定義
在工程實際問題的最佳化設計中,所列的目標函式往往很複雜,為了使問題簡化,常常將目標函式在某點鄰域展開成泰勒多項式來逼近原函式。
二元函式的黑塞矩陣
![黑塞矩陣](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/3/a53/wZwpmL2cDO5AzM2MDO2UzM1UTM1QDN5MjM5ADMwAjMwUzLzgzLwMzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/6/d80/wZwpmL1AjN3MDN5MTMzEzM1UTM1QDN5MjM5ADMwAjMwUzLzEzL1UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/7/bde/wZwpmL3QjMzkjM3EDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLxQzL4QzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![黑塞矩陣](/img/4/427/wZwpmL3UDMxUzM2AzN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwczL3czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/f/8e9/wZwpmL2YDOygzNwMDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzAzL1QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
由高等數學知識可知,若一元函式 在 點的某個鄰域內具有任意階導數,則 在 點處的泰勒展開式為: ,其中 , 。
![黑塞矩陣](/img/6/079/wZwpmL3UTO3gDM4cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL1IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/b/0e0/wZwpmLxcDNzITOwUTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
二元函式 在 點處的泰勒展開式為:
![黑塞矩陣](/img/7/3c4/wZwpmL0UjM3ETM4cjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3YzL1gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/5/d86/wZwpmL4EDNwYjN4QDN3UzM1UTM1QDN5MjM5ADMwAjMwUzL0QzL2YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/a/11d/wZwpmL2IzNxMjN2UDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1gzLzMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
其中, 。
將上述展開式寫成矩陣形式,則有:
![黑塞矩陣](/img/7/935/wZwpmL3IDN0cDMxUTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzL4QzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
即:
![黑塞矩陣](/img/4/54f/wZwpmLygTM4kDN0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
其中:
![黑塞矩陣](/img/d/8e8/wZwpmL3ADOxcTNycTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3UzL1gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/f/c5b/wZwpmL4UTOwATMyQjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0YzL2EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/6/079/wZwpmL3UTO3gDM4cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL1IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/f/c5c/wZwpmL2ETNxYDNxYzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/6/079/wZwpmL3UTO3gDM4cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL1IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/f/c5c/wZwpmL2ETNxYDNxYzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
是 在 點處的黑塞矩陣。它是由函式 在 點處的二階偏導數所組成的方陣。
多元函式的黑塞矩陣
![黑塞矩陣](/img/2/cf0/wZwpmLzYzN5QzM5ITOxMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL0EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/f/c5c/wZwpmL2ETNxYDNxYzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
將二元函式的泰勒展開式推廣到多元函式,則 在 點處的泰勒展開式的矩陣形式為:
![黑塞矩陣](/img/4/54f/wZwpmLygTM4kDN0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
其中:
![黑塞矩陣](/img/1/7af/wZwpmLyITO5IjN5ITO2UzM1UTM1QDN5MjM5ADMwAjMwUzLykzL1czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/a/62c/wZwpmL4IDM5gzM1QTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzL3QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/f/c5c/wZwpmL2ETNxYDNxYzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(1) ,它是 在 點處的梯度。
![黑塞矩陣](/img/4/9d3/wZwpmLwUzM1ITM0QDM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0AzLyIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/a/62c/wZwpmL4IDM5gzM1QTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzL3QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/f/c5c/wZwpmL2ETNxYDNxYzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(2) 為函式 在 點處的黑塞矩陣。
![黑塞矩陣](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/2/d94/wZwpmL2czM0QzMyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzLyYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
黑塞矩陣是由目標函式 在點X處的二階偏導數組成的 階對稱矩陣。
對稱性
![黑塞矩陣](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/5/017/wZwpmLxYzM2AzMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/6/3fc/wZwpmL3IDN0czN1UTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/5/017/wZwpmLxYzM2AzMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
如果函式 在 區域內二階連續可導,那么 黑塞矩陣 在 內為對稱矩陣。
![黑塞矩陣](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
原因:如果函式 的二階偏導數連續,則二階偏導數的求導順序沒有區別,即
![黑塞矩陣](/img/c/1ec/wZwpmL3MzN5gDN3MzM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzMzLzMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![黑塞矩陣](/img/6/3fc/wZwpmL3IDN0czN1UTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/1/3df/wZwpmLxUTO3cjMyIDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLwUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/6/3fc/wZwpmL3IDN0czN1UTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
則對於矩陣 ,有 ,所以 為對稱矩陣。
利用黑塞矩陣判定多元函式的極值
定理
![黑塞矩陣](/img/2/cf0/wZwpmLzYzN5QzM5ITOxMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL0EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/9/4a0/wZwpmL4MDNzITM3AjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwYzL0IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
設n多元實函式 在點 的鄰域內有二階連續偏導,若有:
![黑塞矩陣](/img/7/e79/wZwpmL1EDOwQjNxMTO2UzM1UTM1QDN5MjM5ADMwAjMwUzLzkzL1czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
並且
![黑塞矩陣](/img/4/707/wZwpmLxUDMyADOzUTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzL0EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
則有如下結果:
![黑塞矩陣](/img/2/cf0/wZwpmLzYzN5QzM5ITOxMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL0EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/9/4a0/wZwpmL4MDNzITM3AjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwYzL0IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(1)當A正定矩陣時, 在 處是極小值;
![黑塞矩陣](/img/2/cf0/wZwpmLzYzN5QzM5ITOxMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL0EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/9/4a0/wZwpmL4MDNzITM3AjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwYzL0IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(2)當A負定矩陣時, 在 處是極大值;
![黑塞矩陣](/img/9/4a0/wZwpmL4MDNzITM3AjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwYzL0IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(3)當A不定矩陣時, 不是極值點。
![黑塞矩陣](/img/9/4a0/wZwpmL4MDNzITM3AjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwYzL0IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(4)當A為半正定矩陣或半負定矩陣時, 是“可疑”極值點,尚需要利用其他方法來判定。
實例
![黑塞矩陣](/img/1/944/wZwpmL0ADM4EzMzkjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL3IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
求三元函式的極值。
![黑塞矩陣](/img/6/987/wZwpmL3IDOwkTM4cjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3YzL0QzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![黑塞矩陣](/img/9/834/wZwpmLxUDMwMDO5QzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0czL3EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
解:因為,故該三元函式的駐點是。
![黑塞矩陣](/img/b/deb/wZwpmLxQTM3IDM4czN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3czLxIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
又因為,
![黑塞矩陣](/img/4/58a/wZwpmLxEjMxUTO0QzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL3IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
故有:
![黑塞矩陣](/img/9/834/wZwpmLxUDMwMDO5QzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0czL3EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![黑塞矩陣](/img/b/976/wZwpmL0UTO3cDN0EDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLxQzL3AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
因為A是正定矩陣,故是極小值點,且極小值。