Linear Programming Introduction, Lecture Notes - Mathematics, Study notes of Linear Programming

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Linear Programming: Chapter 1

Introduction

Robert J. Vanderbei

October 17, 2007

Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/∼rvdb

Resource Allocation

maximize c 1 x 1 + c 2 x 2 + · · · + cnxn subject to a 11 x 1 + a 12 x 2 + · · · + a 1 nxn ≤ b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 nxn ≤ b 2 ... am 1 x 1 + am 2 x 2 + · · · + amnxn ≤ bm x 1 , x 2 ,... , xn ≥ 0 ,

where

cj = profit per unit of product j produced bi = units of raw material i on hand aij = units of raw material i required to produce one unit of product j.

Shape Optimization (Telescope Design)

The problem is to design and build a space telescope that will be able to “see” planets around nearby stars (other than the Sun).

Consider a telescope. Light enters the front of the telescope. This is called the pupil plane. The telescope focuses all the light passing through the pupil plane from a given direction at a certain point on the focal plane, say (0, 0).

focal plane

light cone (^) pupil plane

However, the wave nature of light makes it impossible to concentrate all of the light at a point. Instead, a small disk, called the Airy disk, with diffraction rings around it appears.

These diffraction rings are bright relative to any planet that might be orbiting a nearby star and so would completely hide the planet. The Sun, for example, would appear 1010 times brighter than the Earth to a distant observer.

By placing a mask over the pupil, one can design the shape and strength of the diffraction rings. The problem is to find an optimal shape so as to put a very deep null very close to the Airy disk.

Airy Disk and Diffraction Rings

A conventional telescope has a circular openning as depicted by the left side of the figure. Visually, a star then looks like a small disk with rings around it, as depicted on the right.

The rings grow progressively dimmer as this log-plot shows:

Electric Field

Consider “tinting” the front opening of the telescope using a nonuniform tint given by A(r).

In such a situation, the image plane electric field of a star is a rotationally sym- metric real function E(ρ):

E(ρ) = 2π

∫ D/ 2

0

A(r)J 0 (2πrρ)rdr

The intensity of the light at radius ρ from the center of the image plane is given by the square of the electric field.

Maximizing Throughput

We maximize the “area” under A(r) (make the tinting as bright as possible) subject to very strict contrast constraints:

maximize

∫ D/ 2

0

A(r)rdr

subject to − 10 −^5 E(0) ≤ E(ρ) ≤ 10 −^5 E(0), for ρmin ≤ ρ ≤ ρmax 0 ≤ A(r) ≤ 1 , for 0 ≤ r ≤ D/ 2

The first constraint guarantees 10 −^10 light intensity throughout a desired annulus of the focal plane, and the remaining constraint ensures that the tinting is really a tinting.

Finite Impulse Response (FIR) Filter Design

• Audio is stored digitally in a computer as a stream of short integers: uk, k ∈ Z.
• When the music is played, these integers are used to drive the displacement of the speaker from its resting position.
• For CD quality sound, 44100 short integers get played per second per channel.

0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 -

FIR Filter Design—Continued

• A finite impulse response (FIR) filter takes as input a digital signal and con- volves this signal with a finite set of fixed numbers h 0 ,... , hn to produce a filtered output signal:

yk =

∑^ n

i=−n

h|i|uk−i.

• Sparing the details, the output power at frequency ν is given by

|H(ν)|^2

where

H(ν) =

∑^ n

k=−n

h|k|e^2 πikν^ = h(0) + 2

∑^ n

k=

hk cos(2πkν),

• Similarly, the mean absolute deviation from a flat frequency response over a frequency range, say L ⊂ [0, 1], is given by

1 |L|

L

|H(ν) − 1 | dν

Conversion to a Linear Programming Problem

minimize

0

t(ν)dν

subject to t(ν) ≤ Hw(ν) + Hm(ν) + Ht(ν) − 1 ≤ t(ν) ν ∈ [0, 1]

