GAMS: General Algebraic Modeling System Linear and Nonlinear Programming

1. GAMS:General Algebraic Modeling SystemLinear and Nonlinear Programming

2. The lectures are based on: --McCarl, Bruce A., andThomas 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.

3. 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

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:

6. 2. Linear Programming ♣ Assumptions of LP

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

8. -- 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 nvariables and m constraints, the dual problem will have mvariables and nconstraints ♣ Duality in LP

9. Primal and Dual Algebra Primal Dual

10. Primal Example Dual

11. Summary

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

13. 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