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‫But now we discussed that when we wanted to find a solution that minimizes the cost function.

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‫And at the same time, we take constraints into account, then this gradient method here, it was not

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‫suitable anymore.

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‫And the reason for that was very simple.

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‫The gradient method makes the gradient of the cost function zero.

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‫That's why you have here this condition here.

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‫And from that, you compute your solution.

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‫Delta, you global.

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‫But if you look at this mini example here, then you can clearly see that the only place where the gradient

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‫is zero is this place here in the center of these circles.

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‫But that's also in the red region.

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‫The minimum of the cost function, considering the constraints, is in this corner here where the green

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‫area starts.

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‫That's where you're Jay means superscript see is and its corresponding Delta Delta zero and Delta Delta

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‫One.

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‫But is your gradient zero in this corner of here?

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‫No, it's not.

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‫It's something else.

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‫A gradient is like a derivative in one D.

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‫It's like a slope, and it's definitely not zero in this corner here.

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‫That's why the gradient method does not work when you also want to include constraints in your optimization.

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‫In fact, I told you that finding the solution that also respects your constraints by hand is very hard

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‫and impractical.

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‫You can minimize your cost function subject to constraints by hand when you have one or two independent

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‫variables, let's say Delta Delta zero and Delta Delta One, but not when you have more like 20, for

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‫example, which was our case control engineers don't do it by hand.

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‫Instead, they use programs called solvers.

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‫We discussed that soldiers are programs created by very smart people, and the task of solvers is to

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‫find that unique combination of independent variables that gives you the smallest possible value for

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‫your cost function.

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‫Your Jay means superscript.

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‫See the green solution that also respects your constraints.

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‫So this delta you global superscript see it will be found by a solver.

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‫The idea behind solvers is that you give it information about your cost function and your constraints,

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‫and in return they give you your solution.

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‫Delta, your global superscript see that minimizes your cost function and respects all your constraints.

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‫And so in the concourse, I taught you how to transmit information about your cost function and your

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‫constraints to solvers correctly so that they would give you the right solutions.

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‫There are many solvers out there, some of them are more advanced and some of them are less advanced.

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‫The algorithms that they use to perform these optimization tasks are subject on its own.

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‫We use kewpie solvers in Python.

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‫This is how you imported it.

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‫And then from the kewpie service library, there was a function solve on this called kewpie.

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‫And so we use it in order to compute our solution Delta you global that respects our constraints.

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‫And together with kewpie solvers, we also used another solver type called Sirieix OPPT that's proved

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‫to work better in windows and everything that you can see here inside the parentheses.

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‫We'll get to that very soon.

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‫And so I chose kewpie solvers for two reasons.

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‫Firstly, it works just like a very popular quadratic solver in MATLAB called Quad Proc.

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‫So if you know how to use kewpie solvers, you also know how to use quite proc.

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‫And the second reason is that it's built specifically for optimization problems that have a quadratic

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‫positive, definite cost function.

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‫Subject to linear constraints.

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‫So the cost function is quadratic.

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‫It's also positive definite because a double bar is positive definite.

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‫And then the constraints there are linear.

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‫That's exactly our case.

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‫Our cost function is quadratic.

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‫H double bar is positive definite and our constraints are linear in the core.

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‫As I explained to you that if you have constraints, for example, these ones here, and if you could

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‫rewrite them in this form here where this matrix here and this vector here, where they are constant,

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‫meaning that you don't have independent variables inside this matrix, nor in this vector.

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‫And then also, if this vector here is like this and not something like this where you have Delta Delta

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‫zero squared here in Delta Delta One, where for example, this one here where you have Delta Delta

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‫Zero and then the square root of Delta Delta One for as long as you have this form here, that's when

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‫you have linear constraints.

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‫And that's our case.

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‫That was our case in the car course.

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‫And that's the case here in this drone course as well.

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‫And kewpie solvers and quite prog in MATLAB are specifically built for these kind of optimization problems

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‫in which you have a quadratic cost function.

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‫Your double is positive definite and the constraints are linear.

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‫And so here as well, we will use kewpie solvers in the next video.

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‫Let's see how we transmitted information about our cost function and our constraints to our solver in

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‫order to find our solution that minimizes the cost function and respects all our constraints.

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‫Thank you very much.