− ≤ Hw(ν) ≤ , ν ∈ W

− ≤ Hm(ν) ≤ , ν ∈ M

− ≤ Ht(ν) ≤ , ν ∈ T

Specific Example

filter length: n = 14

integral discretization: N = 1000

Demo: orig-clip woofer midrange tweeter reassembled

Ref: J.O. Coleman, U.S. Naval Research Laboratory, CISS98 paper available: engr.umbc.edu/∼jeffc/pubs/abstracts/ciss98.html

Historical Data

Year US US S&P Wilshire NASDAQ Lehman EAFE Gold 3-Month Gov. 500 5000 Composite Bros. T-Bills Long Corp.

• ρmin = O = {(ξ, 0) : ξ 0 ≤ ξ ≤ ξ 1 }
• ρmax =
  • • • 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -
        • • • -20 -15 -10 -5
• 1973 1.075 0.942 0.852 0.815 0.698 1.023 0.851 1. Bonds Bonds
• 1974 1.084 1.020 0.735 0.716 0.662 1.002 0.768 1.
• 1975 1.061 1.056 1.371 1.385 1.318 1.123 1.354 0.
• 1976 1.052 1.175 1.236 1.266 1.280 1.156 1.025 0.
• 1977 1.055 1.002 0.926 0.974 1.093 1.030 1.181 1.
• 1978 1.077 0.982 1.064 1.093 1.146 1.012 1.326 1.
• 1979 1.109 0.978 1.184 1.256 1.307 1.023 1.048 2.
• 1980 1.127 0.947 1.323 1.337 1.367 1.031 1.226 1.
• 1981 1.156 1.003 0.949 0.963 0.990 1.073 0.977 0.
• 1982 1.117 1.465 1.215 1.187 1.213 1.311 0.981 1.
• 1983 1.092 0.985 1.224 1.235 1.217 1.080 1.237 0.
• 1984 1.103 1.159 1.061 1.030 0.903 1.150 1.074 0.
• 1985 1.080 1.366 1.316 1.326 1.333 1.213 1.562 1.
• 1986 1.063 1.309 1.186 1.161 1.086 1.156 1.694 1.
• 1987 1.061 0.925 1.052 1.023 0.959 1.023 1.246 1.
• 1988 1.071 1.086 1.165 1.179 1.165 1.076 1.283 0.
• 1989 1.087 1.212 1.316 1.292 1.204 1.142 1.105 0.
• 1990 1.080 1.054 0.968 0.938 0.830 1.083 0.766 0.
• 1991 1.057 1.193 1.304 1.342 1.594 1.161 1.121 0.
• 1992 1.036 1.079 1.076 1.090 1.174 1.076 0.878 0.
• 1993 1.031 1.217 1.100 1.113 1.162 1.110 1.326 1.
• 1994 1.045 0.889 1.012 0.999 0.968 0.965 1.078 0.

Risk vs. Reward

Reward—estimated using historical means:

rewardj =

T

∑^ T

t=

Rj (t).

Risk—Markowitz defined risk as the variability of the returns as measured by the historical variances:

riskj =

T

∑^ T

t=

Rj (t) − rewardj

However, to get a linear programming problem (and for other reasons) we use the sum of the absolute values instead of the sum of the squares:

riskj =

T

∑^ T

t=

∣Rj (t) − rewardj

Portfolios

Fractions: xj = fraction of portfolio to invest in j.

Portfolio’s Historical Returns:

R(t) =

j

xj Rj (t)

Portfolio’s Reward:

reward(x) =

T

∑^ T

t=

R(t) =

T

∑^ T

t=

j

xj Rj (t)

Portfolio’s Risk:

risk(x) =

T

∑^ T

t=

|R(t) − reward(x)|

T

∑^ T

t=

j

xj Rj (t) −

T

∑^ T

s=

j

xj Rj (s)

T

∑^ T

t=

j

xj

Rj (t) − 1 T

∑^ T

s=

Rj (s)

T

∑^ T

t=

j

xj (Rj (t) − rewardj )