Quadratic Optimization Problems

Zubair Khalid

Department of Electrical EngineeringSchool of Science and Engineering

Lahore University of Management Sciences

https://www.zubairkhalid.org/ee563_2020.html

Convex Optimization

Outline

• Quadratic Program (QP)

• Quadratically Constrained Quadratic Program (QCQP)

• Examples

Section 4.4

Quadratic Program (QP)Section 4.4

Quadratic (Convex)

Affine

Interpretation:

Formulation:

Quadratically Constrained Quadratic Program (QCQP)

Interpretation:

Formulation:

represents an Ellipsoid for each i

Examples1. Linear with Quadratic Constraint

T ' ' (A -II)_ c) J

m l fl ; '"' ,. 2. ~ c., I/<.

"'n" , 1' , YW" i 2 t... <::

.s "'~ ( d -:(,.. 7L1A1l~l, Y't -'lz

A f-5++ 'J;:_ -- A L

-'I'L II A - ''2-c_ If 2-

~;. A (} _112 -Jf

-liz_ - 112 -I

YA it\ I~; 2L cIA (} 7(_ -A A c.. - A c.

- --s Ill .hi~- d: t; 'J TJ ~I (.f..,c.I.J~ /fA- ''z c_ It_ I A- '~e-ll~

(3,JJ _..,- ----

Examples2. Least-Squares:

3. Constrained Least-Squares:

Examples4. Linear Program with Random Cost

c_ T~ C Cos-i

A~-< b --...

~')1...:::- -i,

c.. - (...9") + -

E [ e-J ~ E- [ (c - c:) ( c - c:) -rJ =-

I & . f-r ow

1 . L ·' I 7 c_o. )I we.. 11"'1 LKvpo ,~y,,/e..

II\ nc:no; I / I "'-" nJ o ro/1 .,. HS 1 C .7

: --7 rv11 i\ i 1'11 d (._ E [ c I I'\. J ::= c. ,11.

I I I I I I I I I I I I

W\ if'timi2-L -;;; -s -t y ( ....,.. ) {....... ,..._ o VM... c '?<.

ExamplesVtJJt,( c."T~)=- E. (c.T~- E[c'"-))2. :::. E (c.''l{- ~~)( '-~- c~)

A Not-for-Profit University

-=- ~.,- £ (c-C:)(c-z)jlL:::. )(..'Zx

c::_ TIL -t f 1-I Z /(__ A-~ ~ _b

~,(_ ::: ~

6LP

Feedback: Questions or Comments?

Email: zubair.khalid@lums.edu.pk

Slides available at: https://www.zubairkhalid.org/ee563_2020.html(Let me know should you need latex source)