定義
![共軛函式](/img/0/392/wZwpmL4cDMyYTNzkTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL5kzLxIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/7/975/wZwpmL1YjN4IDM3kjMzATN1UTM1QDN5MjM5ADMwAjMwUzL5IzL1AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
設函式 ,定義函式 為
![共軛函式](/img/b/0ce/wZwpmLyQTOxAjM4MjMzATN1UTM1QDN5MjM5ADMwAjMwUzLzIzL0YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/5/3a5/wZwpmLyUjM1UzM1EzMzATN1UTM1QDN5MjM5ADMwAjMwUzLxMzLxgzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/8/01c/wZwpmL0gzN4UzN4MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzLzAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/5/fe7/wZwpmLxATO3IDN3UjMzATN1UTM1QDN5MjM5ADMwAjMwUzL1IzL3czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
此函式稱為函式 的 共軛函式。使上述上確界有限,即差值 在 有上界的所有 構成了共軛函式的定義域。圖1描述了此定義。
![圖1](/img/8/ced/wZwpmL1ATMxgzNxUjMzATN1UTM1QDN5MjM5ADMwAjMwUzL1IzLwgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/f/f37/wZwpmL0AjNxYTO4AjMzATN1UTM1QDN5MjM5ADMwAjMwUzLwIzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/e/5e3/wZwpmL2YTO5cDM3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL1YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/1/127/wZwpmL3QDM1MDM0EjMzATN1UTM1QDN5MjM5ADMwAjMwUzLxIzL2UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/d/488/wZwpmL1UTMwIDO3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL4YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/4/ead/wZwpmLzMjN4EzNwkjMzATN1UTM1QDN5MjM5ADMwAjMwUzL5IzL2AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
圖1中,函式 以及某一 。共軛函式 是線性函式 和 之間的最大差值,見圖中虛線所示。如果 可微,在滿足 的點 處差值最大。
![共軛函式](/img/3/35c/wZwpmLwEDO3YzMyAjNxMzM1UTM1QDN5MjM5ADMwAjMwUzLwYzLxAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/2/047/wZwpmL4YzMwUTN3ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLygzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/3/35c/wZwpmLwEDO3YzMyAjNxMzM1UTM1QDN5MjM5ADMwAjMwUzLwYzLxAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/f/4ea/wZwpmLxcTNwQzNwYzMzATN1UTM1QDN5MjM5ADMwAjMwUzL2MzLwIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/f/27e/wZwpmLwYTN1IzM4EzMzATN1UTM1QDN5MjM5ADMwAjMwUzLxMzLzMzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
顯而易見, 是凸函式,這是因為它是一系列 的凸函式(實質上是仿射函式)的逐點上確界。無論f是否是凸函式, 都是凸函式。(注意到這裡當 是凸函式時,下標 可以去掉,這是因為根據關於擴展值延伸的定義,對於 )。
基本性質
Fenchel不等式
![共軛函式](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/2/047/wZwpmL4YzMwUTN3ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLygzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
從共軛函式的定義我們可以得到,對任意 和 ,如下不等式成立
![共軛函式](/img/9/308/wZwpmL0ADN0UzN5kTMzATN1UTM1QDN5MjM5ADMwAjMwUzL5EzLxUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
上述不等式即為 Fenchel不等式(當 可微的時候亦稱為 Young不等式)。
![共軛函式](/img/3/e54/wZwpmL3YzN5ETNyQjMzATN1UTM1QDN5MjM5ADMwAjMwUzL0IzL0IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/0/554/wZwpmLwcDOzEjN5kjMzATN1UTM1QDN5MjM5ADMwAjMwUzL5IzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
以函式 為例,其中 ,我們可以得到如下不等式
![共軛函式](/img/2/584/wZwpmL3UTMwcTO4AjMzATN1UTM1QDN5MjM5ADMwAjMwUzLwIzLzczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
共軛的共軛
![共軛函式](/img/0/29c/wZwpmLwQTNxQTM3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL2MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/7/006/wZwpmL2EzMyIzM3AzMzATN1UTM1QDN5MjM5ADMwAjMwUzLwMzL4IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/b/ff3/wZwpmL4cDM0AjM4MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL2UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/7/006/wZwpmL2EzMyIzM3AzMzATN1UTM1QDN5MjM5ADMwAjMwUzLwMzL4IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
上面的例子以及“共軛”的名稱都隱含了凸函式的共軛函式的共軛函式是原函式。也即:如果函式f是凸函式且f是閉的(即 是閉集),則 。例如,若 ,則我們有 ,即f的共軛函式的共軛函式還是f。
