Interior Point Method for Optimization

[@hoytvolker3]

I came here through the chapters of your book, keep up the good work!

[@charleszhu2945]

I like your examples! They make hard things much easier! Thanks a lot!

[@srinivasd3778]

Great Video... I have been looking for it for a long time. Thank you!

[@Mohammad-fv1zb]

super courses,tutorial and website for students in every place in the world specially for who they have no access to courses like this.

[@asdfqwer2988]

2:45 why is x greater than 0?

[@apm]

That is a standard form for any inequality constraint. It could be f(z) > g(z) that is converted to x = f(z)-g(z) and x>0. More info is at apmonitor.com/me575/index.php/Main/InteriorPointMethod

[@asdfqwer2988]

@@apm Thank you very much, Sir. Now, I have three more questions. (1)In 10:12, is -\mu X_k^{-1}e missing in the first element of the vector b, which is under the blue word 'linear system'? (2)Since we have already found KKT solution with Newton-Raphson method, why do we still need step length? I think the KKT solution of the barrier problem is the optimal solution of the barrier problem. (3)Since the optimal solution is obtained when \mu is small enough, why not choose a small enough \mu directly, such as 10^-10, so that the problem can be solved much faster?

[@apm]

@@asdfqwer2988 the equation is correct. That term is brought over to the other side of the equation with Sigma. See equation 11 in this paper: cepac.cheme.cmu.edu/pasilectures/biegler/ipopt.pdf

[@asdfqwer2988]

@@apm Thank you very much. The right side of equation 11 in the paper is the derivative of the barrier function with respect to x, but the equation in the vedio is the derivative of the f with respect to x.

[@apm]

@@asdfqwer2988 I recommend the paper as the correct source.

[@paektian]

Thank you. I hope you can clarify my doubt: Do we iterate the algorithm until convergence for a fixed mu, and then run again the whole thing for a smaller mu? Or do we update mu as well upon at each iteration of x, lambda, and z? It is not clear to me looking at the algorithm flow chart (12:28) when to update mu. Thank you.

[@apm]

Yes, mu decreases and then the iterations start again to solve the sub-problem. Check out this webpage for additional details and a paper citation and PDF: apmonitor.com/me575/index.php/Main/InteriorPointMethod

[@alexisdasiukevich5417]

Where does the "x >= 0" come from at 2:45?

[@apm]

That's part of the general form for Interior Point solvers. You can translate any general optimization problem into that form with slack variables and rearrangement.

[@lukenuculaj5275]

Hello, great video! This helped me get comfortable with the Interior Point Method. Question though: in the scenario where there are inequality constraints that require the incorporation of slack variables, would we have to change the objective barrier function to include slack variable "s"? If so, how?

[@apm]

It becomes one of the inequality constraints (x>0). See apmonitor.com/me575/index.php/Main/InteriorPointMethod and apmonitor.com/wiki/index.php/Main/SlackVariables

[@yunjoonjung7594]

This is a great video. Thank you so much for posting it.

[@apm]

Thanks for you Stack Overflow questions as well!

[@snakesnake9899]

Thanks for your video! For the graph @ around 7:10, when mu is 1, 2, 5, and 10, my calculated x values that minimize the augmented obj function are 0.366, 0.618, 1.158, and 1.791. I just let the 1st derivative to be zero and solve the equation because the aug obj function is convex. They seem to be a little bit different to your color-coded graph. What could be the reason? Thanks again!

[@apm]

How different were the values? It may just be a numerical solution issue.

[@shubhambhoir6753]

Great tutorial. Thanks so much !!!

[@matthewjames7513]

Thanks for the video! How good are barrier functions? Are there other more accurate aways to incorporate inequality constraints?

[@apm]

There are other methods such as Active Set SQP. See the KKT condition exercises at apmonitor.com/me575

[@sammykmfmaths7468]

i an applying in my Phd research ... thank you

[@nahuelpiguillem2949]

thank u very much, just right to the bones and clear

[@KaramAbuGhalieh]

Hello again , in the slide where we took the derivative of the barrier problem @7:37 , what is the c(x) term, how does it appear and is it the same as constraints we defined? thanks in advance

[@apm]

+Karam AbuGhalieh, c(x) are the constraints that were given in the original optimization statement. More information on the KKT conditions is here: apmonitor.com/me575/index.php/Main/KuhnTucker

[@9553954961]

in 11:59 , what are z(L,0) and z(U,0) for initializing lambda?

