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CS3331: Numerical Methods
Quiz 5
June. 13, 1997
- 1.
- (4%) We want to use steepest descent (or gradient descent)
to minimize an objective function , where
.
- (a)
- List the simple formula of steepest descent
for minimizing .
- (b)
- List the normalized version of steepest descent
for minimizing .
- 2.
- (7%) Let E(x)=x2-2x.
- (a)
- (2%) Write down the simple steepest descent
formula for xn+1 (in terms of xn and ).
- (b)
- (3%) Find the general solution of xn+1
(in terms of x0 and ).
- (c)
- (2%) What's the range for for
the convergence of the simple steepest descent formula?
- 3.
- (5%) Let E(x) = x3-x2.
- (a)
- (2%) Write down the simple steepest descent
formula for xn+1 (in terms of xn and ).
- (b)
- (3%) What's the range for for the
convergence of the simple steepest descent formula?
(Suppose that we want to find the minimum between
x=0 and x=1.)
- 4.
- (12%)
Let E(x,y) = x2-2xy+y2.
- (a)
- (2%) Write down the simple steepest descent
formula for
(in terms of xn, yn, and ).
- (b)
- (2%) To find the gradient path, we can assume that
a given point
on the gradient path has a velocity vector
equal to the negative gradient vector of E(x,y).
In other words, we can have the formula
Find the matrix A.
- (c)
- (4%) Solve the above equation by assuming that
. What is , , , and
?
- (d)
- (2%) What is c1 and c2 if
x(0)=1 and y(0)=0?
- (e)
- (2%) So what's the function of the gradient path
when expressed as a y=f(x)?
- 5.
- (2%) Plot the paths of gradient descent on the following
contour plot of the "peaks" function. The points labeled
with "*" is the starting locations.
Figure:
The function of "peaks" and its contour. Please plot the ideal
gradient paths on top of the contour plot.
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J.-S. Roger Jang
3/20/1998