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The cbc.exe ist not our builded exe! When compiling the solution "cbc" the cbc.exe in 'C:\Users\firda\OneDrive\Desktop\coin-or\Cbc\MSVisualStudio\v17\Debug' is built. We, however, want to include cbc into our projects (using the lib files and so on). Regarding your problem: Do you have any problems compiling the solution "CBC Test"? If there aren't any error, a executable should appear in CBC Test/Debug/. I suppose there are some compilation errors. Maybe there are some issues with your shortcuts... Try to compile by rightclicking on the solution, then choose "Build" or "Compile". In addition I recommend to follow the newer guide, which is linked in the video description. The way I include all the header files in this video here is not the best way. In the newer video I show how to do it properly. And this video here is two years old, so there might be some changes, which causes your problems. If you have any more question, I'd be happy to help.

@@franksmathematics9779 I very appreciate your respond, thanks Franks. I will see your other videos

This sound like a linker error - Does VS know where to find the binaries and the h-files? Without the header files you cannot use the binary as the linker does not where to "enter". I am not sure if something has changed in the newest version, but the procedure (building the libary and linking it correctly into your project) should be the same, as it is a standard procedure. However, I will check on this - hopefully on this weekend, because after that I am traveling for a month. But I am glad that you where able to build the v17 version - this is the first step!

@@franksmathematics9779 Thanks Franck, that would be great if could verify this. I am using the VS2002 community edition and had no problem building the binaries, it is just when i include them in my project that I get errors.

@@Megapixelrealestate I had no trouble using the newest version v2.10.12 in Release Mode using x64, using VS2022 Community (17.11.2). The process is nearly the same - however I just chosed a different approach to link the h-Files to VS. You can find the update here: th-cam.com/video/mMutgRhZKlQ/w-d-xo.html I hope this helps you! Best, Frank

@@franksmathematics9779 Thanks for verifying Franck. The problems I am getting must be related to bugs in my code, at least I know its not a version problem.

@@Megapixelrealestate Good luck hunting down the bugs!

There was no specific reason why I picked the 32bit version. Maybe just for fun? I really dont remember it anymore :D If you are free to choose, pick the 64 bit version! However it is weird that you can compile it in Debug, but not in release. Have you checked on your linker details (see the video for details)? Are they different in Debug and Release?

@@franksmathematics9779 I can in fact compile cbc in release without errors, the errors arise in my own project to which I have added the compiled binaries.

Oh that is weird - i just checked on the master branch - there v10, v14, v16 and even v17 are present. But you are right: In the stable/2.10 there are some folders missing - but v17 is there. So you can either switch to master branch, or use the v17 version with the newest VS version. Or maybe try the v16 version from master branch with the stable/2.10 branch - maybe this is working...

Yes, you are correct, this is a typo. u in U representes the constraints which should apply to u, therefore it should be \min_\limits_{u \in U}. Thanks for pointing this out!

Hi, the matrix P is our preconditionier! Around minute 12 I give an example how to use the Jacobi method as a preconditioner, which will result in a certain matrix P. The idea/derivation of the matrix P was given early in the video. The choice of an efficient P (in terms of: less iterations in the CG method) however is not that easy, since this heavily depends on the underlying problem. You can easily derive different matrices P for different splitting methods like Gauss-Seidel, SSOR and so on. I hope this helps in understanding the idea - if you have any more question, please do not hesitate!

@@franksmathematics9779 Thanks for your explanation. I am now somehow able to apply it to my computations :D Now, I have another question: in your example, did you apply the preconditioner to both z^k and z^(k+1)? If yes, does it mean that the number of steps of the Jacobi method (minute 14:05) applies to both z^k and z^(k+1)? Look forward to hearing from you. Thank you very much.

@@KebutuhanKirimFile2 Glad you made it work! The number of Jacobi-steps is denoted by l, which is in the definition of the matrix P. So the formal correct answer to your question would be: "You only apply the matrix P to z_k once to obtain z_{k+1}!". In practice you will not compute the inverse of P, but rather apply l steps of the Jacobi method to z_k and call the output z_{k+1}! Best, Frank

@@franksmathematics9779 Thanks for your reply but maybe I asked it wrongly. Let me clarify it once again. First question: So, in your example, for the value of I = 5, it gives a total of 130 iterations of CG. Does it mean that the Jacobi loops here are in fact performed 5 x 130 = 650 times? Second question: how's about the actual CPU time for different values of I? I can understand that increasing the value of I will decrease the number of iterations of CG, but in fact doing the Jacobi loops for larger I values will also consume time (memory accesses, arithmetic operations, etc.)? Look forward to hearing from you. Thank you very much!

