3. 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
4. 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
5.
6.
7. 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
8. 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
9. 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
10. 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/
11. 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
12. Optimization Problem Types
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
13. Optimization Problem Types
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.
14. 3. Advanced Methods
Swarm Intelligence Based Algorithms
Bio-inspired (not SI-based) algorithms
Physical and chemistry based algorithms
others
16. 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
18. 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.
19. 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)
https://www.youtube.com/watch?v=ATd0MZQGN7I&list=PPSV
24. 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