Part 4 Nonlinear Programming

1. Part 4 Nonlinear Programming 4.3 Successive Linear Programming

2. Basic Concept of Linearization Constants

3. Approach 1:Direct Use of Linear Programs The simplest and most direct use of the linearization construction is to replace the general nonlinear problem with a complete linearization of all problem functions at some selected estimate solution. The linearized problem takes the form of a linear program (LP) and can be solved as such.

4. Case 1.1The linearly constrained case Nonlinear Linear

5. Case 1.1The approximate LP problem Feasible point Linear

7. Bounded Line Search

8. Equivalent Approximation

9. Frank-Wolfe Algorithm

11. Remark

12. Case 1.2The general case

13. Direct Linear Approximation

14. Remark In order attain convergence to the true optimum, it is sufficient that at each iteration an improvement be made in both the objectivefunctionand constraint infeasibility. This type of monotonic behavior will occur if the problem functions are mildly nonlinear.

15. Approach 2Separable Programming The motivation for this technique stems from the observation that a good way of improving the linear approximation over a large interval is to partition the interval into subintervals and construct individual linear approximation over each subinterval, i.e., piecewise linear approximation.

16. Case 2.1Single-Variable Functions

17. Line Segment in Interval k Linear!

18. Line Segment in Interval k

19. Generalized Formula for a Single-Variable Function

20. Case 2.2Multivariable Separable Functions

21. General Formula for a Multi-Variable Function

22. General Formula for a Multi-Variable Function

23. Restricted Basis Entry Prior to entering one lambda into the basis (which will make it nonzero), a check should be made to ensure that no more than one other lambda associated with the same variable xi is in the basis. If there is one such lambda in the basis, it has to be adjacent.

24. Example

25. Nonlinear Linear Nonlinear Linear

28. Homework Slack variable