@@KebutuhanKirimFile2 Yes you are right, multiplying (number of Jacobi iterations in each step) * (number of CG steps) gives the final amount of performed Jacobi iterations. Regaring your second question: I do not have the data available anymore. There should be some sort of "sweet spot" where the computation time is minimal. Maybe I will redo the calculation and measure the CPU time - this sound like an interesting question. Do you have made any research into this topic/area and can provide the results?

Hi, you could start directly with the definition of A-orthogonality for sure - but I thought it is helpful to actually show where the idea of A-orthogonality arises here (starting with the optimization problem -> going to the contour lines which are ellipses -> they can be described in the eigen space of A -> put this into a definition -> The contour lines of f can be described as a circle in the eigen space of A - roughly spoken). For the construction of the CG method it is not needed. Best, Frank

Thank you! Part 3 is already structured in my head, but I haven't had the time to start working on it. So I am afraid it might take some time...

There is no mathematical reason for it. The only reason is that the 1/2 cancels with the factor 2 coming from the norm when you compute the derivate. Since one mainly works with the derivate, this makes some terms "look nicer". Good luck with your master thesis - you picked an interesting topic!

Ah dang it. You mean on the slide starting at 5:00? Thats true, I made a typo there. Thanks for pointing this out, I will add some notifications there!

My formulation is indeed not correct there - I should have noted the definition of maximality there. Nevertheless the set B obtained by Zorn's lemma will give us the correct set. Thanks for pointing it out.

Thanks! I am working on it - it is 75% finished. However due to a new job and some other changes in my life I didn't had much time to work on my youtube channel. But this channel is not dead, I promise!

Can you specify your question? As I showed in this video for infinite dimensional spaces a Hamel basis is always uncountable, while a Schauder basis is countable by definition, hence they cannot be the same. A Schauder basis might be a subset of a hamel basis. The problem here is: You cannot explicitly construct a Hamel basis. In finite dimensions however everything is the same, there is not need to differentiate between Schauder and Hamel basis. I hope I could answer your question, if not feel free to ask again!

​@@franksmathematics9779ok, I get it.Thanks.

Hi, can you please explain your question a little bit? Do you want to add an additional time-dependent source/force inside the ODE? In this case no special changes have to be made: Just stick with the descretization and add the pointwise evaluation of your time-dependent source.

In the code given by you, the final desired end effector position is (x,y)=(2,2). Instead of this, if I want to make the end effector to follow a sinusoidal trajectory, say x=0.5sin(wt), y= 0.5cos(wt), what change should I make in the code.@@franksmathematics9779

Thank you!

Thank you! I am quite busy at the moment but I am still working on the next part(s). Stay tuned!

Hi, the problem is that your problem u''(x) = -sin(x) and u'(0) = 1 and u'(pi) = -1 has no unique solution. It is the same argument as in the video: If w(x) is a solution to your problem, so is w(x) + c with an arbitrary chosen but constant c (just plug it in the problem and compute the derivative). If the exact analysis has no unique solution, how can we expect the numerical approximation to have an unique solution? We simply cannot expect this behaviour. Here this will result in an irregular system matrix. I hope this helps to clarify the problem.

@@franksmathematics9779 Thank you so much

Unfortunately I do not have a textbook available where this is described in the way I did. I did a short check on my textbooks but they also do not cover the preconditioned CG method. Most of the stuff I present here is taken from papers - If papers are okay I can give you some names...

​@@franksmathematics9779 If you can, i would be appreciated thank you very much

@@heesangyoo1536 It took me a while but I found the source where most of the results are taken from - but unfortunately it is written in german and I am afraid there is no english translation... I hope this helps at least a little bit: Kanzow - Numerik linearer Gleichungssystem (Numeric of linear systems of equations) Chapter 5.4 - Das präkonditionierte CG-Verfahren (The preconditioned CG-Method) Here is the link: link.springer.com/book/10.1007/b138019 If I stumble something else I will let you know...

