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1、作 业 二用 DFP 算 法 求 解 min f (x) = x 2 + 2x 2 - 2x x- 4x ， 取 x= (11)T ，10H 0 = 01 。121 210一、求解：g(x) = (2x1- 2x2- 4,-2x1+ 4x2)T , g= (-4,2)T , p= -H 0 g0= (4,-2)T00(1) 求迭代点 x1令j (a) =f (x+ap ) = 40a2 - 20a- 3，得j (a) 的极小值点a= 1 ，0x1所以得：0= x00+a0 p0= (2,0.5)T , g0041= (- 1,-2)T ,0s0 = x1- x0= (1,-0.5)T , y=。

2、 g1- g0= (3,-4)T .H y yT Hs sT1 8438于是，由 DFP 修正公式有 H = H- 0 0 00 + 0 0 =0000010yT H yyT s100 3841下一个搜索方向为 p1= -H1 g1= 1 (8,6)T 5(2) 求迭代点 x2令j(a) = f (x +ap ) = 8a2 - 4a- 5.5 ，得j(a) 的极小值点a = 41115115于是得： x= x +ap = (4,2)T , g = (0,0)T ，所以： x* = x= (4,2)T , f * = -8 ，211 1222因 Hesse 阵G(x) = G = - 2T为正。

3、定阵， f (x) 为严格凸函数，所以 x* 为整体- 24 极小点。二、DFP 算法迭代步骤如下：(1) 给定初始点 x0 ，初始矩阵 H 0 (通常取单位阵)，计算 g0 ，令 k=0，给定控制误差e。(2) 令 pk = -Hk gk 。a(3) 由精确一维搜索确定步长ak ， f (xk +ak pk ) = min f (xk +apk )0(4) 令 xk +1 = xk +ak pk 。*k +1(5) 若 gk e，则 x = x停；否则令 sk= xk +1 - xk， yk= gk +1 - gk 。(6) 由 DFP 修正公式得 Hk +1 。令 k=k+1，转步骤(2)。

4、三、DFP 算法 matlab 程序实现function best_x,best_fx,count=DFP(x0,ess) syms x1 x2 t;f=x1*x1+2*x2*x2-2*x1*x2-4*x1;fx=diff(f,x1);%求表达式 f 对 x1 的一阶求导fy=diff(f,x2);%求表达式 f 对 x2 的一阶求导fi=fx fy;%构造函数 f 的梯度函数%初始点的梯度和函数值g0=subs(fi,x1 x2,x0);f0=subs(f,x1 x2,x0); H0=eye(2);%输出 x0,f0,g0x0 f0 g0xk=x0;fk=f0; gk=g0; Hk=H0; 。

5、k=1;while(norm(gk)ess)%迭代终止条件|gk| x0=1 1;ess=1e-6; best_x,best_fx,count=DFP(x0,ess)程序运行结果：x0 =11f0 =-3g0 =-42*第 1 次寻优xk =2.00000.5000fk =-5.5000gk =-1-2Hk =0.84000.38000.38000.4100*第 2 次寻优xk =42fk =-8gk =00Hk =1.00000.50000.50000.5000结果如下：best_x =42best_fx =-8count = 2可以看到，最优点 x* = (4,2)T , f * = -8。

6、 ，迭代次数 2 次，与前面结果一致。“”“”At the end, Xiao Bian gives you a passage. Minand once said, people who learn to learn are very happy people. In every wonderful life, learning is an eternal theme. As a professional clerical and teaching position, I understand the importance of continuous learning, life is di。

7、ligent, nothing can be gained, only continuous learning can achieve better self. Only by constantly learning and mastering the latest relevant knowledge, can employees from all walks of life keep up with the pace of enterprise development and innovate to meet the needs of the market. This document is also edited by my studio professionals, there may be errors in the document, if there are errors, please correct, thank you。