Lecture 9 – Nonlinear Programming Models

Topics

• Convex sets and convex programming

• First-order optimality conditions

• Examples

• Problem classes

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

General NLP

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 = x1 + (1–)x2 S for all such 0 ≤ ≤ 1.

x1

x2

x1x1x2

x2

S = {(x1, x2) : (0.5x1 – 0.6)x2 ≤ 1

2(x1)2 + 3(x2)2 ≥ 27; x1, x2 ≥ 0}

(Nonconvex) Feasible Region

x

1

x

2

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.

Max f (x1,…,xn)

s.t. gi(x1,…,xn) bi

i = 1,…,mx1 0,…,xn 0

is a convex program if f is concave and each gi is convex.

Convex Programming

Min f (x1,…,xn)

s.t. gi(x1,…,xn) bi

i = 1,…,mx1 0,…,xn 0

is a convex program if f is convex and each gi is convex.

x11 2 3 4 5

1

2

3

4

5

x2

Maximize f (x) = (x1 – 2)2 + (x2 – 2)2

subject to –3x1 – 2x2 ≤ –6

–x1 + x2 ≤ 3

x1 + x2 ≤ 7

2x1 – 3x2 ≤ 4

Linearly Constrained Convex Function with Unique Global Maximum

(Nonconvex) Optimization Problem

First-Order Optimality Conditions

Minimize { f (x) : gi(x) bi, i = 1,…,m }

Lagrangian: L(x,μ) = f (x) + μi gi (x) −bi( )

i=1

m

∑

∇L(x,μ) =∇f (x) + μi∇gi (x)

i=1

m

∑ =0

Optimality conditions

• Stationarity:

• Complementarity: μigi(x) = 0, i = 1,…,m

• Feasibility: gi(x) bi, i = 1,…,m

• Nonnegativity: μi 0, i = 1,…,m

Commercial optimization software cannot guarantee that a solution is globally optimal to a nonconvex program.

Importance of Convex Programs

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,μ) = f(x) + μ(g(x) – b)

we want L(x,μ) = f(x) + μg(x) = 0

μg(x) – b) = 0

g(x) – b ≤ 0, μ 0

For a convex program, all local solutions are global optima.

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) = r2h

s.t. 2r2 + 2rh = s

r 0, h 0

r

h

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.

Example: Cylinder Design

h =s 2r

2

2rVolume = V = r2

[

s 2r 2

2r] =

rs 2

r 3

dVdr

= 0 r = (s

6)1/2

h = s

2r r =2(s

6)1/2

V = r 2h = 2(

s 6)

3/2

r = (s

6)1/2

h = 2(s

6)1/2

Is this a global optimal solution?

Solution by Substitution

V(r ) = rs 2

r 3 dV(r) dr =

s2

3r 2 d2V(r )

dr 2 r

d2V dr 2

0 for all r 0

Thus V(r ) is concave on r 0 so the solution is a global maximum.

Test for Convexity

• 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.

Advertising (with Diminishing Returns)

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

1

23456789

10111213141516171819

A B C D E F G H I J K L M N O

Nonlinear Model Name: Adv100 Objective Terms Solver: Excel Solver

72.111 Type: NLP1 Linear: 0 Type: Nonlinear2 Change Goal: Max NonLinear 1: 72.111 Sens.: Yes Comp. Time 00:00

TRUE Objective: 72.111NonLinear 2: 0 Status OptimalTRUE Solve100 Variables 1 2

Change Relation Name: X1 X2Values: 69.231 30.769

Lower Bounds: 0 0

Linear Obj. Coef.: 0 0Nonlinear Obj. Terms: 8.3205 5.547Nonlinear Obj. Coef.: 6 4

ConstraintsNum. Name Value Rel. RHS Linear Constraint Coefficients

1 Con1 100 <= 100 1 12 Con2 0 <= 10000 0 0

Let rj = random variable associated with return on stock j

μj = expected return on stock j

jj variance of return for stock jWe are also concerned with the covariance terms:

ij = cov(ri, rj)

If ij > 0 then returns on i and j are positively correlated.

If ij < 0 returns are negatively correlated.

Portfolio Selection with Risky Assets (Markowitz)

• Suppose that we may invest in (up to) n stocks.

• Investors worry about (1) expected gain (2) risk.

Decision Variables: xj = # of shares of stock j purchased

R(x) = μjxj

n

j =1Expected return of the portfolio:

V(x) = ijxixj

Variance (measure of risk): n

i =1

n

j =1

V(x) = 11x1x1 + 12x1x2 + 21x2x1 + 22x2x1

= 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.

Example: If x1 = x2 = 1, we get

11 12

21 22

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

1 −1−1 1

⎡

⎣⎢

⎤

⎦⎥

Nonlinear optimization models …Let pj = price of stock j

b = our total budget

risk-aversion factor (when 0 risk is not a factor)

Consider 3 different models:1) Max f (x) = R(x) – V(x)

s.t. pj xj b, xj 0, j = 1,…,n

where 0 is determined by the decision maker

n

j =1

If , then buying stock 2 is just like

buying additional shares of stock 1.

11 12

21 22

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

1 11 1

⎡

⎣⎢

⎤

⎦⎥

2) Max f (x) = R(x)

s.t. V(x) , pjxj b, xj 0, j = 1,…,n

where 0 is determined by the investor. Smaller

values of represent greater risk aversion.

n

j =1

3) Min f (x) = V(x)

s.t. R(x) , pj xj b, xj 0, j = 1,…,n

where 0 is the desired rate of return

(minimum expectation) is selected by the investor.

n

j =1

Decision variables: Let (xj, yj), j = 1,…,n, be the

incremental horizontal and vertical displacement of

each link, where n 10.Constraints:

xj2 + yj

2 = 1, j = 1,…,n, each link has length 1

x1 + x2 + • • • + xn = 10, net horizontal displacement

y1 + y2 + • • • + yn = 0, net vertical displacement

Hanging Chain with Rigid Links

10ft

x

y

1 ft

each link

What is equilibrium shape of chain?

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

12 y1 + (y1 + 1

2 y2) + (y1 + y2 + 1

2y3) + • • •

+ (y1 + y2+ • • • + yn-1 + 12 yn)

= (n 1 + 12 ]y1 + (n 2 +

12 )y2

+ (n 3 + 12)y3 + • • • +

32 yn-1 +

12 yn

Summary

Min (n j + ½)yj n

j =1s.t. xj

2 + yj2 = 1, j = 1,…,n

x1 + x2 + • • • + xn = 10

y1 + y2 + • • • + yn = 0

Is a local optimum guaranteed to be a global optimum?

Consider a chain with 4 links:

These solutions are both local minima.

Constraints xj2 + yj

2 = 1 for all j yield a nonconvex

feasible region so there may be several local

optima.

No!

Direct Current Network

Problem: Determine the current flows I1, I2,…,I7 so that the total content is minimized

Content: G(I) = 0

I

v(i)di for I ≥ 0 and G(I) = I

0

v(i)di for I < 0

Ω

2010Ω Ω100v I

I

I1 3

4

10 20Ω

I2

I5

I6 I7

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

Minimize Z = –100I1 + 5I22 + 5I3

2 + 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

-100 I

I

I1

3

4I2

I5

I6 I71

2 3 5

64

5

5

10

10

0 0

2 2

22

Network diagram:

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) . . . (xn

atn), 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.