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GAMS: General Algebraic Modeling System

Linear and Nonlinear Programming

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The lectures are based on:

-- McCarl, Bruce A., and Thomas H. Spreen. Applied Mathematical Programming Using Algebraic Systems. Available at: http://agecon.tamu.edu/faculty/mccarl/regbook.htm

-- Paris, Q. An Economic Interpretation of Linear Programming. Iowa State University Press: Ames, Iowa, 1991.

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Outline

1. Types of Programming Problems the general mathematical programming problem different types of mathematical programming problems

2. Linear Programming (LP) assumptions of LP an example of LP problem duality in LP (primal & dual) programming without using algebraic modeling

3. Nonlinear Programming (NLP) price endogenous model comparison with LP

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1. Types of Programming Problems

♣ The general mathematical programming problem: Optimize Subject to (s.t.)

-- X is a vector of decision variables -- F(X) is called the objective function -- G(X) must belong to S1 -- the variables individually must fall into S2

( )F X

1( )G X S

2X S

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♣ Different Types of Mathematical

Programming Problems:

F(X) G(X) X

a linear programming problem

linear linear non-negative

an integer programmingproblem

some areintegers

a quadratic programmingproblem

quadratic linear non-negative

a nonlinear programmingproblem

nonlinear nonlinear non-negative

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2. Linear Programming♣ Assumptions of LP

Definition Example

Proportionality Relationship between Outputs and inputs are proportional

If the net return per unit of xj produced is cj , then 100 units of xj have 100cj return

Additivity Every function is the sum of individual contribution of respective activities

a1x1(wheat)+a2x2(corn) =available land

Divisibility All decision variables are continuous (can take on any non-negative value including fractional ones)

x1=12, x2=3.8

Certainty All parameters are known constants

a1=5, a2=2

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♣ An Example of LP Problem

1 2

1 2

1 2

1 2

40 30 ( )

. . 120 ( )

4 2 320 ( )

, 0

Max x x profits

s t x x land

x x labor

x x

GAMS program and results: see handout 1!

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♣ Duality in LP

-- Economic theory indicates: scarce resources have value

-- In LP models, scarce resources are allocated, so they should be, valued -- Whenever we solve an LP problem, we solve two problems: the primal resource allocation problem, and the dual resource valuation problem

-- If the primal problem has n variables and m constraints, the dual problem will have m variables and n constraints

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Primal and Dual Algebra

Primal

j jj

ij j ij

j

c X

. . a X b 1,...,

X 0 1,...,

Max

s t i m

j n

Dual

ii

ij ji

i

Min b

s.t. a c 1,...,

Y 0 1,...,

i

i

Y

Y j n

i m

'

. .

0

Max C X

s t X b

X

'

'. .

0

Min bY

s t AY C

Y

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Example

1 2

1 2

1 2

1 2

40 30 ( )

. . 120 ( )

4 2 320 ( )

, 0

Max x x profits

s t x x land

x x labor

x x

1 2

1 2 1

1 2 2

1 2

( ) ( )

120 320

. . 4 40 ( )

2 30 ( )

, 0

land labor

Min y y

s t y y x

y y x

y y

Primal

Dual

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Summary

Primal Dual

maximize minimize

≤constraint y≥0

x≥0 ≥constraint

=constraint y free

x free =constraint

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GAMS program and results for dual

problem: see handout 2!

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Important Economic Interpretation

-- y1 is associated with the first primal constraint (land), y2 with the second primal constraint (labor)-- dual variables (yi) can be interpreted as the marginal value (or shadow price) of each resource

Thus,

☺ variable y1 gives the marginal value of the 1st resource

(land)

☺ variable y2 gives the marginal value of the 2nd resource (labor)

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☺ the value of the land and labor used in producing a

unit of x1 should be greater than or equal to the marginal revenue contribution of x1 (40, price of x1)

☺ the marginal value of land plus twice the marginal

value of labor should be greater than or equal to the profit earned by producing x2 (30, price of x2)

1 24 40y y

1 22 30y y

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♣ Programming without using algebraic modeling -- take the above primal problem as an example: see handout 3! -- only good for small sized problems ♠ for medium or large sized models (more than 30 rows and/or columns) ♠ to concisely state problems in an abstract general fashion

→ use algebraic modeling!!!

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Before we finish LP, remember that the

default LP solver is Cplex!!!

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3. Nonlinear Programming

turn our attention to continuous, certain, nonlinear optimization problems

relax the LP additivity and proportionality assumptions

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focus on nonlinear objective function It is much more difficult for nonlinear solvers

to deal with nonlinear constraints general NLP solver is Minos!!!

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♣ price endogenous model

-- a quadratic programming problem

-- originally motivated by Enke and Samuelson

-- fully developed by Takayama and Judge (1973)

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-- general form:

maximizes the integral of the area underneath the demand curve minus the integral underneath the supply curve s.t. a supply-demand balance

-- the resultant objective function value of the general

form is commonly called consumers' plus producers'

surplus

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2 2

2 2

( ) ( )

1/ 2 1/ 2

6 0.15 0.1

. . 0

,

d d s s

d d d d s s s s

d d d d s s s s

d d s s

d s

d

Max P dQ PdQ

a b Q dQ a b Q dQ

a Q b Q a Q bQ

Q Q Q Q

s t Q Q

Q

0sQ

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GAMS program and results: see handout 4!

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♣ Comparison with LP

Two important changes:

the objective function equation contains the nonlinear squared terms

in the SOLVE statement we indicate that the

problem is a nonlinear programming problem by saying SOLVE USING NLP