Lecture 9 – Nonlinear Programming
Models
Topics
• Convex sets and convex programming
• First-order optimality conditions
• Examples
• Problem classes
General NLP
Minimize f(x)
s.t. gi(x) (, , =) bi, i = 1,…,m
x = (x1,…,xn)T is the n-dimensional vector of
decision variables
f (x) is the objective function
gi(x) are the constraint functions
bi are fixed known constants
Convex Sets
Definition: A set S  n is convex if every point on the line
segment connecting any two points x1, x2  S is also in S.
Mathematically, this is equivalent to
x0 = lx1 + (1–l)x2  S for all l such 0 ≤ l ≤ 1.
x1

x2


x1
x1

x2
x2


(Nonconvex) Feasible Region
S = {(x1, x2) : (0.5x1 – 0.6)x2 ≤ 1
x2
2(x1)2 + 3(x2)2 ≥ 27; x1, x2 ≥ 0}
x1
Convex Sets and Optimization
Let S = { x  n : gi(x)  bi, i = 1,…,m }
Fact: If gi(x) is a convex function for each i = 1,…,m then S is a
convex set.
Convex Programming Theorem: Let x  n and let f (x) be a
convex function defined over a convex constraint set S. If a
finite solution exists to the problem
Minimize { f (x) : x  S }
then all local optima are global optima. If f (x) is strictly
convex, the optimum is unique.
Convex Programming
Min f (x1,…,xn)
Max f (x1,…,xn)
s.t. gi(x1,…,xn)  bi
i = 1,…,m
x1  0,…,xn  0
s.t. gi(x1,…,xn)  bi
i = 1,…,m
x1  0,…,xn  0
is a convex program if
f is convex and each gi
is convex.
is a convex program if
f is concave and each
gi is convex.
Linearly Constrained Convex Function
with Unique Global Maximum
Maximize f (x) = (x1 – 2)2 + (x2 – 2)2
x2
5
subject to –3x1 – 2x2 ≤ –6
4
–x1 + x2 ≤ 3
3
x1 + x2 ≤ 7
2
2x1 – 3x2 ≤ 4
1
1
2
3
4
5
x1
(Nonconvex) Optimization Problem
First-Order Optimality Conditions
Minimize { f (x) : gi(x)  bi, i = 1,…,m }
Lagrangian: L(x,)  f (x)   i gi (x)  bi 
m
i1
Optimality conditions
m
• Stationarity: L(x,)  f (x)   igi (x)  0
i1
• Complementarity: igi(x) = 0, i = 1,…,m
• Feasibility: gi(x)  bi, i = 1,…,m
• Nonnegativity: i  0, i = 1,…,m
Importance of Convex Programs
Commercial optimization software cannot guarantee
that a solution is globally optimal to a nonconvex
program.
NLP algorithms try to find a point where the gradient of the
Lagrangian function is zero – a stationary point – and
complementary slackness holds.
Given
L(x,) = f(x) + (g(x) – b)
we want
L(x,) = f(x) + g(x) = 0
g(x) – b) = 0
g(x) – b ≤ 0,   0
For a convex program, all local solutions are global
optima.
Example: Cylinder Design
We want to build a cylinder (with a top and a bottom)
of maximum volume such that its surface area is no
more than s units.
Max V(r,h) = pr2h
s.t.
2pr2 + 2prh = s
r
h
r  0, h  0
There are a number of ways to approach this problem.
One way is to solve the surface area constraint for h and
substitute the result into the objective function.
Solution by Substitution
2
s

2p
r
s  2pr 2
rs
2
Volume
=
V
=
pr
 pr 3
[
] =

h=
2
2pr
2pr
dV
s 1/2
s
s 1/2
= 0 r=(
)
h = 2pr  r = 2( p)
dr
6p
6
V =
pr 2h
s 3/2
= 2p ( p)
6
s 1/2
r = ( p)
6
s 1/2
)
h = 2(
6p
Is this a global optimal solution?
Test for Convexity
dV(r)
s
rs
3
 pr  dr = 2  3pr 2 
V(r ) =
2
d2V(r )
dr 2
d2V 0 for all r 0
dr 2
Thus V(r ) is concave on r 0 so the solution is a
global maximum.
 6pr
Advertising (with Diminishing Returns)
• A company wants to advertise in two regions.
• The marketing department says that if $x1 is spent in
region 1, sales volume will be 6(x1)1/2.
• If $x2 is spent in region 2, sales volume will be 4(x2)1/2.
• The advertising budget is $100.
Model: Max f (x) = 6(x1)1/2 + 4(x2)1/2
s.t.
x1 + x2  100, x1  0, x2  0
Solution: x1* = 69.2, x2* = 30.8, f (x*) = 72.1
Is this a global optimum?
Excel Add-in Solution
A
B
C
D
L
K
J
I
H
G
F
E
Objective Terms Solver: Excel Solver
Name: Adv100
0 Type: Nonlinear
Linear:
Type: NLP1
Goal: Max NonLinear 1: 72.111 Sens.: Yes
0
Objective: 72.111NonLinear 2:
1 Nonlinear Model
2 72.111
Change
2
3
4 TRUE
Solve
5 TRUE
Variables
100
6
Name:
Change Relation
7
Values:
8
Lower Bounds:
9
10
11
Linear Obj. Coef.:
12
Nonlinear Obj. Terms:
13
Nonlinear Obj. Coef.:
14
Constraints
15
RHS
Rel.
Num. Name Value
16
100
<=
100
Con1
1
17
10000
<=
0
Con2
2
18
19
2
1
X2
X1
69.231 30.769
0
0
0
8.3205
6
0
5.547
4
Linear Constraint Coefficients
1
1
0
0
M
N
O
Comp. Time 00:00
Status Optimal
Portfolio Selection with Risky Assets (Markowitz)
• Suppose that we may invest in (up to) n stocks.
• Investors worry about (1) expected gain (2) risk.
Let rj = random variable associated with return on stock j
j = expected return on stock j
sjj variance of return for stock j
We are also concerned with the covariance terms:
sij = cov(ri, rj)
If sij > 0 then returns on i and j are positively correlated.
If sij < 0 returns are negatively correlated.
Decision Variables: xj = # of shares of stock j purchased
n
Expected return of the portfolio: R(x) =  jxj
j =1
n
Variance (measure of risk): V(x) = 
n
sijxixj
i =1 j =1
Example:
s 11 s 12   1 1