[@apm]

Those are upper and lower Lagrange multipliers for the inequality constraints.

[@abdullahimohammad9513]

Can this method be applied to a nonconvex constrained optimization problem?

[@apm]

Yes, but it will only find a local minimum.

[@ThiagoSoares-zm1yx]

At the page 14 you explain how to initialize the variable lambda (solving a linear system) but in the rigth-hand side there are matrices Z_L,0 and Z_U,0. What are these matrices?

[@apm]

The IPOPT paper has a good explanation of this matrices. They are diagonal matrices with z_i = mu/x_i

[@rafaelfernandomacuriorella5861]

These appear when we have a lower and upper bound for the constraint of x.

[@cvanaret]

Hi! I'm familiar with the implementation of IPOPT, but hadn't heard of APOPT and BPOPT. Is there a technical report that I could read that explains their differences wrt IPOPT?

[@apm]

Sure, here are some references in the Wikipedia article: en.wikipedia.org/wiki/APOPT

[@ajpenner34]

Very good introduction to this topic, I will definitely be going to the course page you suggested

[@TheLewischen]

Amazing tutorial!!!

[@apm]

Thanks, check out more optimization content at apmonitor.com/me575

[@beatlekim]

You saved my life!

[@Effesianable]

Thank you for the great video! You seem to use the words 'interior point methods' and 'barrier methods' interchangeably. Are these methods exactly the same (i.e. synonyms) or does one subsume the other? Thx

[@apm]

The barrier term is mu ln(x). The interior point method refers to the solution method where the barrier term is included with the objective function. As the video shows, we don't solve the barrier function in the original form but instead transform it into a form that can be solved with sparse symmetric linear solvers to find a new search direction. Interior point method is the more commonly used term for this approach.

[@Effesianable]

I see, thank you very much for the clarification!

[@alielectricalelectronicsan2092]

@@apm Respected sir i watch a lot off videos on youtube And you is the first one how completly describe the interor point method . You is the first one how want. To get every person and people knowledge properly from your video. Now i am start working on your this example. As my assignment on this. The assignment is write a code on matlab on interor point method. I am a beginer of matlab but i know what is linear program and how to solve minimize and maximize problems in the matlab. Now i am working on my second assignment as i mention above. Write a code on matlab on interor point method on matlab. Now i watch your videos but i am confuse . Why. 1 ) = i am not know what is our question where from we start . Beacuse there are 2 videos that you are posting with the same name interor point method. One video is a file type in which you teach uss. 2 )= secondly i see another video in which you write some thing with green pointer on the black borad . My mind say i think you may explain in this video according to the first video in which you shows data with matrix as written in black words i think you provide the solution of this black video questions with green video with expand properly. 3 ) = The third is you also says to some peopels to check and seen barier example 1. Baries function 1 Barrier function 2 and other. 4 )= i am not see where is its matlab code that you written sir in the video i see just the out puts and not know where i get from this complet code. Sir in all analysis i want to know and My Main point is that i want to know If a function is given us as f min ( ) . Or. f max ( ). Then what are the main steps then we further to solve this problem with interor point method way. Step 1 , step 2 ,step 3. My means that the main basic steps. ??. Thanks sir i hope you will guide me with best wayy and i know you answer of her each student with best way. Sir one thing that i am saying to you is you can not provide your email adress this is the drawback of this chanel .as i want to write this my all question that i want i asking to you Write with small mobile pad for comment is difcult for me. Thanks i hope you will provide me information and help with best way. Thanks . I pray for you to live long 🙏 And spread your knowledge for world students for long time thanks again.

[@apm]

@@alielectricalelectronicsan2092 Here is example code: apmonitor.com/me575/index.php/Main/InteriorPointMethod Best wishes on your assignment.

[@abdullahimohammad9513]

Thank you for this video, you have really saved my ass. What does 'n' stand for in the barrier expression?

[@apm]

'n' is the number of variables

[@abdullahimohammad9513]

@@apm Thank you very much

[@sephgeodynamics9246]

Awesome, thank you sir!

[@juliasim1691]

Prof. Hedengren ¿Could you please mention how ipopt method is compared with Mosek, KNITRO or Gurobi? Thanks in advance.

