Math 287 - Nonlinear Optimization - Spring 2014
Answers and Solutions for Practice Problems, Topic 2
NOTE: Answers and solutions provided here may or may not contain full details. In
your own homework problems you are always expected to provide full detai
Math 287
Nonlinear Optimization
Spring 2015
Quadratic Interpolation:
(1) Implements 4PBUF.
(2) At each iteration, given ak < ck < bk , nd quadratic
qk (x) through (ak , f (ak ), (ck , f (ck ), (bk , f (bk ). Let
dk minimize qk (x). We hope qk has a minimi
Math 287
Nonlinear Optimization
Spring 2015
Regula falsi, continued.
Termination criteria: (i) |g(ck )| < ;
(ii) |ck+1 ck | < ;
(iii) k > K (wont diverge but may converge very
slowly).
But not |bk ak | < since usually bk ak 0.
ak never changes,
bk+1 = ck
Math 287
Nonlinear Optimization
Spring 2015
1. INTRODUCTION
Optimization models
Optimization = nd optimal = best = maximum or minimum
General optimization problem: max/min f (x)
objective function, real-valued
subject to x = (x1 , x2 , . . . , xn ) S
fe
Math 287
Nonlinear Optimization
Spring 2015
How do we nd minimizers? For continuous function of one variable on closed interval, need to
look at (a) critical points (f = 0), and (b) boundary points (ends). Worry about (b) later;
for now, what about minimi
4. [12] The function g(x) = XZ + 2x - 4 is known to have a root (a place where g(x) = 0) in the interval
[0,2]. Perform the specified number of iterations of each of the following methods to obtain a better idea
of where the root is. In each case state th
Math 287 - Nonlinear Optimization - Spring 2015
Solution for problem due 12th March
3A. The equation x2 + x1 x2 + x2 = 12 denes an ellipse with axes along the lines y = x. This
2
1
ellipse intersects the parabola x2 = 2(x1 + 1)2 8 in four places.
(a) Draw
Math 287 - Nonlinear Optimization - Spring 2015
Solutions for problems due 5th February
2A. In the early days of computers, computers did not have built-in division algorithms. Some
computers did division by a by computing the reciprocal 1/a using Newtons
Math 287 - Nonlinear Optimization - Spring 2015
Solutions for problems due 22nd January
1D. Find the critical point of the quadratic f (x, y) = 2x2 8xy +10y 2 4y +19 and use deniteness
properties of the Hessian matrix to say as much as you can about wheth
Math 287 - Nonlinear Optimization - Spring 2015
Answers and Solutions for Practice Problems, Topic 2
NOTE: Answers and solutions provided here may or may not contain full details. In
your own homework problems you are always expected to provide full detai
Math 287 - Nonlinear Optimization - Spring 2015
Solution for problem due 15th January
1B. (a) Formulate the problem of nding the best even quadratic (quadratic symmetric about the
y-axis) y = a0 + a2 x2 through the the points (0, 7), (1, 6) and (2, 1) as
Math 287 - Nonlinear Optimization - Spring 2015
Solutions for problems due 29th January
1H. A function f : Rn R is ane if f (x + y) = f (x) + f (y) whenever x, y Rn and
, R with + = 1. (Note that there is no other restriction on the value of or ; in
parti
Math 287 - Nonlinear Optimization - Spring 2015
Solution for problem due 12th February
2F. The function
f (x) = 4 4x + 9x2 12x3 + 6x4 2 cos(1 2x) sin(1 2x)
has a local minimizer somewhere between 0 and 1. Find this minimizer by the following methods. Use
Math 287 - Nonlinear Optimization - Spring 2015
Solutions for problems due 19th February
2D. In quadratic interpolation, we interpolate a quadratic q(x) through three points (x1 , y1 ),
(x2 , y2 ) and (x3 , y3 ) by setting q(x) = y1 p1 (x) + y2 p2 (x) + y
Math 287 - Nonlinear Optimization - Spring 2015
Answers and Solutions for Practice Problems, Topic 3
NOTE: Answers and solutions provided here may or may not contain full details. In
your own homework problems you are always expected to provide full detai
Math 287
Nonlinear Optimization
Spring 2015
In fact, Newtons Method can be thought of as steepest descent after change of coordinates to make
contours more uniform.
Newton = Scaling + Steepest Descent: Let S be matrix so that
S symmetric,
S positive denit
Math 287 - Nonlinear Optimization - Spring 2014
Answers and Solutions for Practice Problems, Topic 1
NOTE: Answers and solutions provided here may or may not contain full details. In
your own homework problems you are always expected to provide full detai
Math 287 - Nonlinear Optimization - Spring 2014
Solutions for problems due 23rd January
1B. (a) Formulate the problem of nding the best even quadratic (quadratic symmetric about the
y -axis) y = a0 + a2 x2 through the the points (0, 8), (1, 5) and (2, 1)
Math 287 - Nonlinear Optimization - Spring 2014
Solution for problem due 30th January
1H. Let r 0, and let T : Rn Rm be a linear transformation. Prove that the set S = cfw_x
Rn | T (x) r is a convex set, where x = x2 + x2 + . . . + x2 is the usual lengt
Math 287 - Nonlinear Optimization - Spring 2014
Solutions for problems due 6th February
1I. Suppose f is a convex function on Rn . Prove that the set S of global minimizers of f is a
convex set.
Solution: Let x, y S and [0, 1]. Let z = x + (1 )y ; we must
Math 287
Nonlinear Optimization
Spring 2014
2. METHODS OF 1-DIMENSIONAL UNCONSTRAINED OPTIMIZATION
1-dim. optimization special case, useful in itself;
also generalize ideas to multi-dim. methods;
also use 1-dim. methods for line search as part of multi
Math 287
Nonlinear Optimization
Spring 2014
Convexity
Many variations: pseudoconvex, quasiconvex, etc. Important because can guarantee local minimizer
is global minimizer.
Three related concepts: convex combinations, convex sets and convex functions.
Conv
Math 287
Nonlinear Optimization
Spring 2014
(B) Methods using f but not f
To get anywhere, must assume f unimodal on interval [a, b]:
decreases strictly on [a, x ], then increases strictly on
[x , b].
f is strictly decreasing on interval I : x1 < x2 f (x
Math 287
Nonlinear Optimization
Spring 2014
1. INTRODUCTION
Optimization models
Optimization = nd optimal = best = maximum or minimum
General optimization problem: max/min f (x)
objective function , real-valued
subject to x = (x1 , x2 , . . . , xn ) S
f
Math 287
Nonlinear Optimization
Spring 2015
Convexity
Many variations: pseudoconvex, quasiconvex, etc. Important because can guarantee local minimizer
is global minimizer.
Three related concepts: convex combinations, convex sets and convex functions.
Conv