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Key Concepts in Linear Programming and Non-Linear Programming

This document covers essential concepts in linear programming (LP), including duality, optimal solutions, unbounded and infeasible solutions, and multiple optima. It discusses important derivatives like reduced cost, and conditions for optimality relating to basic and non-basic variables. The guide also touches on multi-input, multi-output scenarios, spatial equilibrium, sequencing, storage, and preferences. Additionally, it outlines challenges in non-linear programming, addressing issues like badly behaved functions and the importance of proper specification and scaling.

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Key Concepts in Linear Programming and Non-Linear Programming

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  1. Linear Program MAX CBXB + CNBXNB s.t. BXB + ANBXNB = b XB , XNB ≥ 0

  2. Important LP Equations

  3. Important LP Derivatives

  4. Duality

  5. Duality

  6. Unbounded Solution

  7. Infeasible Solution

  8. Multiple Optima

  9. Degeneracy

  10. Complementary Slackness • derived from duality

  11. Reduced Cost • Negative derivative of objective function with respect to a variable • At optimality: • Zero for all basic variables • Non-negative for all non-basic variables (max) • Non-positive for all non-basic variables (max)

  12. Multi-input, Multi-output

  13. Mixing / Blending

  14. Spatial Equilibrium (GAMS Ex.)

  15. Sequencing

  16. Sequencing

  17. Storage

  18. Lexicographic preferences

  19. Weighted Preferences

  20. Well behaved, Separable Function

  21. Well behaved, Separable Function

  22. Well behaved, Separable Function

  23. Disequilibrium – Known Life

  24. Disequilibrium – Unknown Life

  25. Equilibrium ‑ Unknown Life

  26. Fixed Costs

  27. Fixed Capacity

  28. Minimum Habitat Size

  29. Minimum Habitat Size

  30. Warehouse

  31. Mutual exclusive products

  32. Either-Or-Active constraints

  33. Distinct Variable Values

  34. Badly behaved non-linear functions

  35. Badly behaved non-linear functions

  36. Non-linear Programming • Specification often straightforward • Solving more difficult • scaling (manual vs. computer) • lower bounds to avoid division by zero and other illegal operations • local versus global extremes

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