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Part 4 Nonlinear Programming

Learn how to handle equality constraints in nonlinear programming through methods such as substitution, Lagrange multipliers, and Kuhn-Tucker conditions. Explore the necessary and sufficient conditions for optimality and the use of penalty and barrier methods.

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Part 4 Nonlinear Programming

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  1. Part 4 Nonlinear Programming 4.1 Introduction

  2. Standard Form

  3. An Intuitive Approach to Handle the Equality Constraints One method of handling just one or two equality constraints is to solve for 1 or 2 variables and eliminate them from problem formulation by substitution.

  4. Use of Lagrange Multipliers to Handle m Equality Constraints and m+n Variables

  5. Equivalent Formulation

  6. Choice of Decision Variables For a given optimization problem, the choice of which variables to designate as the decision variables is not unique. It is only a matter of convenience to make a distinction between decision and state variables.

  7. First Derivation of Necessary Conditions (i)

  8. First Derivation of Necessary Conditions (ii)

  9. First Derivation of Necessary Conditions (iii)

  10. Second Derivation of Necessary Conditions (i)

  11. Second Derivation of Necessary Conditions (ii)

  12. Second Derivation of Necessary Conditions (iii)

  13. Second Derivation of Necessary Conditions (iv)

  14. Second Derivation of Necessary Conditions - General Formulation

  15. Derivation with Lagrange Multipliers

  16. Example: Solution:

  17. Sensitivity Interpretation

  18. Generalized Sensitivity

  19. Problems with Inequality Constraints Only

  20. One Constraint and One Variable

  21. Two Possibilities at Minimum *

  22. One Constraint and Two Variables Area of improvement

  23. J Inequality Constraints and N Variables

  24. 2-D Case

  25. Kuhn-Tucker Conditions: Geometrical Interpretation At any local constrained optimum, no (small) allowable change in the problem variables can improve the value of the objective function. lies within the cone generated by the negative gradients of the active constraints.

  26. General Formulation

  27. Active Constraints

  28. Kuhn-Tucker Conditions

  29. Kuhn-Tucker Necessity Theorem

  30. Sensitivity

  31. Constraint Qualification • When the constraint qualification is not met at the optimum, there may or may not exist a solution to the Kuhn-Tucker problem. • The Kuhn-Tucker necessity theorem helps to identify points that are not optimal. On the other hand, if the KTC are satisfied, there is no assurance that the solution is truly optimal.

  32. Second-Order Optimality Conditions

  33. Necessary and Sufficient Conditions for Optimality If a Kuhn-Tucker point satisfies the second-order sufficient conditions, then optimality is guaranteed.

  34. Basic Idea of Penalty Methods

  35. Example

  36. Exact L1 Penalty Function

  37. Equivalent Smooth Constrained Problem

  38. Barrier Method

  39. Generalized Case

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