s 21 s 22   1 1 
If x1 = x2 = 1, we get
V(x) = s11x1x1 + s12x1x2 + s21x2x1 + s22x2x1
= 1 + (1) + (1) + 1 = 0
Thus we can construct a “risk-free” portfolio (from
variance point of view) if we can find stocks “fully”
negatively correlated.
s 11 s 12  1 1
If 

 , then buying stock 2 is just like
s 21 s 22  1 1
buying additional shares of stock 1.
Nonlinear optimization models …
Let pj = price of stock j
b = our total budget
b  risk-aversion factor (when b  0 risk is not a factor)
Consider 3 different models:
1) Max f (x) = R(x) – bV(x)
n
s.t.
pj xj b, xj  0, j = 1,…,n
j =1
where b  0 is determined by the decision maker
2) Max f (x) = R(x)
s.t.
n
V(x) , pjxj b, xj  0, j = 1,…,n
j =1
where   0 is determined by the investor. Smaller
values of  represent greater risk aversion.
3) Min f (x) = V(x)
s.t.
n
R(x) , pj xj b, xj  0, j = 1,…,n
j =1
where   0 is the desired rate of return
(minimum expectation) is selected by the investor.
Hanging Chain with Rigid Links
10ft
1 ft
x
each link
y
What is equilibrium shape of chain?
Decision variables: Let (xj, yj), j = 1,…,n, be the incremental
horizontal and vertical displacement of each link,
where n  10.
Constraints:
xj2 + yj2 = 1, j = 1,…,n, each link has length 1
x1 + x2 +
•••
+ xn = 10, net horizontal displacement
y1 + y2 +
•••
+ yn = 0, net vertical displacement
Objective: Minimize chain’s potential energy
Assuming that the center of the mass of each link is at
the center of the link. This is equivalent to minimizing
1 y + (y + 1 y ) + (y + y + 1 y ) +
1
1
1
2
2 3
2 2
2
+ (y1 + y2 +
1
1
= (n  1 + 2 ]y1 + (n  2 + 2 )y2
1
+ (n  3 + 2 )y3 +
•••
•••
•••
1
+ yn-1 + 2 yn)
3
1
+ 2 yn-1 + 2 yn
Summary
n
Min  (n  j + ½)yj
j =1
s.t. xj2 + yj2 = 1, j = 1,…,n
x1 + x2 +
•••
+ xn = 10
y1 + y2 +
•••
+ yn = 0
Is a local optimum guaranteed to be a global optimum?
No!
Constraints xj2 + yj2 = 1 for all j yield a nonconvex
feasible region so there may be several local optima.
Consider a chain with 4 links:
These solutions are both local minima.
Direct Current Network
10 
20
I2
I4
I3
I1
100v
I6
I5
10
20 
I7
Problem: Determine the current flows I1, I2,…,I7 so that the total
content is minimized
Content: G(I) =
0
I
0
v(i)di for I ≥ 0 and G(I) = I v(i)di for I < 0
Solution Approach
Electrical Engineering: Use Kirchoff’s laws to find
currents when power source is given.
Operations Research: Optimize performance measure
in network taking flow balance into account.
Linear resistor: Voltage, v(I ) = IR
Content function, G(I ) = I 2R/2
Battery: Voltage, v(I ) = –E
Content function, G(I ) = –EI
Network Flow Model
Network diagram:
2
2
5 I2
2
3
10 I 4
5
5 I 32
-100I1
1
0 I6
4
2
10 I5
0 I7
6
Minimize Z = –100I1 + 5I22 + 5I32 + 10I42 + 10I52
subject to I1 – I2 = 0, I2 – I3 – I4 = 0, I5 – I6 = 0,
I5 + I7 = 0, I3 + I6 – I7 = 0, –I1 – I6 = 0
Solution: I1 = I2 = 50/9, I3 = 40/9, I4 = I5 = 10/9,
I6 = –50/9, I7 = –10/9
NLP Problem Classes
• Constrained vs. unconstrained
• Convex programming problem
• Quadratic programming problem
f (x) = a + cTx + ½ xTQx, Q  0
• Separable programming problem
f (x) = j=1,n fj(xj)
• Geometric programming problem
g(x) = t=1,T ctPt(x), Pt(x) = (x1at1) . . . (xnatn), xj > 0
• Equality constrained problems
What You Should Know About
Nonlinear Programming
• How to identify a convex program.
• How to write out the first-order optimality
conditions.
• The difference between a local and global
solution.
• How to classify problems.