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Presentation: Problem No:1.8 B.Charitha Reddy EE19BTECH11001 Electrical Engineering IIT Hyderabad

This document provides a summary of a problem analyzing the convexity of the function f(x)=x1x2. It presents the problem statement, solution showing the condition for convexity, and plots the function. The solution shows that f(x) is neither convex nor concave as the sign of (a1-a2)(b1-b2) cannot be determined. A plot of the function is provided in the code linked in the document.

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Charitha Reddy
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88 views8 pages

Presentation: Problem No:1.8 B.Charitha Reddy EE19BTECH11001 Electrical Engineering IIT Hyderabad

This document provides a summary of a problem analyzing the convexity of the function f(x)=x1x2. It presents the problem statement, solution showing the condition for convexity, and plots the function. The solution shows that f(x) is neither convex nor concave as the sign of (a1-a2)(b1-b2) cannot be determined. A plot of the function is provided in the code linked in the document.

Uploaded by

Charitha Reddy
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Presentation

Problem no:1.8
B.Charitha Reddy
EE19BTECH11001
Electrical Engineering
IIT Hyderabad.

September 1, 2019

1/8
1 Problem

2 Solution
Convexity

3 Plot

2/8
Problem

Problem Statement

Use the relation

f (λx + (1 − λ)y ) 6 λf (x) + (1 − λ)f (y ) (1.1)

where

06λ61 (1.2)

to examine the convexity of f(x)


where f(x)=x1 x2

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Solution Convexity

Convex Optimization

Condition for convexity:

λf (x) + (1 − λ)f (y ) − f (λx + (1 − λ)y ) > 0 (2.1)

where

06λ61 (2.2)

f (x) = x1 x2 (2.3)
   
a a2
Consider two points on the given function 1 ,
b1 b2

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Solution Convexity

λf (x) + (1 − λ)f (y) − f (λx + (1 − λ)y) (2.4)

 
λa1 + (1 − λ)a2
=⇒ λa1 b1 + (1 − λ)a2 b2 − f (2.5)
λb1 + (1 − λ)b2

=⇒ λa1 b1 + (1 − λ)a2 b2 − (λa1 + (1 − λ)a2 )(λb1 + (1 − λ)b2 ) (2.6)

=⇒ λ(1 − λ)a1 b1 + λ(1 − λ)a2 b2 − λ(1 − λ)(a2 b1 + a1 b2 ) (2.7)

=⇒ λ(1 − λ)(a1 b1 + a2 b2 − a2 b1 − a1 b2 ) (2.8)

=⇒ λ(1 − λ)(a1 − a2 )(b1 − b2 ) (2.9)

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Solution Convexity

We know that λ(1 − λ) > 0


But we cannot determine the sign of (a1 − a2 )(b1 − b2 )
So,f(x) is neither convex nor concave .

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Plot

Plot
The code in
https://github.com/bojjacharitha/success/blob/master/optimization.py

plots Fig.

20
10
0
10
20

24
4 2 0 2 4 4 20

eps./figs/bhagavan37 / 8
Plot

Plot

20
10
0
10
20

4 4
2 2
0 0
2 2
4 4

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