[@apm]

+Simulation & Control There is a survey of the capabilities for different solvers on Wikipedia: en.wikipedia.org/wiki/MOSEK (see bottom). One of the main differences is that IPOPT solves general nonlinear optimization problems while Mosek and Gurobi solve Mixed Integer Linear Programming problems and have support for some quadratic objectives / constraints as well. KNITRO is a collection of 3 nonlinear programming solvers (2 Interior Point, 1 Active Set) and it also supports Mixed Integer solutions as well. I'm developing two solvers: APOPT (active set) and BPOPT (interior point) for MINLP problems as well. To decide which solver is best for your application, I'd recommend several benchmark studies like are shown here: www.mdpi.com/2227-9717/3/3/701/html (see Figure 10).

[@nuraffikah6409]

Hello Sir, I want to ask what is the difference between the exterior point method and interior point method?

[@apm]

I haven't heard of an exterior point method. Could you give additional details? Interior point methods are named because the algorithm keeps the variables within the interior of the inequality constraints.

[@nuraffikah6409]

APMonitor.com is it interior point method also known as ( barrier method) and exterior function method known as penalty method.

[@apm]

Thanks for your explanation. I just hadn't heard of penalty methods called exterior point methods.

[@nuraffikah6409]

APMonitor.com thank you sir for your respond.

In all the examples, because you include the condition for x is greater than or equal to 0, thus using ln(x) has no problem. What happens if the region for x contains both negative and positive values? Which function can be used for both negative and positive x's?

I got the answer after watching your example here: kzbin.info/www/bejne/pYfUoZSQd5lrm68

[@9553954961]

how are we initializing the value of \mu and decrease it for each iteration? here you assigned 1 as initial value , how it would change over each iteration? thanks

[@apm]

The initial barrier parameter value of \mu depends on whether you would like a strong contribution from the barrier term or a small contribution initially. For problems where you have a good initial guess, you may want to choose a smaller value. If you don't have a good initial guess, it can make the initial iterations less productive because the problem is more ill-conditioned.

[@ziaurahman1891]

what is the difference between barrier and trusted region

[@apm]

A barrier method adds a turn to the objective function to stay away from constraints. I trust region method modifies the line search to stay within a certain trusted region for the next step. Both methods have advantages and disadvantages.

[@millamulisha]

Do you know of any guaranteed, or even reliable, closed formulae for 'mu' and 'alpha' as functions of the iteration step which converge for all, say, quadratic programs or similarly for other classes of programming problems? Thanks in advance! :)

[@apm]

+mattmilladb8 I haven't seen this type of guarantee in closed form. Most solvers are adaptive on those parameters because every problem is different. You could reduce the mu value to approach zero on the first step but that would make the linear system for the KKT conditions ill-conditioned. Some of the papers related to IPOPT (one of the best interior point solvers) are given here: projects.coin-or.org/Ipopt/wiki/IpoptPapers

[@millamulisha]

+APMonitor.com Thank you for the reference. I only ask because you'd need either a constant step size or one which is a function of the iteration you're on to prove convergence or for complexity bounds. For instance in Karmarkar's Projective Scaling Method the 'alpha' needs to be ~1/3 to prove that the algorithm operates with polynomial time complexity less than either the simplex method or affine scaling. Would really be helpful to have a function like mu(k) or alpha(k) where k is the index referencing the current iteration (given that one expects mu to at least change as one iterates) even if either the functions works only within a subset of programming problems. That said I really found the video to be quite helpful, so thanks again! :)

[@millamulisha]

+APMonitor.com Not to barrage you with questions but do you think an application of Nesterov's momentum trick (Nesterov's Accelerated Gradient Descent) would speed up the convergence of the naive Newton-Rhapson method in this implementation? Thanks again! :)

[@apm]

Nesterov's momentum trick may help if you are using a steepest descent method - it seems to be common in fitting neural network models. In most Interior Point solvers there are two methods to "accelerate" convergence, especially when you are close to the optimal solution. The first is to use exact 2nd derivatives from automatic differentiation (versus a limited memory BFGS method). The second method is the second order correction that comes with checking convergence criteria at the new trial point.

[@smarteeful]

I just went through this video and checked the source code. Can you please tell me how can I create/define my own problem without connecting to apmonitor server?

[@apm]

+Muhammad Bilal, you can download and install the APMonitor server to your own computer (available in Linux or Windows). apmonitor.com/wiki/index.php/Main/APMonitorServer Modify the source code to point to 127.0.0.1 (or the local Intranet address where you have it installed) instead of byu.apmonitor.com

You're awesome!

