QP, QCQP and SOCP - Examples

Zubair Khalid

Department of Electrical EngineeringSchool of Science and Engineering

Lahore University of Management Sciences

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

Convex Optimization

Examples1. Robust Quadratic Program

&f: minimize

subject to x -< b Gx == !lJ

I I

~~ Si~tL I I I I I I

p ~o b lc..~ • u ..L I -----· * 'ft C...L t .. :t. "' i t~f't.J e $ P I minimize

I 'il I

* f i:> .SfA ~ec::l L- e A-A...o~ I I

* Pt: y CSLf ·--

minimize lfo(x) == supPEY ~xT Px + qT x + n

subject to Ax -< b Gx == lJJ

6l ; r:-.,.. Y ~ { P., r 2. , ••• , P k j , f" ,..,.,..,.. I "'r a.. bu.~ h t t­

or ti WI; a-~.,_, 0 ~ r ..t~WL,. . i (/

~!"" LUMS A Not-for-Profit University

I I I I I

~ * cJs;.,.J I I I I I I I I I I I I I I I I I I I I I I

I

I ~1"f? :T: 1-L_ .lt+~~+~ -c , L;;:..f,. .. ,/<:...

A 't- ~ ~ I ~~ = ~

vr----------------~~ Q_(_fLf

Examples2. Sum of Norms

p -yYlifli yf'l i z t.. L. ~ ~; ')l + h dl2- ,

?(_ '-~'

YY\ 11\iYn i z e.... ('){., t:)

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ki~1t : c,,~({ :

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

we. cnt pu '$ (fC,; 7..+~ ;/~; 1oA.-w! r

X,~ [~]c R11+ P

A~ [ ~;.

C::. 0 . . . 0

0 .

'

Examples3. Maximum of Norms

--m•'l( I ~; -~.- + h i 112., i::.l, ... , K

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Examples4(a). Formulation of Hyperbolic constraints as Second-order constraints4(b). Maximizing Harmonic Mean

Equivalence:

f..11pu. boLe Cons1UA,..-t :-

~ w ~1~ wff ~ ,"i. '= R~

{_2-~ )l--t (~- ~)· ~(1 +~)Z-4 ,.)~ ~ (j-+ 1:)l-- Q - ~) 1---Z- --

:=-) ~ w._ ~ it 1~ => vJ~~ ~ ~

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Examples4(b). Maximizing Harmonic Mean

z I -1 . a:r..., ' 1.~ ' ' "'- t>i

K

:z -'-

l1

< -t· - l

' ~ (-ti.) ( 'fr7t- b;J

L

II qr!"-'-i - .{:j II~ ~ a. f 'X..- I;, +-ii

vr-------.J' .Soc..

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)