Introduction to optimization

Introduction to Optimization
Dr. Varun Kumar
Dr. Varun Kumar (IIIT Surat) Introduction to Optimization 1 / 30

Outlines
1 Introduction to Optimization
2 Motivation for Learning an Optimization:
3 Scope of Optimization:
4 Application of Optimization
5 References

Introduction to Optimization
⇒ Optimization is process for getting the fruitful or desirable outcome
at minimum cost.
⇒ It is a branch of mathematics that can be applied in the various field
of engineering.
⇒ Optimization does not correspond only global maxima and minima.

Un-constrained vs constrained optimization
Un-constrianed optimization
⇒ When there is no constraints or restriction on input (decision
variable).
⇒ In such a scenario, for finding the optimum (maximum or minimum)
value of output (objective function) for observed input is the problem
of un-constrained optimization. Ex-
Q Let y = f (x) = x2 + 3x + 17, find the x, so that y became optimum.
⇒ Here,
1 y = f (x) → Output (objective function) that depends on single
input/decision variable x.
2 x has no restriction, i.e, −∞ < x < ∞ or x ∼ (−∞, ∞).
3 From above expression, y → ∞ at x → −∞ or ∞
4 Hence, optimum → minima not maxima
5 For finding the optimum, we need to calculate the 1st
and 2nd
order
derivative of the above expression.

Constrained optimization
⇒ When there is a constraints or restriction on the input/ decision
variable → x.
⇒ Under given restriction, finding the input (x) and optimum output (y)
are the problem of constrained optimization.
Q Let a given objective function has been given below,
y= f (x) = x2
1 + x2
2 + 2x1 + 3x2 + 17
s.t 2x1 + 3 ≥ 0, x2 ≥ 0, 6x1 + 5x2 = 0
Find optimum y and corresponding x.
⇒ Here, s.t→ subject to
⇒ 2x1 + 3 ≥ 0, x2 ≥ 0, 6x1 + 5x2 = 0 are a constraint or restriction on
x1 and x2.
Note: In most of the real world application, the nature of the
problem lies in constrained optimization.

Prerequisite to this course:
1 Linear algebra
2 Calculus
3 Trigonometry
4 Statistics

Motivation:
Motivation:
1 Does maximization and minimization always be the best solution for
any problem?
2 Can we reduced our search space for getting the optimum output?
3 How do we model the real world problem that could be considered as
optimization problem?
4 What may be the different ways for finding the optimum solution?

Scope of optimization:
1 Super market chain and e-commerce industry
2 Wireless communication
3 Banking or finance
4 Machine learning
5 Civil engineering
6 Mechanical engineering
7 Chemical reaction
8 Production engineering
9 Agriculture
10 Industry 4.0
Many more

Example

1 Optimization in supermarket chain and e-commerce

Ex- A supermarket chain
⇒ For successful operation and better growth
→ Need a better optimization technique
Important supermarket chain
Future group (Big-Bazar)
D-Mart
V-Mart
Key feature and components of supermarket chain:
1 Hundreds of stores all over a country
2 Selling thousands of goods
3 Millions of customers
4 Details of each transaction: date, customer identification code, goods
bought and their amount, total money spent
5 Consumption of gigabytes or more data every day

Continued–
Challenges of Supermarket Chain:
1 To increase the selling product in regular interval
2 Minimum latency in software platform.
3 To follow adaptive pattern of marketing, which changes in time and
by geographic location
Many more

Usage of optimization
Usage of optimization:
1 Stored data (details of transaction): Need a rigorous analysis and
turned into information.
2 This information helps for short and long term prediction.
Note: Behavioral pattern of selling a product has not a fixed algorithm
and does not follow any mathematical expression.

2 Tele-communication:

Continued–
Tele-communication:
Challenges of Tele-communication
Reduction of recurring cost (fuel cost + maintenance)
Improvisation of quality of service (QoS) at low power and less
bandwidth.
QoS → High data rate, minimum latency, better link reliability
Call patterns are analyzed for network optimization and maximizing the
quality of service (QoS).

Key Wireless Resources:

Continued–
Shannon’s data rate
R = B log2

1 +
S
I + N

B → Available bandwidth S → Signal power
I → Interference N → Noise
Multi-objective optimization
1st objective → Maximizing the data rate
2nd objective → Minimizing the available bandwidth
3rd objective → Minimizing the signal power
4th objective → Minimizing the interference+noise
Note: It cannot be removed in any practical system.

Continued–
Wireless resources allocation using optimization: why ??
Network Densfication
Frequency planning
Maximum number of user support in localized area under given
wireless resource constraints.
Many more

Banking and Finance

3 Finance
Finance:
Financial institution (Bank) has a strategically importance in any country.
Business Model of Financial Institution:
Give Take
Output Input → Profit Output Input → Loss

Continued–
Aim of Financial Institution:
Maximizing the profit: Saving account (4%) On loan (14-18%)
Minimizing the loss:
Technology driven network
Reduction of operational expenditure
Earlier prediction of defaulter customer
Developing a mechanism for preventing non-performing assets (NPA)
Finding a potential robust customer using optimization
Source of income.
Current debt on customer.
Banks analyze customer’s past data (repay history of previous loan).
Bank also analyzes the loan amount.
To build models to use in credit applications.
Predict the chance for willful defaulter.
Many more

Optimization in civil engineering

Optimization in civil engineering
⇒ Measurement of tensile strength of a material that has been used for
bridge construction.
⇒ Design is an important aspect for making a bridge under different
geographic location.
⇒ As per the design of bridge, optimum load bearing capacity
measurement.

4 Optimization in manufacturing sector

Manufacturing:
Industrial manufacturing plays a vital role in country’s economy.
Aim of industrial manufacturing:
To increase the production yield.
Minimize the production loss.
Automatic fault detection and correction mechanism.
Develop proper supply chain.
Optimization in industrial manufacturing:
⇒ To develop a mathematical models that are used for
Optimization → Based on the past data
Control
Troubleshooting

5 Optimization for empowering the health infrastructure
Medical
Hospital, doctor and medical equipment collectively empower our health
infrastructure.
Optimization for medical diagnosis:
A simple disease can have multiple reasons. Ex-
Fever
Sex → male/female
Age
Frequency of occurrence
Intake of food supplement
Past history of some disease related to the respective patient, etc
Based on the above input, learning programs helps in medical diagnosis.

6 Optimization in field of ML and AI
Science:
Large amounts of data can only be analyzed fast enough by computers in
the field of
Physics
Astronomy
Biology and many more
World wide web :
It is constantly growing, and searching for relevant information cannot be
done manually.
⇒ Extensive usage of AI in search engine, like Google, Bing.

Optimization in industry 4.0
Industry 4.0
Industry ⇒ Key source of employment generator
Industry ⇒ Cause of socioeconomic upliftment
Quality of successful industrialization
1 Mass production capability
2 Technology driven network
3 Large scale expansion capability
4 Scientific temperament
5 Better future prediction
.

Conclusion
Conclusion
1 Optimization is the part of our day to day routine activity.
2 It is widely accepted in the various field of engineering.
3 Apart from the engineering application, it can also be applied in other
non-engineering application.
4 It gives a direction that how can we achieve maximum gain at
minimum input.
5 It gives a direction that how can we minimize the detrimental effect
under fixed input constraints.

Introduction to optimization

In this document

More Related Content

What's hot

Similar to Introduction to optimization

More from VARUN KUMAR

Recently uploaded

Introduction to optimization