基本介紹
![全局極小點](/img/0/a91/wZwpmLyUDN4EDO5gDM5IDN0UTMyITNykTO0EDMwAjMwUzL4AzLzYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
定義1 若對於任意的,都有
![全局極小點](/img/9/3b1/wZwpmL0cTOwgzNyITN0MTN1UTM1QDN5MjM5ADMwAjMwUzLyUzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![全局極小點](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/8/bbe/wZwpmLzITM2cDM2kTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL5UzL4QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![全局極小點](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
則稱為的一個 全局極小點。若上述不等式嚴格成立且,則稱為的一個 嚴格全局極小點。
![全局極小點](/img/c/ad0/wZwpmL1UDOyQDO2UjN0MTN1UTM1QDN5MjM5ADMwAjMwUzL1YzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
定義2 若對於任意的,都有
![全局極小點](/img/9/3b1/wZwpmL0cTOwgzNyITN0MTN1UTM1QDN5MjM5ADMwAjMwUzLyUzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![全局極小點](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/d/dd9/wZwpmL4UjN0EjMwITN2IDN0UTMyITNykTO0EDMwAjMwUzLyUzL4IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/8/bbe/wZwpmLzITM2cDM2kTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL5UzL4QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![全局極小點](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
則稱為的一個 局部極小點,其中為某個常數。若上述不等式嚴格成立且,則稱為的一個 嚴格局部極小點。
由上述定義可知,全局極小點一定是局部極小點,反之不然.一般來說,求全局極小點是相當困難的,因此,通常只求局部極小點(在實際套用中,有時求局部極小點已滿足了問題的要求) 。
全局極值
![全局極小點](/img/f/188/wZwpmL4ETN1EjN0IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzL0czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/d/75f/wZwpmLyIzN2QDM5YTM0EDN0UTMyITNykTO0EDMwAjMwUzL2EzL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/5/758/wZwpmL3UTN4QTM4kTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL5UzL1QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/3/ca0/wZwpmLzEDM4EDNwgjN0MTN1UTM1QDN5MjM5ADMwAjMwUzL4YzL1gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![全局極小點](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/d/75f/wZwpmLyIzN2QDM5YTM0EDN0UTMyITNykTO0EDMwAjMwUzL2EzL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/8/bbe/wZwpmLzITM2cDM2kTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL5UzL4QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![全局極小點](/img/5/250/wZwpmLwAzNwYzNxQzMxMzM1UTM1QDN5MjM5ADMwAjMwUzL0MzLwMzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/4/25f/wZwpmL2ATN1MDN3EjN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxYzLyIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![全局極小點](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
全局極值(global extreme value)是實值函式在某區域取得極小或極大的值。設X為R中的集合,f為X上的實值函式,對於,如果對所有的均滿足(),則稱為在X上的 全局極小(大)點,為 全局極小(大)值。若對所有的,且,均有(),則稱為在X上的 嚴格全局極小(大)點,為 嚴格全局極小(大)值 。
凸函式的全局極小點
凸函式的極值(extreme value of convex function)是凸函式在某點鄰域(或某區域)內取得的極小值或極大值。凸函式的極值有以下性質:
![全局極小點](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
1.若為定義在凸集S上的凸函式,則它的任一局部極小點就是它在S上的全局極小點,而且它的極小點形成一個凸集。
![全局極小點](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/b/f17/wZwpmLzAzM2AzM2ITOwMzM1UTM1QDN5MjM5ADMwAjMwUzLykzLzUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/0/fef/wZwpmL3YzN3UzNzUjMxMzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL3AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![全局極小點](/img/9/262/wZwpmL2YDMwgjN0ATN0MTN1UTM1QDN5MjM5ADMwAjMwUzLwUzL1gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![全局極小點](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
2.設為定義在凸集S上的可微凸函式,若存在點,使得對於所有的,有,則是在S上的全局極小點。
3.定義在凸集上的凸函式的駐點(梯度為0的點),就是其全局極小點。全局極小點並不是惟一的,但若為嚴格凸函式,則其 全局極小點是惟一的 。