟏)

Python can be used to optimize parameters in a model to best fit data, increase
profitability of a potential engineering design, or meet some other type of
objective that can be described mathematically with variables and equations.
Mathematical optimization problems may include equality constraints (e.g. =),
inequality constraints (e.g. <, <=, >, >=), objective functions, algebraic
equations, differential equations, continuous variables, discrete or integer
variables, etc.
Python

Example: Non-linear problem solving by Python
Objective: 𝑥2 − 3 = 0
Constraint:𝑥2 + 𝑦2 = 20
y= 𝑥2
Variable= (x,y)

Reference
 https://apmonitor.com/che263/index.php/Main/ExcelSolver
 https://www.youtube.com/watch?v=vOEN65nm4YU&list=PLG6ePePp5vvYVEjRanyndt7Z
SqTzillom&index=1
 Arora, J. (2012). Introduction to Optimum Design. In Introduction to Optimum
Design. https://doi.org/10.1016/C2009-0-61700-1
 Zemmari, A., & Benois-Pineau, J. (2020). Optimization methods. SpringerBriefs
in Computer Science, 21–33. https://doi.org/10.1007/978-3-030-34376-7_4

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