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The polytope algorithm for multivariate unconstrained optimization, a method for finding the minimum of a function when derivatives are not available. The algorithm is detailed step by step with examples in both matlab and hand calculations.
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Number of functionevaluationsnincreases as e^ , where n is numberof dimensions.
The Polytope Algorithm • This is a direct search method. • Also known as “simplex” method. • In n dimensional case, at each stage wehave n+1 points
Polytope Example • For n = 2 we have three points at eachstep.^ x^1 x 3
a(c - x ) cc - x^33 x^2 (worst point)
x r
Detailed Polytope Algorithm 1. Evaluate F(
MATLAB Example Polytope >> banana = @(x)10*(x(2)-x(1)^2)^2+(1-x(1))^2; >> [x,fval] = fminsearch(banana,[-1.2, 1],optimset('Display','iter'))
Polytope Example by Hand^ (^
)^ (^
) (^
(^2222222) ) (^22)
(^5). (^0) ) (^1) () 2 ( 3
(^5). (^0) ) (^1) ( (^25). 0 ),(
+−−++− −−+− +=^ yx
yx yxyxF
Polytope Example: Step 1 • Polytope is i 1
2 3 x (0,0)i^
(0,0.5)
(0.433,0.25) F( x )^ 9.7918i
7.^
-^ Worst point is
x ,^ c^ = ( x +^ x )/2 = (0.2165,0.375)^123
-^ Relabel points:
x fl^ x ,^ x fl^ x^3113
-^ x =^ c^ +^ a( c -^ x r^
) = (0.433,0.75) and F( 3
x )=3.6774r^
-^ F( x )< F( x ) sor^1
x is best point so try to expand.r^
-^ x =^ c^ +^ b( x - c e^ r^
) = (0.5413,0.9375) and F(
x )=3.1086e
-^ F( x )< F( x ) so accept expander^
After Step 1 2 1 1
2
After Step 2 2 1 1
Polytope Example: Step 3 • Polytope is i 1
2 3 x (0.5413,0.9375)i^
(0.433.0.25)^
(0.9743,0.6875) F( x )^ 3.1086i
-^ Worst point is
x ,^ c^ = ( x +^ x )/2 = (0.7578,0.8125)^213
-^ Relabel points:
x fl^ x ,^ x fl^ x ,^3123
x fl^ x^12
-^ x =^ c^ +^ a( c -^ x r^
) = (1.0826,1.375) and F( 3
x )=3.1199r^
-^ F( x )>F( x ) so polytope is too big. Need to contract.r^2 •^ x =^ c^ +^ g( x - c c^ r^
) = (0.9202,1.0938) and F(
x )=2.2476c^
-^ F( x )<F( x ) so accept contraction.c^ r^
Polytope Example: Step 4 • Polytope is i 1
2 3 x (0.9743,0.6875)i^
(0.5413,0.9375)
(0.9202,1.0938) F( x )^ 2.0093i
-^ Worst point is
x ,^ c^ = ( x +^ x )/2 = (0.9472,0.8906)^213
-^ Relabel points:
x fl^ x ,^ x fl^ x^3223
-^ x =^ c^ +^ a( c - x r^
) = (1.3532,0.8438) and F( 3
x )=2.7671r^
-^ F( x )>F( x ) so polytope is too big. Need to contract.r^2 •^ x =^ c^ +^ g( x - c c^ r^
) = (1.1502,0.8672) and F(
x )=2.1391c^
-^ F( x )<F( x ) so accept contraction.c^ r^
After Step 4 2 1 1
2
After Step 5 2 1 1
2
Polytope Example: Step 6 • Polytope is i 1
2 3 x (0.9743,0.6875)i^
(1.1502,0.8672)
(0.9912,0.9355) F( x )^ 2.0093i
-^ Worst point is
x ,^ c^ = ( x +^ x )/2 = (0.9827,0.8117)^213
-^ Relabel points:
x fl^ x ,^ x fl^ x^3223
-^ x =^ c^ +^ a( c -^ x r^
) = (0.8153,0.7559) and F( 3
x )=2.1314r^
-^ F( x )>F( x ) so polytope is too big. Need to contract.r^2 •^ x =^ c^ +^ g( x - c c^ r^
) = (0.8990,0.7837) and F(
x )=2.0012c^
-^ F( x )<F( x ) so accept contraction.c^ r^