Hi, in the end most of this problems work in the way as you described: Getting rid of the state variable y and reduce this problem to an optimization problem which only depends on the control u. However to do so there is a lot of theory (especially theory of ODEs like solvability, reachability, controllability and so on) which is skipped in this video, as the focus was on the numerical implementation. If you think of a more basic approach (do not use the magic black box fmincon) like a (semismooth) Newton method you have an update step of the form u_{k+1} = u_k + s(u_k) where s is a (more or less complex) function. And here now comes the fun part: To compute s(u_k) you have to solve the ODE, or a linearized form of it - sometimes even multiple times. Thats one way to solve it in a reduced order, I hope this helps. Edit: In my video about the Semi-smooth Newton Method I did a reformulation of a constrained optimization problem (and nothing more is this problem in the video) to a fixed point equation, which is solved by a Newton method. A similar approach can be done here: th-cam.com/video/zOIMH6fNFOA/w-d-xo.html

@@franksmathematics9779 Thanks for your comprehensive reply. I would assume that solving the reduced order problem with fmincon (if, for the sake of simple implementation, my choice is that at the moment) is only possible once we just have the input constraints and not state constraints. In this case, the state equation can be easily integrated into the objective function recursively (I think this approach works). In other cases where we have state constraints or state-dependant constraints on the input, we must also include the states as the decision variables. Am I right?

Yes this is correct. (depending on the type of constraints) Without state constraints your approach should work. You just need a straightforward implementation of a ODE solver (this could be some recursion of course). Are you going to implement this? If so, keep me updated! State constraints, especially inequalities in ODE/PDE optimization are always ugly. As far as I know there is no direct solver or technique how to include them into a reduced formulation. I once wrote a paper how to tackle them with some sort of augmented Lagrange function, which however is again some sequence of optimization problems...

@@franksmathematics9779 I work with LTI state equation and would like to consider different objectives with and without state constraints. I will share the results.

@@franksmathematics9779 Feasible Direction Algorithm handles state-dependant input constraints in the reduced format. The effort is needed to convert state constraints to state depending input constraints

This depends. In my area of research (functional analysis / optimal control) it is very common to use (x,y) for inner products.

@@franksmathematics9779 How do you write a tuple of vectors x,y instead of (x,y) ? By the way, using (x,y) for vector tuples is common in many optimization books. I am quite surprised it's not the case in your optimal control related field.

@@ncroc Fair enough, good point. To be more precise I usually add an index like (x,y)_X to indicate that I use the inner product in the space X. However I usually drop this index when it is clear from the context. I have to admit that this may lead to some missunderstandings with tupels if you are using tupels along with inner products.

In my defense: English is not my mother tongue ;) But yes, you are right, but I hope it is clear from the context.

Thank you! I am quite busy at the moment but I am still working on the next part(s). Stay tuned!

Hi Stuart. Sure, here you go: github.com/FranksMathematics/CBC_ReleaseParallel_Win86 This Repository includes the Visual Studio Files including the compiled files. Please note, that I only changed the Profile "ReleaseParallel" under x86. If you want it for another architecture you have to change it accordingly.

@@franksmathematics9779 Thanks I figured out my issue I was trying to link the x64 version of cbc with the win32 version of pthreads

Hi, i am not sure what do mean: Given an uncountabe infinite set of linear independent vectors (in some vector space V) - you then construct a new set containing all linear combination of these vectors? Your question is then: Does this set form a vectorspace? It is clearly a subspace of V, so all you have to check is the subspace property. If it is a vectorspace you can apply the results from this video. However dealing with uncountable sets is tricky, as you cannot simply write them down. Hopefully this answers your question? If not, please clarify your question please! Best, Frank

Hi, i suppose you talk about the Hamilton-Jacobi-Bellmann equation? Thats an interesting topic/question for sure. I have dealt with these equations before and I will definitely put this on the list. At the moment I am preparing a series about optimal control problems and how to solve them. The HJB-equations would definitely fit there!

@@franksmathematics9779 sure thank you