2. Local and Local-Global Approximations Local algebraic approximations – Variants on Taylor series Local-Global approximations – Variants on “fudge factor”

3. Local algebraic approximations Linear Taylor series Intervening variables Transformed approximation Most common: y i =1/x i

4. Beam example Tip displacement Intervening variables y i =1/I i

5. Reciprocal approximation It is often useful to write the reciprocal approximation in terms of the original variables x instead of the reciprocals y

6. Conservative-convex approximation At times we benefit from conservative approximations All second derivatives of g C are non-negative Convex linearization obtained by applying the approximation to both objective and constraints

7. Three-bar truss example

8. Stress constraint on member C Stress in terms of areas Stress constraint Using non-dimensional variables What assumption on stress?

9. Results around (1,1). x1x1 x2x2 ggLgL gRgR gCgC 0.75 0.36350.27830.36350.3850 1.000.750.42270.34260.4493 1.250.750.42050.40700.50080.5137 0.751.00-0.0856-0.0417-0.0631-0.0417 1.251.000.06190.08700.07410.0871 0.751.25-0.3786-0.3617-0.3191-0.2977 1.001.25-0.2440-0.2974-0.2334 1.25 -0.1819-0.2330-0.1819-0.1690

10. Problems local approximations What are intervening variables? There are also cases when we use “intervening function” in order to improve the accuracy of a Taylor series approximation. Can you give an example? AnswersAnswers What is conservative about the conservative approximation? Why is that a plus? Why is it useful that it is convex? AnswersAnswers

11. Local Approximations pros and cons Derivative based local approximations have several advantages – Derivatives are often computationally inexpensive – Derivatives are needed anyhow for optimization algorithms – These approximations allow rigorous convergence proofs There are some disadvantages too – They can have very small region of acceptable accuracy – They do not work well with noisy functions

12. Global approximations Can be based on more approximate mathematical model Can be based on same mathematical model with coarser discretization Can be based on fitting a meta-model (surrogate, response surface) to a number of simulations Pro and cons complement those of local approximations: Wider range, noise tolerance, but more expensive, and less amenable to math proofs

13. Combining local and global approximations Can use derivatives to combine the two models The combined approximation matches the value and slope at x 0.

14. Example Approximating the sine function as a quadratic polynomial

15. Overall comparison.

16. Without linear.

17. Problems local-global Given the function y=sinx, compare the linear, reciprocal, and global local approximation about x 0 =  /3, where the global approximation is y S =2x/ 