1 Introduction Optimization: Produce best quality of life with the available resources Engineering design optimization: Find the best system that satisfies given requirements Analysis versus design –Analysis: determine performance of given system –Design: Find system that satisfies given requirements. Design involves iterations in which many design alternatives are analyzed.

2 Objective function: measures performance of a design or a decision Constraints: Requirements that a design must satisfy Numerical optimization can be the only practical approach for most real-life problems

3 General optimization problem statement Find design (decision) variables, X To minimize objective function, F(X) so that –g(X) no greater than zero (inequality constraints) –h(X)=0

4 Example: tubular column optimization Design a column to minimize the mass so that the column does not fail under a given applied axial load Three failure modes--three constraints: yielding, Euler buckling, local buckling May not have unique optimum At the optimum some constraints are active, i.e. applied stress is equal to failure stress

5 Active constraints g2=0 g3=0 x1 x2 Feasible region Weight increases Optimum

6 A taxonomy of optimization problems DeterministicNon deterministic One objective Multiple objectives Static Dynamic

7 Taxonomy Deterministic: know values of all input variables Non deterministic: Only probability distribution of input variables known Static: Solve one optimization problem Dynamic: Solve sequence of optimization sub problems (e.g. chess) Single objective Multiple objectives

8 Iterative optimization procedure Most real life optimization problems solved using iterations Two steps is each iteration –Find search direction –One dimensional search -- find how far to go in a given direction

9 Necessary and sufficient conditions, unconstrained minimization Gradient =0 at X* Gradient =0 at X*, Hessian pos. def. at X*, X* local min Gradient =0 at X*, Hessian pos. def. everywhere, X* global min

10 Necessary condition for local optimum, constrained minimization. Example g1g1 g2g2 FF A A is local minimum, there is no feasible and usable sector -F-F g1g1 g2g2 FF -F-F Feasible sector B is not a local minimum, the feasible sector and the usable sector intersect B Feasible sector g 2 =0 g 1 =0 F=constant g1=0 g 2 =0

11 Necessary condition for local optimum, constrained minimization (continued) For the example, there exist two non negative numbers 1 and 2 such that:  F+ 1  g  g 2 =0 General case: There are non negative numbers j  0, j=1,…,m  F+  j  g j +  k+m  h k+m =0 where the first sum is for j=1,…,m and the second for k=1,…,l

12 Design space convex, K-T conditions satisfied Local optimum Global optimum Sufficient conditions global optimum

13 Sensitivity analysis Allows one to find the sensitivity derivatives of the optimum solution and the optimum value of the objective function with respect the a problem parameter without solving the optimization problem many times. Useful for finding important constraints and important design variables. Very high sensitivity of objective function wrt design parameters; poor design

14 Equations for sensitivity analysis Sensitivity derivatives of design variables A: second order derivatives of objective function and active constraints (size nxn) B: columns are gradients of constraints (size nxm) c: second order derivatives of objective function and constraints wrt design variables and design parameter (size nx1) d: derivatives of constraints wrt design parameter (size mx1)  X and  : derivatives of design variables at optimum and Lagrange multipliers wrt parameter

15 Equations for sensitivity analysis (continued) Chain rule for sensitivity derivatives of objective function Sensitivity derivatives useful for predicting effect of small changes in problem parameters on solution