Operations Research 10B: Hessian Matrix, Convex & Concave Functions

Textbooks:
amzn.to/2VgimyJ
amzn.to/2CHalvx
amzn.to/2Svk11k
In this video, I'll talk about Hessian matrix, positive semidefinite matrix, negative semidefinite matrix, and convex and concave functions.
----------------------------------------
Smart Energy Operations Research Lab (SEORL): binghamton.edu/seorl
UA-cam CHANNEL: / yongtwang

[@YongWang]

Hi Guys, please comment and let me know what you think about this Operations Research Open Course. Your feedback is really appreciated. If you enjoy the video, please subscribe and share. All my replies here are only related to the content in my own videos. I am afraid I won't be able to answer other questions. Thanks for your understanding.

[@cheimaasaid3147]

Thank you 🤗

[@ambalikasarkar8325]

at slide timing 6: 30-second orde derivative of x2 should be -2 not that only 2. please check

[@ShinjiCarlos]

@@ambalikasarkar8325, it's right, my friend. :)

[@muhammadseyab9032]

So nice videos sir

[@user-py3xy4px1k]

@@ShinjiCarlos it is -2 * -x ,, so it is 2

[@danmcgloven8169]

Great explanation. Btw, Prof. Ahmad Bazzi provides more insights on convex optimization !

[@mykolalazarenko]

The amazingly simple explanation with great examples! Thank you very much!

[@lynguyen709]

Thank you very much for your detailed explanation and concrete examples! I really appreciate it!

[@210187859]

Very clear explanation of the hessian matrix and some examples. Thank you Mr. Wang

[@YongWang]

Hi Mark, glad it helped!

[@underlecht]

Clear, short explanation. Perfect. Thank you.

[@user-xl7vu3bq1b]

very simple ,very good explanation ,excellent examples . thank you Dr wang

[@YongWang]

+طلعت الخولي Thanks for the comment

[@tstoof]

Really clear explanation with very helpful examples. Thank you very much!

[@FrederikSchuetz]

Very helpful! Thanks a lot for putting the time and effort in for this video!

[@YongWang]

Thanks for the comment, Frederik

[@SheevStalin309]

Thanks a lot for your video, I have an optimisation exam tomorrow morning this was very helpful!

[@YongWang]

Glad it helped, Jordan

[@vipinsagark7355]

superbly explained in an understandable way to all

[@kevingries687]

Thank you for sharing Mr. Wang!

[@YongWang]

Glad to share. Hope you like it.

[@mrpuppydog100]

Super well explained thanks so much for this explanation

[@tsunningwah3471]

十分好的影片

[@nanyang8816]

Very very helpful! Thanks a lot!

[@poonamthakur9280]

Very helpful. Thankyou Mr. Wang!

[@YongWang]

You are welcome, Poonam.

[@mariamtahir3699]

Sir Your concepts are amazing

[@TheHumbernator]

Very well explained, thanks!

[@YongWang]

thanks, humbers

[@hayatullahtayeeq2609]

Thanks so much, I really used it.

[@admnejat7698]

thank you.It is really clear.

[@zurzakne-etra7069]

this was super helpful!

[@Masce16]

Mr Wang, how the determinant condition of Hessian 2x2 (third computation on the last examples) changes in case of nxn symmetric matrix?

[@muhammadseyab9032]

So nice presentation

[@solomon8441]

thank you sir, very informative.

[@gourabchanda2102]

Sir thank you so much! I am a beginner and this is the most lucid explanation that I have ever come across.

[@YongWang]

Thanks

[@user-py3xy4px1k]

@@YongWang I have problem in Matlab and I think you can solve it, please give me your email

[@chukwuebukaapeh3482]

Please, I'd appreciate if you do something on Marquardt Method of forcing the Hessian Matrix to be positive definite. It's something I really need help on

[@upomarahman6654]

Thank you so much

[@FuriouGladiator]

Maaan so helpful tnx alot !

[@_PraveenSingh_]

Is the hessian matrix always symmetric if the 2nd order P.DEs of function is continuous, like always? Is there a way to determine the convexity of a function if the Hessian matrix is not symmetric?

[@orangeraven3]

Thank you for the clear explanation. One precision: on the screen around 2 m 49 s the third partial derivative appears as df/dx1 = 4x3 when in fact it's the derivative with respect to x3, so it should be df/dx3 = 4x3. Otherwise all good!