可微函式
可微函式f的共軛函式亦稱為函式f的 Legendre變換。(為了區分一般情況和可微情況下所定義的共軛,一般函式的共軛有時稱為 Fenchel共軛。)
![共軛函式](/img/b/ff3/wZwpmL4cDM0AjM4MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL2UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/5/3a5/wZwpmLyUjM1UzM1EzMzATN1UTM1QDN5MjM5ADMwAjMwUzLxMzLxgzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/7/8b3/wZwpmL4EjMzQjMzMzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL4EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/7/8b3/wZwpmL4EjMzQjMzMzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL4EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/5/3a5/wZwpmLyUjM1UzM1EzMzATN1UTM1QDN5MjM5ADMwAjMwUzLxMzLxgzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/7/8b3/wZwpmL4EjMzQjMzMzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL4EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
設函式f是凸函式且可微,其定義域為 ,使 取最大的 滿足 ,反之,若 滿足 , 在 處取最大值。因此,如果 ,我們有
![共軛函式](/img/2/f59/wZwpmLzUjN2UDMzMzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzLxAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/f/b0e/wZwpmL0ADMxIzMxMzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/1/127/wZwpmL3QDM1MDM0EjMzATN1UTM1QDN5MjM5ADMwAjMwUzLxIzL2UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
所以,給定任意y,我們可以求解梯度方程 ,從而得到y處的共軛函式 。
![共軛函式](/img/d/3c6/wZwpmL0cDOwYTOzMzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL3czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/a/942/wZwpmLwEDN3MTM2UjMzATN1UTM1QDN5MjM5ADMwAjMwUzL1IzLwIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
我們亦可以換一個角度理解。任選 ,令 ,則
![共軛函式](/img/0/8c1/wZwpmLygzNxYjMzkjMzATN1UTM1QDN5MjM5ADMwAjMwUzL5IzLyUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
伸縮變換和複合仿射變換
![共軛函式](/img/3/79f/wZwpmL2ETN1QzM4MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL4EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/b/842/wZwpmLygjM3cDMxkjMzATN1UTM1QDN5MjM5ADMwAjMwUzL5IzLwAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
若a>0以及b∈R, 的共軛函式為 。
![共軛函式](/img/e/e42/wZwpmLzITOyUDN0ATMzEzM1UTM1QDN5MjM5ADMwAjMwUzLwEzLyUzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/c/21d/wZwpmL4YDNzETM3YzMzATN1UTM1QDN5MjM5ADMwAjMwUzL2MzL1QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/1/37d/wZwpmLycTNzgjN3UjMzATN1UTM1QDN5MjM5ADMwAjMwUzL1IzL2UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
設 非奇異, ,則函式 的共軛函式為
![共軛函式](/img/3/cae/wZwpmLxATM2YDNxAzMzATN1UTM1QDN5MjM5ADMwAjMwUzLwMzL1UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/3/3c6/wZwpmLxYDN1QTN0YzMzATN1UTM1QDN5MjM5ADMwAjMwUzL2MzL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
其定義域為 。
獨立函式的和
![共軛函式](/img/a/11c/wZwpmLzMDN0QDM0MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL0QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/2/a23/wZwpmLwITM2EDN2cjN1IDN0UTMyITNykTO0EDMwAjMwUzL3YzL4YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![共軛函式](/img/5/e6b/wZwpmL0AjMxgjM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL2QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![共軛函式](/img/7/27c/wZwpmLwcTMyAjMyEjMzATN1UTM1QDN5MjM5ADMwAjMwUzLxIzLzIzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/b/9e6/wZwpmL0IzM2kzMxAjMzATN1UTM1QDN5MjM5ADMwAjMwUzLwIzL1EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
如果函式 ,其中 和 是凸函式,且共軛函式分別為 和 ,則
![共軛函式](/img/5/f55/wZwpmLzQjMzQjM2AzMzATN1UTM1QDN5MjM5ADMwAjMwUzLwMzL4EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
換言之,獨立凸函式的和的共軛函式是各個凸函式的共軛函式的和。(“獨立”的含義是各個函式具有不同的變數。)
舉例分析
考慮R上一些凸函式的共軛函式。
仿射函式
