Complexity Issues

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Complexity Issues
http//www.math.iastate.edu/wu/math597.html
http//www.math.iastate.edu/wu/Math597HW0000

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

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Local Optimization
A necessary condition for a point x to be a
minimizer of f is the gradient of f at x is
equal to zero.
x is a local minimizer of f if for any x in a
small neighborhood of x, f (x) gt f (x).

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Newtons Method
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Newton Step
Matlab
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Solving Linear Systems
Hs -g
LTs t
H LLT
LLTs -g
Lt -g
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Solving Linear Systems
h11s1 h12s2 h1nsn -g1
h21s1 h22s2 h2nsn -g2
. . .
hn1s1 hn2s2 hnnsn -gn
g1 g2 g .
. . gn
Hs -g
l11t1
-g1 l21t1 l22t2
-g2 .
. . ln1t1 ln2t2
lnntn -gn
t1 t2 t .
. . tn
Lt -g
l11s1 l21s2 ln1sn -t1
l22s2 ln2sn
-t2 . .
.
lnnsn -tn
s1 s2 s .
. . sn
LTs t
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Total Calculations
H LLT
n3 / 3
Lt -g
n2
LTs t
n2
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Further Reading
Numerical Methods for Unconstrained Minimization
and Nonlinear Equations by John Dennis and Robert
Schnabel
Practical Methods of Optimization by Roger
Fletcher
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Matlab Code for DME Calculation
gtgt gtgt for i 1 n for j 1 n
DX (i,j) sum ((X (i,) X (j,))
. 2) DX (i,j) sqrt (DX
(i,j)) end end gtgt gtgt for i 1
n for j 1 n DY
(i,j) sum ((Y (i,) Y (j,)) . 2)
DY (i,j) sqrt (DY (i,j)) end
end gtgt gtgt dme sqrt (sum (sum ((DX DY)
. 2))) / n gtgt
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Matlab Code for RMSD Calculation
gtgt gtgt xc sum (X) / n yc sum (Y) / n gtgt gtgt
XX (,1) X (,1) - xc (1) gtgt XX (,2) X
(,2) - xc (2) gtgt XX (,3) X (,3) - xc
(3) gtgt gtgt YY (,1) Y (,1) - yc (1) gtgt YY
(,2) Y (,2) - yc (2) gtgt YY (,3) Y (,3)
- yc (3) gtgt gtgt C YY XX gtgt U, S, V
svd ( C ) gtgt Q U V gtgt gtgt rmsd sqrt
(sum (sum ((XX YY Q) . 2)) / n) gtgt
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fid fopen ('protein.pdb','r') numlines 0
ENDT while ENDT END numlines
numlines 1 A (numlines, 179) fscanf
(fid, 'c', 1,79) A (numlines, 80) fscanf
(fid, c\n, 1,1) ENDT A (numlines,
13) end fclose (fid) for i 1 numlines
if A (i,14) 'ATOM' Astart i
break end end for i Astart numlines if
A (i,14) 'ATOM' Alast i
end end coords A (AstartAlast, 3154) fid
fopen ('coords.dat','w') for i 1
Alast-Astart1 fprintf (fid, 'c', coords
(i,)) fprintf (fid, '\n') end
READ PDB
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fid fopen ('protein.pdb','r') numlines 0
ENDT while ENDT END numlines
numlines 1 A (numlines, 179) fscanf
(fid, 'c', 1,79) A (numlines, 80) fscanf
(fid, c\n ,1,1) ENDT A (numlines,
13) end fclose (fid) for i 1 numlines
if A (i,14) 'ATOM' Astart i break
end end for i Astart numlines if A
(i,14) 'ATOM' Alast i
end end fid fopen ('coords.dat','r') coords
fscanf (fid, 'f f f\n', 3,inf) coords
coords' fclose (fid) for i Astart Alast
if A (i,14) 'ATOM' A (i,3138)
sprintf ('8.3f',coords(i-Astart1,1)) A
(i,3946) sprintf ('8.3f',coords(i-Astart1,2))
A (i,4754) sprintf ('8.3f',coords(i-As
tart1,3)) end end fid fopen
('protein.pdb','w') for i 1 numlines
fprintf (fid, 'c', A (i,)) fprintf (fid,
'\n') end fclose (fid)
WRITE PDB
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Homework Assignment 3 Due 6pm, Wed, September 24th
1. In Newtons method for local minimization, let
the Hessian matrix be approximated by an identity
matrix. The method then becomes a so-called
gradient method. Consider a one-dimensional
function f . The gradient method searches for a
local minimizer of f with the following
iterative formula xk1 xk ?k g
(xk), where xk and xk1 are the iterates in kth
and k1th iterations, g (xk) is the first
derivative of f at xk, and ?k is a parameter to
be determined in every iteration so that the
sequence of the iterates can be guaranteed to
converge globally to a minimizer of f . Design a
strategy for choosing ?k and use it to write a
Matlab code for the one-dimensional version of
the gradient method.
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Homework Assignment 3 Due 6pm, Wed, September 24th

• Test your program on the following problem
• min f (x)
• f (x) 10 x argtan (x) 5 ln (x2 1)
• with ?0 1 and x0 2.0, and observe the
  convergence behavior of the algorithm with or
  without your global convergence strategy. Turn in
  your program and a short description on your test
  results.