Interior Point Method for Optimization

Interior point methods or barrier methods are a certain class of algorithms to solve linear and nonlinear convex optimization problems. Violation of inequality constraints are prevented by augmenting the objective function with a barrier term that causes the optimal unconstrained value to be in the feasible space.

[hoytvolker]

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

[Mohammad-fvzb]

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

[charleszhu]

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

[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.

[srinivasd]

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

[lukenuculaj]

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?

[snakesnake]

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!

[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?

[yunjoonjung]

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

[ThiagoSoares-zmyx]

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?

[ajpenner]

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

[matthewjames]

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

[abdullahimohammad]

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

[sammykmfmaths]

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

[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

[abdullahimohammad]

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

[alexisdasiukevich]

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

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

[nahuelpiguillem]

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

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?