[@YongWang]

Hi orangeraven3, thank you for pointing out this error. Yes, it should be ∂f/∂x3=4*x3 at 2:45

[@badisanaveen7300]

for finding the given function is convex or not. Is we have to do 2nd order derivatives for objective function is sufficient or we have to consider the constraints also or no need

[@ryanalfeldt1169]

On the 3x3 hessian example, you knew the final z equation was all greater than zero because the coefficients were all positive and the z’s were squared, what if my z’s aren’t all squared but all my coefficients are positive? Is it still positive semi definite?

[@user-is6ps6zb4y]

Thank you so much!!!

[@guliyevshahriyar]

Thank you

[@nitinkumarmittal4369]

Thank you for a clear explanation. I believe, there's a typo at 2:49 for partial derivative of f w.r.t to x_3

[@lakshaydahiya8986]

Thanku very much sir...that was easy

[@YongWang]

You are welcome, Lakshay.

[@sophiehall38]

I finally get it, thanks!

[@YongWang]

Thanks, Sophie

[@xiaoqingzhou2669]

Thank you so much for uploading, the given examples were sooo clear and easy to understand!! May I ask how to determine the function whether its quasiconvex or quasiconcave? Thanks!

[@YongWang]

Hi ZXQ, thanks for the comment. The cases of quasiconvex or quasiconcave functions are not covered in this introductory course. But if you are interested, you may refer to web.mit.edu/14.102/www/notes/lecturenotes1007.pdf

[@mahmoudfaouzichaoubi5855]

Thank you..

[@abdalrhmanabdallah9515]

very useful video

[@muhammadsuleman-bf5oh]

Appreciable

[@venkatmandal4569]

thanx sir...very nice.

[@YongWang]

u r welome, venkat

[@chrisarnold769]

Very much appreciated! Had a hard time understanding my professor.

[@YongWang]

thanks, chris

[@muhammadseyab9032]

Thnaks alot sir

[@YongWang]

Glad you like it

[@openpo191]

thank you! I have a question: how to prove if the function is non convex (or even nonsmooth) function?

[@YongWang]

Hi Open, the definitions of convex and concave functions are discussed in a previous video in this course: ua-cam.com/video/a_gRfwHUlhQ/v-deo.html you may try to prove by contradiction.

[@danielhhoskins]

excellent

[@YongWang]

glad to hear it

[@alirezahosseinzadehkashani1225]

Thanks alot

[@YongWang]

You are welcome, alireza

[@hariomkumawat3214]

Thank you sir, I am from INDIA

[@bandaphilemon397]

Awesome

[@user-gq4pr6xw2m]

thanks！

[@AJ-et3vf]

Awesome video sir! Thank you! How often do non-symmetric hessian matrices occur? It's something of interest to me because I've coded finite-difference approximations to the Hessian matrix in Python. All of the functions I've tested often have symmetrix Hessian matrices. Hope to see one that doesn't have symmetric hessian matrix.

[@sheeevum]

hessians are always symmetric bc mixed partial derivatives are the same when taken in any order

[@pushpasahu5413]

Super

[@YongWang]

thanks, pushpa

[@nhik0803]

how can i solved 4x4?

[@tchanabachir8254]

thanks a lot sir

[@YongWang]

You are welcome, Tchana.

[@adistiimister1135]

Can I know the source or the literature?

[@YongWang]

Hi Adisti, the textbook (if this is what you meant) is introduced in the first video in this course.

[@HridoyKhan-bx3wm]

if All are zero.. then what will be??

[@ss00mm11]

Excellent video. Do you have a video on the KKT conditions?

[@YongWang]

Yes, ss00mm11. It's 10C Nonlinear Convex Programming & KKT Conditions in this course playlist (ua-cam.com/video/pA-xwiwyBz4/v-deo.html)

[@ss00mm11]

Thank you!

[@YongWang]

@@ss00mm11 you are welcome

[@Fabrishio]

d2f/dx1x2 = 0 no?

[@Fabrishio]

@ 6:35

[@YongWang]

hi fabrishio, ∂2f/(∂x1)(∂x2)=-2

[@VipinBalakrishnan]

There is a mistake at 2:49 df/dx3=4x3

[@rohithgopal453]

its correct only......df/dx3=4

[@YongWang]

Hi Vipin and rohiith, thank you for pointing out this error. Yes, it should be ∂f/∂x3=4*x3 at 2:45