Chapter 382 (2/2)
”At present, mathematical research in this direction is indeed a blank, so we need to study and fill it in!” Professor Fresnel's eyes swept slowly across their faces. ”So I said yesterday, you should be prepared. This is a tough battle
”Starting from scratch, there is no reference material, and the time limit is Only two months! ”
Professor Fresnel continued, ”I won't say anything about encouragement. I just hope you two don't forget the purpose of coming here. If you want to quit, I'm always welcome.”
”Here's the extra words. Now let's talk about the subject.”
Professor Fresnel asked them to find a place to sit down, moved over to a laptop, opened a PPT, and pointed out, ”this is a short research process I did.”
”I am the leader of this project, and your two tasks are to assist me in solving some links that are not too difficult.”
Cheng Nuo and hull nodded to show that they knew.
The ability of the two of them is not enough to support the framework of the project.
Professor Fresnel went on to explain, ”the proposed name of this project is called Fritz John necessary optimality conditions on Riemannian manifolds. First of all, we must understand what Riemannian manifolds are and what Fritz John's necessary optimality conditions are! ”
”The concept of Riemannian manifolds goes without saying, and Fritz John's necessary optimality conditions should be unfamiliar to you.” He first looked at Cheng Nuo, ”Cheng Nuo, do you understand this concept?”
Cheng Nuo replied without thinking, ”the so-called Fritz John necessary optimality condition is the necessary optimality condition of minf (x), St. {g (x) ≤ 0, H (x) = 0, X ∈ M”Yes, that's Fritz John's necessary optimality condition. You can also see that this Fritz John necessary optimality condition, if we study it directly, will not only have a lot of variables and the function equation is not easy to define, but also has the problem of complex formula in the process of derivation. ”
”So we need to change our thinking.”
Professor Fresnel turned to the next page of PPT, which only wrote a line of formula:
F: m → R, G: m → R ^ L, H: m → R ^ n
Cheng Nuo glanced at it and suddenly realized, ”Lipschitz function?”
Professor Fresnel glanced at Cheng Nuo with a trace of appreciation. ”To be exact, it's a local Lipschitz function!”
Lipschitz function means that if f (x) satisfies on the interval I for any two different real numbers X1 and X2 of the definition domain D: ‖ f (x1) - f (x2) ‖ = k ‖ x1-x2 ‖ holds, there must be f (x) uniformly continuous on the interval I.
in Cheng Nuo's mind, he has roughly understood what the breakthrough point of Professor Fresnel in this project is.
Professor Fresnel continued his theoretical explanation, ”in this formula, we can treat m as an m-dimensional Riemannian manifold.”
”You two should have read the paper by Aton on MP in Hilbert space?”
They both nodded at the same time.
”That's good. By analogy, we can extend the MP problem from linear space to Differential Manifolds, and differential manifolds are nonsmooth. Then we can construct the following framework.”
The next PPT is shown in front of the two.
”The first step is to establish a nonsmooth analysis tool on Riemannian manifolds, that is, to define generalized directional derivatives and generalized gradients on the manifolds.”
Second, we discuss the properties of generalized gradient
In the third step, on the basis of the first two steps, the fritz John type optimality conditions for the problem (MP) on Riemannian manifolds are discussed
The framework has already been set up by Professor Fresnel.
And Cheng Nuo in the see that orderly process steps, there is a sense of urgency.
Originally, this project should be done like this!