=s=1
f (x0 , y 0 ) f (x0 , y 0 ) + 0 d0 ) = f (1, 2) f (1, 2) + (6, 32) = f (1, 2) f (5, 30) =
19 (75 + 810000) = 810056 < f (x0 , y 0 )T d0 = 0.1 1 (6, 32)T (6, 32) = 106
= s = 0.1
f (x0 , y 0 ) f
CSE 241 Algorithms and Data Structures
Fall 2010
Homework 4: Solutions
1. (5 pts) Thanks!
2. (15 points)
(a) (6 pts) The vertices are visited in the order S, A, C, D, E, B. In the breadth-rst search t
CSE 543T Homework 2 Solutions
Exercise 2.1.6
First of all, notice that since f (x) 0, xi and ai , f (x) = 0 is clearly not the maximum for f (x).
Therefore, 0 < x < 1 i. The original problem is equiva
Motivation for Gradient Methods
We have the closed-form information of
f(x)
How to utilize this information
What information can we get?
Why is gradient so important?
Easy to find the descending
CSE543T: Algorithms for Nonlinear Optimization
Lecture 2: Unconstrained Optimization
Definitions and Conditions
Unconstrained Optimization
Minimize f(x)
where x is defined in X
What do we know? (Assu
CSE 241 Algorithms and Data Structures
Fall Semester, 2010
Homework 3: Solutions
1. (10 pts) The algorithm to take the union of two skiplists uses the ideas of the merge method used within mergesort
t
CSE 241 Algorithms and Data Structures
Fall Semester, 2010
Homework 2: Solutions
1. (10 points) Hand simulation of hashing.
(a) Slots 1, 2, 3, 6, and 7 are null (i.e. an empty list). Slot 0 points to
How to handle constraints
From outside the feasible set
From inside the feasible set
Any way else?
6.252 NONLINEAR PROGRAMMING
LECTURE 9: FEASIBLE DIRECTION METHODS
LECTURE OUTLINE
Conditional Gra
In the last Chapter
minimize f(x)
subject to xX, where X is convex
Necessary condition for a local minimum:
f(x*)(x x*) >= 0 for any xX
Problems:
Not working for nonconvex problems
Difficult to i
Summary of Lagrangian Theory
A necessary condition for constrained local
minimum
A unified treatment for equality and
inequality constraints
A dual view of constrained optimization
Constraints and
LEAST-SQUARES PROBLEMS
m
minimize
f (x) =
1
2
g (x)
2
=
gi (x)
1
2
2
i=1
subject to x
n,
where g = (g1 , . . . , gm ), gi :
n
ri .
Many applications:
Solution of systems of n nonlinear equations
wi
Conjugate Gradient Methods applied
to Nonquadratic Problems
min f(x), where f(x) is a general function
use the same algorithm
xk+1 = xk + k dk
Where k = argmin f(xk + dk)
dk = - gk + k dk-1
Approx