无约束优化算法：单纯形法_无约束优化方法
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单纯形法
1.算法原理
单纯形法的基本思想是：
设x,x,...,x(0)(1)(n)是Rn中的n1个点，构成一个当前的单纯形，xmax,xmin定义如下：
f(xmax)maxf(x(0)),f(x(1)),...,f(x(n)) f(xmin)minf(x(0)),f(x(1)),...,f(x(n))
记x为这个单纯形除去xmax外的所有顶点的形心，1n(i)xxxmax
ni0取xmax关于x的反射点x(n1)，x(n1)x(xxmax)构成新的单纯形，反复上述过程，直到达到停止条件。
2.函数fminsearch
1）函数语法
xfminsearch(fun,x0)
xfminsearch(fun,x0,options)[x,fval]fminsearch(...)[x,fval,exitflag]fminsearch(...)[x,fval,exitflag,output]fminsearch(...)函数输入：
fun：目标函数
x0：迭代初始点
options：函数参数设置
函数输出： x：最优点
fval：最优点对应的函数值 exitflag：函数停止信息
1：函数收敛正常停止
0：迭代次数，目标函数计算次数达到最大数-1：算法被输出函数停止 output：函数运算信息 2）函数使用
（1）目标函数程序BanaFun.m
functionfBanaFun(x)(不含导数解析式)
f100*(x(2)x(1)^2)^2(1x(1))^2
NelderMeadSimplex函数不需要导数信息。
（2）算法参数设置：SimplexUnc.m
optionsoptimset('LargeScale','off','gradobj','off','MaxFunEvals',250,'display','iter')
（3）函数调用运算：SimplexUnc.m
optionsoptimset('LargeScale','off','gradobj','on','MaxFunEvals',250,'display','iter')x[1.9,2]
[x,fval,exitflag,output]fminsearch(@BanaFun,x,options)
3）计算结果
Iteration
Func-count
min f(x)
Procedure
0
267.62
236.42
initial simplex
67.2672
expand
12.2776
expand
12.2776
reflect
12.2776
contract inside
6.76772
contract inside
6.76772
reflect
6.76772
contract inside
6.76772
contract outside
6.62983
contract inside
6.55249
contract inside
6.46084
contract inside
6.46084
reflect
6.46084
contract inside
6.45544
contract outside
6.42801
expand
6.40994
expand
6.32449
expand
6.28548
expand
6.00458
expand
6.00458
reflect
5.43287
expand
5.43287
reflect
4.63434
expand
4.63434
reflect
4.63434
contract inside
4.63434
contract outside
4.31027
expand
4.31027
contract inside
4.00991
expand
4.00991
reflect
112
114
115
117
119
120
3.55664
3.55664
3.23438
3.23438
2.9515
2.82878
2.56426
2.54453
2.43615
2.34358
2.28129
2.21473
2.08627
2.08627
1.86677
1.86677
1.80424
1.58432
1.58432
1.27128
1.27128
1.05673
1.05673
0.816708
0.816708
0.816708
0.760575
0.601009
0.601009
0.516477
0.516477
0.516477
0.416316
0.416316
0.416316
expand reflect reflect
contract inside expand reflect reflect
contract outside reflect reflect reflect reflect reflect reflect reflect
contract inside reflect expand reflect expand reflect reflect
contract inside expand
contract inside contract inside reflect expand
contract inside reflect
contract inside reflect expand
contract inside reflect
122
0.345716
reflect
124
0.345716
contract inside
126
0.285909
expand
128
0.281068
reflect
130
0.22878
reflect
132
0.22878
contract inside
134
0.203104
expand
136
0.148
expand
138
0.0999997
expand
140
141
143
145
147
148
150
152
154
156
158
160
161
163
165
166
168
170
172
174
176
178
180
182
184
185
187
189
191
193
195
197
199
201
203
0.0999997
0.0999997
0.0217142
0.0217142
0.0217142
0.0217142
0.0217142
0.0217142
0.0191193
0.00610404
0.00610404
0.00261955
0.00261955
0.000256151
0.000256151
0.000256151
0.000256151
0.00020711
0.00010357
2.09236e-005
2.09236e-005
1.80497e-006
1.80497e-006
1.80497e-006
1.80497e-006
1.80497e-006
3.74217e-007
3.74217e-007
3.26526e-007
8.07652e-008
1.66554e-008
1.66554e-008
1.66554e-008
5.57089e-009
1.86825e-009
contract inside reflect expand
contract inside contract inside reflect
contract inside contract inside reflect expand
contract outside reflect reflect reflect
contract inside reflect
contract inside contract inside contract inside contract inside contract inside reflect
contract inside contract inside contract inside reflect
contract inside contract inside contract inside contract inside contract inside contract inside contract inside contract outside contract inside
205
1.86825e-009
contract outside
112
207
5.53435e-010
contract inside
113
208
5.53435e-010
reflect
114
210
4.06855e-010
contract inside
Optimization terminated: the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-004
and F(X)satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-004
x =
1.0000
1.0000
fval =
4.0686e-010
exitflag =
output =
iterations: 114
funcCount: 210
algorithm: 'Nelder-Mead simplex direct search'
meage: [1x196 char]
