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Advanced Linearization Techniques for Nonlinear Programming Problems

Learn about advanced linearization approaches in nonlinear programming, such as successive linear programming and separable programming. Understand how to transform nonlinear problems into linear programs for optimization. Explore methods like the Frank-Wolfe algorithm and piecewise linear approximation. Discover strategies for achieving convergence to the true optimum in nonlinear optimization.

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Advanced Linearization Techniques for Nonlinear Programming Problems

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  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

  6. Bounded Line Search

  7. Equivalent Approximation

  8. Frank-Wolfe Algorithm

  9. Remark

  10. Case 1.2The general case

  11. Direct Linear Approximation

  12. 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.

  13. 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.

  14. Case 2.1Single-Variable Functions

  15. Line Segment in Interval k Linear!

  16. Line Segment in Interval k

  17. Generalized Formula for a Single-Variable Function

  18. Case 2.2Multivariable Separable Functions

  19. General Formula for a Multi-Variable Function

  20. General Formula for a Multi-Variable Function

  21. 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.

  22. Example

  23. Nonlinear Linear Nonlinear Linear

  24. Homework Slack variable

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