Argmax (and argmin)

1. Argmax: argumentum maximi

Max argument?

650) This. width = 650; "src =" http://upload.wikimedia.org/math/0/7/a/07ac6e0bdcac327cea6524f406a068bd.png "class =" Tex "alt =" X _ \ mathrm {max }=\ Arg \ max F (x ): \ leftrightarrow f (x _ \ mathrm {max}) =\ Max _ {x \ In D} f (x ). "Style =" border: 0px; "/>

In my opinion, the variable value is assigned to the formula when the maximum value is obtained using argmax.

Argmin

Example:

650) This. width = 650; "src =" http://upload.wikimedia.org/math/9/6/4/96464fbff8cab1f80057380a7f818d27.png "class =" Tex "alt =" \ Arg \ Max _ {x \ In \ r} (x (10-X) = 5, "Style =" border: 0px; "/>

In parentheses, the maximum value of f (x) is 25. In this case, the variable X is 5 and assigned to the formula. The result of the formula is 5.

However, some folie examples can be used in other ways.

For example, to determine a required y value, but to take the Y value into account, the value of X must be taken into account.

That is, it is feasible to replace X (max) in the previous formula with Y.

For example, if the maximum value of f (x) is found, the value of Y is

Argmax (f (x) can be used as the so-called "basis"

I think that's why the literal translation is called "max (min) argument ".

I hope I do not understand it... The Chinese wiki does not have this. It seems that I do not know the Chinese.

At least it is correct to apply it to folie ~

2. What does Arg min mean?

The most common understanding: indicates the variable value when the target function is set to the minimum value.
From Wikipedia

In mathematics,Arg Max(OrArgmax) Stands forArgument of the maximum, That is to say, the set of points of the given argument for which the value of the given expression attains its maximum value:^{[NOTE 1]}

• 650) This. width = 650; "class =" Tex "alt =" \ underset {x }{\ operatorname {Arg \, Max }}\, f (x ): =={ X \ | \ forall Y: f (y) \ le f (x) \} "src =" http://upload.wikimedia.org/math/9/7/f/97f02cc438c1ae150557c1f03646ef02.png "/>

In other words,

• 650) This. width = 650; "class =" Tex "alt =" \ underset {x }{\ operatorname {Arg \, Max }}\, f (x) "src =" http://upload.wikimedia.org/math/ B/B /8/bb8a151866d443a017aeb9cd27e00921.png "/>

Is the set of valuesXFor whichF(X) Has the largest valueM.For example, ifF(X) Is 1 |X|, Then it attains its maximum value of 1X= 0 and only there, So 650) This. width = 650; "class =" Tex "alt =" \ underset {x }{\ operatorname {Arg \, Max }}\, (1-| x |) = \ {0 \} "src =" http://upload.wikimedia.org/math/ B /d/d/bdd336de74d67016573666fdf25c026a.png "/>.

Equivalently, ifMIs the maximumF,Then the ARG Max is the level set of the maximum:

• 650) This. width = 650; "class =" Tex "alt =" \ underset {x }{\ operatorname {Arg \, Max }}\, f (x) = f ^ {-1} (m) =\{ X \ | \ f (x) = M \} "src =" http://upload.wikimedia.org/math/2/a/0/2a0ddd1952e05962222cce4ccf80751d.png "/>

If the maximum is reached at a single value, then one refers to the pointTheArg Max, meaning we define the ARG Max as a point, not a set of points. So, for example,

• 650) This. width = 650; "class =" Tex "alt =" \ underset {x \ In \ BBB {r }{\ operatorname {Arg \, max }}( X (10-X) = 5 "src =" http://upload.wikimedia.org/math/8/4/0/840becd07367e5214a1fcb2675d6eeac.png "/>

(Rather than the singleton set {5}), since the maximum valueX(10X) Is 25, which happens whenX= 5.^{[NOTE 2]}

However, in case the maximum is reached at least values,Arg MaxIsSetOf points.

Then, we have for example

• 650) This. width = 650; "class =" Tex "alt =" \ underset {x \ in [\ Pi] }{\ operatorname {Arg \, Max }}\, \ cos (x) = \ {0, 2 \ Pi, 4 \ pi \} "src =" http://upload.wikimedia.org/math/0/c/9/0c99d783a9fe097e42c3c38b2a1c0dd5.png "/>

Since the maximum value of COS (X) Is 1, which happens on this interval whenX= 0, 2 π or 4 π. on the whole real line, the ARG Max is 650) This. width = 650; "class =" Tex "alt =" \ {0, 2 \ Pi,-2 \ Pi, 4 \ Pi, \ dots \}. "src =" http://upload.wikimedia.org/math/7/7/d/77d0a94143189ec90c51a4d8c66d9789.png "/>

Arg min(OrArgmin) Is defined analogously.

Note also that functions do not in general attain a maximum value, and hence will in general not have an arg max: 650) This. width = 650; "class =" Tex "alt =" \ underset {x \ In \ BBB {r }{\ operatorname {Arg \, Max }}\, X "src =" http://upload.wikimedia.org/math/2/8/1/2819cfd4c0287ffe40d49f826a8f02f1.png "/> is undefined,XIsunbounded on the real line. However, by the extreme value theorem (or the classical compactness argument), a continuous function on a compact interval has a maximum, and thus an ARG Max.

1. Http://blog.163.com/htfei_1984/blog/static/677981242011445104058/

2. Http://www.cppblog.com/guijie/archive/2010/12/13/136273.html

This article is from the squirrel blog, please be sure to keep this source http://apinetree.blog.51cto.com/714152/1563227

Argmax (and argmin)