1. L13 Optimization using Excel See revised schedule read 8(1-4) + Excel “help” for Mar 12 Test Answers Review: Convex Prog. Prob. Worksheet modifications Excel optimization Summary 1

2. Trendline in Excel 2 Excel help “trendline” for Wed

3. Theorem 4.9 3 Given: S is convex if: 1. h i are linear 2. g j are convex i.e. H g PD or PSD When f(x) and S are convex= “convex programming problem”

4. “Sufficient” Theorem 4.10, pg 165 4 The first-order KKT conditions are Necessary and Sufficient for a GLOBAL minimum….if: 1. f(x) is convex H f (x) Positive definite 2. x is defined as a convex feasible set S Equality constraints must be linear Inequality constraints must be convex HINT: linear functions are convex!

5. Worksheet Modifications Naming cells Inserting shapes Inserting MS Equation “object” Recording macros Attaching a macro to a shape Creating a SOLVER hot button Visual basic, tools/references/solver 5

6. 6 Figure 6.1 Excel worksheet for finding roots of 2x/3 – sin x : (a) worksheet; (b) worksheet with formulas showing. Excel Applications

7. Solver parameters 7 Figure 6.2 A Solver Parameters dialog box to define the problem.

8. 8 Figure 6.3 A Solver Results dialog box and the final worksheet.

9. 9 Figure 6.4 A Solver Answer Report for roots of 2x/3 – sin x = 0.

10. 10 Figure 6.5 Worksheet and Solver Parameters dialog box for KKT conditions for Example 4.31.

11. 11 Figure 6.6 Solver Results for KKT conditions for Example 4.31.

12. KKT system of NL EQNs Prob 4.59 and 4.122 12

13. 13 Figure 6.7 Excel worksheet and Solver Parameters dialog box for unconstrained problem.

14. Constrained Optimization Prob. 4.69 and 4.122 14

15. Graphical Solution 15 2 1

16. 16 Figure 6.8 Excel worksheet for the linear programming problem.

17. 17 Figure 6.9 Solver Parameters dialog box for the linear programming problem.

18. 18 Figure 6.10 Solver Results dialog box for the linear programming problem.

19. 19 Figure 6.11 Answer Report from Solver for linear programming problem.

20. 20 Figure 6.12 Sensitivity Report from Solver for the linear programming problem.

21. 21 Figure 6.13 Excel worksheet for the spring design problem.

22. Summary KKT pt from a Convex Prog. Prob. Is a global min! Use modifications for “ease of use” Pay attention to layout – Design variables – Parameters – Analysis/Performance “Variables” – Objective function – Constraints May need multiple starting points 22