[@cusc11]

Do all barrier function use ln(x) or is it possible to use other functions? For example, (1/x) is not defined at 0 and as the solver approached 0, there would be a sharp slope. Is ln(x) used because it not only has the asymptote as x approaches 0, but also is undefined for x

[@apm]

+Matt Anderson there are other barrier functions but the nice thing about ln (x) is that it makes a linear set of KKT conditions for Quadratic Programming problems.

[@kumilan75]

amazing

[@mohammadbeitsadi]

This is gonna sound really stupid, but I don't understand where x>=0 comes from. It is not included in the initial problem, why would you add a constraint like that yourself?

[@apm]

+mohammad beit sadi, this derivation of the interior point method need bounds even if they are +infty to -infty. You can convert any inequality to x>0 such as y>z becomes x=y-z and x>0. In practice, the best solvers such as IPOPT and BPOPT do clever things to improve the efficiency of the solution for large scale problems. See apmonitor.com/me575/index.php/Main/InteriorPointMethod for example code in MATLAB (BPOPT).

[@mohammadbeitsadi]

ah, I see. had look at the example on your website and that made things clear. Thanks for your quick response.

[@prateekparihar9319]

sir can you please explain the example 3rd the method and also the programme...

[@apm]

Please see apmonitor.com/me575/index.php/Main/InteriorPointMethod for all of the implementation details.

[@houdalmayahi3538]

Thank you

[@affyjusbron503]

Hello,sir may I know how to find the solution for Penalty method using matlab

[@apm]

I've posted Matlab code for the barrier method or interior Point method at the following link apmonitor.com/me575/index.php/Main/InteriorPointMethod It is the BPOPT solver. You can also use the fmimcon solver that has an interior point option. apmonitor.com/che263/index.php/Main/MatlabOptimization

[@affyjusbron503]

APMonitor.com thank you for reply sir

[@prateekparihar9319]

can you please send me the theory of interior point method with matlab program and algorithm

[@apm]

+prateek parihar, here is the theory: cepac.cheme.cmu.edu/pasilectures/biegler/ipopt.pdf and MATLAB code: apmonitor.com/me575/index.php/Main/InteriorPointMethod

[@prateekparihar9319]

APMonitor.com thank you..

[@JohnOmage]

My cousin research is on optimal power flow analysis of Nigerian power system using Primal-Dual Interior Point Method. She's presently stuck at her Chapter 3, which is modeling optimal equations with Matlab/MatPower Could you help?

[@apm]

There are many examples here apmonitor.com/me575/index.php/Main/InteriorPointMethod There is also the Gekko or APMonitor software that is used for optimal power flow analysis. Unfortunately I can't help with individual projects because I get many requests like this.

[@trevorsong4345]

AWESOME

[@prateekparihar9319]

it is urgently needed... plz can you post me a video explaining the example no. 3 and it's MATLAB program

[@apm]

For the example problem 3 at apmonitor.com/me575/index.php/Main/InteriorPointMethod, you have a double inequality with 0

[@AA-yj9we]

how to code without link this site server. and how to maximize wind energy capture using dynamic programming.for example by optimizing aerodynamic power cofficient

[@apm]

Here is additional source code: apmonitor.com/me575/index.php/Main/InteriorPointMethod If you'd like to run locally without an internet connection, you can use a local server for APMonitor as shown here: apmonitor.com/wiki/index.php/Main/APMonitorServer If you are using Python GEKKO, you can set remote=False as an option to calculate without an Internet connection.

[@muhammadyaqoob8668]

Sir, Thanks for the great tutorial. I have following 2 very basic questions though: 1. What is the main difference between IPOPT (Interior Point Optimizer), BPOPT (Barrier Point Optimizer?), and APOPT (..? Point Optimizer?)? 2. You talked about combining APOPT and BPOPT to get a better solver. May I know if its available to use or not? Many thanks!

[@apm]

APOPT is an active set solver. BPOPT and IPOPT are essential the same except for differences in the way they treat acceptance of a new trial point. IPOPT uses a filter method while BPOPT uses a merit function. There is additional information on these solvers at apm.byu.edu/prism/uploads/Members/minlp_apopt_informs2012.pdf We are working on the combined approach right now along with several other improvements. The IPOPT solver is open source and available from COIN-OR. APOPT is available from apopt.com (AMPL and APMonitor) and also integrated into the GEKKO Python software.

[@ДимаБарбашин-ж9я]

Great tutorial, thank you very much!