![共軛函式](/img/e/c1f/wZwpmL1UTNxAjN3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzLwczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/f/faa/wZwpmL3AzM1gjM3AzMzATN1UTM1QDN5MjM5ADMwAjMwUzLwMzLyEzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/3/35c/wZwpmLwEDO3YzMyAjNxMzM1UTM1QDN5MjM5ADMwAjMwUzLwYzLxAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/5/e40/wZwpmL4MjM1kTM0YjMzATN1UTM1QDN5MjM5ADMwAjMwUzL2IzL4YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![共軛函式](/img/c/a18/wZwpmL0ETOxATO2EjMzATN1UTM1QDN5MjM5ADMwAjMwUzLxIzL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
,作為x的函式,若且唯若y=a,即為常數時 有界。因此,共軛函式 的定義域為單點集 ,且 。
負對數函式
![共軛函式](/img/0/aba/wZwpmLyQjN3UTO4MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzLwMzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/9/8a5/wZwpmL0cDM5YTO5gTMzATN1UTM1QDN5MjM5ADMwAjMwUzL4EzL2YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/f/628/wZwpmLyATO5QTM3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL0QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![共軛函式](/img/0/d23/wZwpmLzITO3ADOwYzMzATN1UTM1QDN5MjM5ADMwAjMwUzL2MzLwAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/d/fc4/wZwpmL0QzNwgjM2kTMzATN1UTM1QDN5MjM5ADMwAjMwUzL5EzL1EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![共軛函式](/img/7/bf0/wZwpmL4YTMxMjM1YzMzATN1UTM1QDN5MjM5ADMwAjMwUzL2MzL2AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/d/08f/wZwpmLwMTO5cTO1UjMzATN1UTM1QDN5MjM5ADMwAjMwUzL1IzL4IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
,定義域為 。當 時,函式 無上界,當y<0時,在 處函式達到最大值。因此,定義域為 ,共軛函式為 。
指數函式
![共軛函式](/img/2/c49/wZwpmL2QjN2YDN3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/1/e98/wZwpmL0UDO3UzNxQTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL1czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/b/a3a/wZwpmL1EzN4gTM3kjMzATN1UTM1QDN5MjM5ADMwAjMwUzL5IzL3EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
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![共軛函式](/img/8/241/wZwpmL4MDN4QTM5UzMzATN1UTM1QDN5MjM5ADMwAjMwUzL1MzLwIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/b/419/wZwpmL0QzN1UjM3IjMzATN1UTM1QDN5MjM5ADMwAjMwUzLyIzLwMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
,當 時,函式 無界。當y>0時,函式 在 處達到最大值。因此 。當 時, 。綜合起來, (我們規定 )。
負熵函式
![共軛函式](/img/8/497/wZwpmL1QTMzkTN3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzL0MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/5/3c8/wZwpmLzITM3kTMwQzMzATN1UTM1QDN5MjM5ADMwAjMwUzL0MzL2QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/c/e4b/wZwpmLzEDO2AjMzUzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLxMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![共軛函式](/img/c/510/wZwpmLwEDN2UzM3MzMzATN1UTM1QDN5MjM5ADMwAjMwUzLzMzLyUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
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![共軛函式](/img/e/e6c/wZwpmL3cDNxkDO3AzMzATN1UTM1QDN5MjM5ADMwAjMwUzLwMzL4czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
,定義域為 (同上面討論, )。對所有 y,函式 關於 在 上有上界,因此 。在 處,函式達到最大值。因此 。
反函式
![共軛函式](/img/3/45c/wZwpmL1YzM2IjN2EDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![共軛函式](/img/f/80e/wZwpmL2ITN3YzNwYzMzATN1UTM1QDN5MjM5ADMwAjMwUzL2MzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![共軛函式](/img/d/863/wZwpmL1QTNzMzMwAjMzATN1UTM1QDN5MjM5ADMwAjMwUzLwIzL3QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![共軛函式](/img/a/f68/wZwpmLyUTOykjMzYjMzATN1UTM1QDN5MjM5ADMwAjMwUzL2IzLzczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![共軛函式](/img/9/a86/wZwpmLwITOzIDNzEjMzATN1UTM1QDN5MjM5ADMwAjMwUzLxIzL1YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![共軛函式](/img/4/407/wZwpmLygTM5kTMyYjMzATN1UTM1QDN5MjM5ADMwAjMwUzL2IzLwgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
, 。當y>0時, 無上界。當y=0時,函式有上確界0;當y<0時,在 處達到上確界。因此, 且 。