EECS416
Homework 5
Due 2/28/13
1. Consider the following problems in HW 4, which have been determined to be convex: Find the respective global
minimizers.
a) Problem14. 2.4 in GNS p.490:For the problem of finding the minimum distance from a point x0 to th
EECS416
1.
Solution to HW 1
012913
B
For convenient assume that:
AB = c
AC = b
And
Coordinates of A = (0,0)
Coordinates of B = (xB,yB)
yB
D
ysin
y
A
E
y
x
F
C
bx
Decision Variables: There are many ways to define decision variables and hence many possib
EECS416
Homework 6
Due 030513
1 This problem explores the concept of regularity and its roles in the KKT conditions.
\
Reconsider P:
2
min f ( x ) ( x1 1) 2 x2
x1 x2
s.t.
2
x ( x2 1) 1
2
1
2
x1 0
In the last HW, you have shown that KKT conditions are bo
EECS416
Solution to HW 6
Problem 1: For Problem P:
min f(x) = (x1 1)2 + x22
s.t.
g1: x1 + x2 2
0
2
2
g2: x1 + (x2 1) 1
g3: x1
a)
At
3/5/13
0
0
0.5
x (2) only g1 is active and g1(x(2) = (0.5,1.5) is not identically zero. Hence, it is clearly
1.5
linea
EECS416
1. The feasible set is
Solution to HW 2
02/05/2013
2
g1 : x12 x2 1 0
g 2 : x1 x2  2 0 X constraint set
g3 :
x2 0
This is shown as the shaded area below. All points of interest are also shown.
1.4
1.2
7 points
g : x + x  2 = 0
1
0.8
2
2
1
2
2
Summary of Topics for Midterm
1. Model building of optimization problems
2. Graphical solutions of problems with 2 variables (This is done to help provide insights
into geometric structures of optimization problems in Rn)
3. Convex functions: definitions,
EECS416
1. a)
Solution to HW 5
2/28/13
P: min f ( x ) 1 ( x x0 )T ( x x0 )
n
2
xR
s.t. h( x ) : a T x b 0
P is clearly a strictly convex problem. If it has a finite stationary point, it will be a unique global minimizer.
Stationarity:
L( x, ) 1 ( x x0 )T
EECS416
1.
Sample Questions for Test 1
Consider a network of roads shown below where all traffics coming to intersection 1 will try to go
to intersection 4 through either intersection 2 or intersection 3.
2
4
V
1
3
The travel time between a pair of nodes
EECS416
1.
5 points
Solution to HW 4
a) To show that
Fall 2012
,
f ( x) x is convex in R cfw_x R  x 0 for all p 1 or p 0 and concave for 0 p 1
p
we look at
f ( x) px p1 and f ( x) p( p 1) x p2 .
Thus ,
i) For p 1, f ( x ) 0 for all x 0 f ( x) is convex i
EECS416
1.
Homework 1
Due 1/29/13
Given a triangle ABC with the coordinates of A, B and C at (0, 0), (10, 20) and (30,0) respectively, find a
parallelogram ADEF (with D on AB, E on BC and F on AC as shown) that has the largest area possible
B
D
A
E
F
C
a)
EECS416
1.
Homework 7
Due 3/26/13
The following problems deal with convergence and rate of convergence of numerical algorithms.
Show that each the following sequences generated by a numerical algorithm converges. In each case,
determine the order of conve
EECS 416
Alternative Case Study:
Due 4/28/2014
Classifying Epilepsy EEG data using SVM
The goal of this study is to try develop a classifier to classify EEG signals to predict
whether a particular patient will experience an onset seizure episode. The tech
EECS416
1.
Sample Questions for Midterm
Consider a network of roads shown below where all traffics coming to intersection 1 will try to go
to intersection 4 through either intersection 2 or intersection 3.
2
4
V
1
3
The travel time between a pair of nodes
EECS416
Supplementary Notes
Optimality Conditions for Convex Constrained Problems: KKT Theorem
We first consider convex problems. Assume f(x): RnR is continuously differentiable in Rn.
