1. GAMS: General Algebraic Modeling System Linear and Nonlinear Programming
The full system documentation is provided electronically with the software and is also available on-line at:      

2. The lectures are based on:
-- McCarl, Bruce A., and Thomas H. Spreen. Applied
Mathematical Programming Using Algebraic Systems.
Available at:
-- Paris, Q. An Economic Interpretation of Linear
Programming. Iowa State University Press: Ames,
Iowa, 1991.

3. Outline 1. Types of Programming Problems 2. Linear Programming (LP)
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

4. 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

5. ♣ Different Types of Mathematical Programming Problems:
F(X)
G(X) X
a linear programming
problem
linear
non-negative
an integer programming
some are
integers
a quadratic programming
quadratic
a nonlinear programming
nonlinear

6. 2. Linear Programming ♣ Assumptions of LP Proportionality Additivity
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

7. ♣ An Example of LP Problem
GAMS program and results: see handout 1!

8. ♣ 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

9. Primal and Dual Algebra

10. Example
Primal
Dual

11. Summary Primal Dual maximize minimize ≤constraint y≥0 x≥0 ≥constraint
y free
x free

12. GAMS program and results for dual
problem: see handout 2!

13. ☺ variable y1 gives the marginal value of the 1st resource
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)

14. ☺
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)

15. ♣ 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!!!

16. Before we finish LP, remember that the
default LP solver is Cplex!!!

17. 3. Nonlinear Programming
turn our attention to continuous, certain, nonlinear optimization problems
relax the LP additivity and proportionality assumptions

18. focus on nonlinear objective function
It is much more difficult for nonlinear solvers to deal with nonlinear constraints
general NLP solver is Minos!!!

19. ♣ price endogenous model
-- a quadratic programming problem
-- originally motivated by Enke and Samuelson
-- fully developed by Takayama and Judge (1973)

20. -- 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

23. GAMS program and results: see handout 4!

24. ♣ 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