A) Equality Constrained Problems
ECP:
min f ( x)
n
xR
s.t. Ax b 0
wher
Copyright 2004 by Vira Chankong
Case Western Reserve University
Handout: CONVEXITY AND GLOBAL SOLUTION
In general, the optimally conditions (5) and (6) can only identify local optimal solutions. A point
passing both tests can be assured to be the best amo
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Copyright 2004 Vira Chankong
Case Western Reserve University
Optimality Conditions
1. Introduction
In this note, we develop optimality conditions and discuss their uses. The basic questions
addressed here are these: Given an optimization problem
P:
min f(
Copyright 2004 by Vira Chankong
Case Western Reserve University
Handout:
OPTIMALITY CONDITIONS FOR UNCONSTRAINED PROBLEMS
We now begin our development of optimality conditions by first considering an unconstrained
optimization problem in which it is desir
Copyright 2004 by Vira Chankong
Case Western Reserve University
Handout: MATHEMATICAL BACKGROUND
In the first part of this course, we develop optimality conditions and discuss their uses. The basic
questions addressed here are these: Given an optimization
EECS416
Case Study
Due Monday 4/28/14
Positron Emission Tomography Image Reconstruction
(Section 1.7.5 of Linear and Nonlinear Programming by GrivaNashSofer, SIAM, 2009)
Positron emission tomography (PET) is a medical imaging technique that helps diagno
rsimplex.m
The revised simplex (matrixbased) code rsimplex.m takes an LP in any form and calls the following function
modules in the sequence shown below.
Given LP
S_form.m
Convert to Standard Form
If necessary
Print: LP infeasible
Note: w* = optimal
Pha
1/23/2016
EECS416 Convex Optimization in Engineering
Analytical Methods for Unconstrained Problems
Goals for Week 3
To begin looking into methods for solving lowdimension
optimization problems:
Graphical methods for 2D problems
Analytical methods for
1/8/16
EECS416 Convex Optimization in Engineering
Course Overview and Model Building
Motivation
1
1/8/16
Motivation
Completing the triad:
Computational Engineering and Science (CES) has emerged as the critical third pillar
for major scientific discovery

Copyright 2014 by Vira Chankong
Case Western Reserve University
CONVEXITY AND GLOBAL SOLUTION
In general, the optimally conditions (5) and (6) can only identify local optimal solutions. A point
passing both tests can be assured to be the best among its cl
1/29/2016
EECS416 Convex Optimization in Engineering
Convex Functions
Goals for Week 4
To look at fascinating properties of convex functions
To learn how to certify convexity of a function
To appreciate why minimizers of convex functions are
always glo
1/19/2016
EECS416 Convex Optimization in Engineering
Modeling and Solution using MATLABLP and MILP
Best Practice in Optimization
Successful practice of optimization requires:
Good modeling skills
Understanding of efficient and reliable algorithms
Selectio
Copyright 2014 by Vira Chankong
Case Western Reserve University
MATHEMATICAL BACKGROUND
In the first part of this course, we develop optimality conditions and discuss their uses. The basic
questions addressed here are these: Given an optimization problem
1/8/16
EECS416 Convex Optimization in Engineering
Course Overview and Model Building
Best Practice in Optimization
Successful practice of optimization
requires:
Good modeling skills
Understanding of efficient and reliable
algorithms
Selection and use o
1/19/2016
EECS416 Convex Optimization in Engineering
Modeling and Solution using MATLABNLP
Economic Dispatch of Electrical Power: Example
A power grid consists of busses (to which online power generators or loads (demands) are connected) and
transmission
1/30/2016
EECS416 Convex Optimization in Engineering
Key Properties of Convex Functions
Goals
To verify key properties of convex functions
To verify why minimizers and stationary points of convex
functions are always global minimizers
1
1/30/2016
Convex