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Slide 2 Survey of gradient based constrained optimization algorithms Select algorithms based on their popularity. Additional details and additional algorithms in Chapter 5 of Haftka and Gurdals Elements of Structural Optimization Slide 3 Optimization with constraints Standard formulation Equality constraints are a challenge, but are fortunately missing in most engineering design problems, so this lecture will deal only with equality constraints. Slide 4 Derivative based optimizers All are predicated on the assumption that function evaluations are expensive and gradients can be calculated. Similar to a person put at night on a hill and directed to find the lowest point in an adjacent valley using a flashlight with limited battery Basic strategy: 1. Flash light to get derivative and select direction. 2.Go straight in that direction until you start going up or hit constraint. 3.Repeat until converged. Some methods move mostly along the constraint boundaries, some mostly on the inside (interior point algorithms) Slide 5 Gradient projection and reduced gradient methods Find good direction tangent to active constraints Move a distance and then restore to constraint boundaries A typical active set algorithm, used in Excel Slide 6 Penalty function methods Quadratic penalty function Gradual rise of penalty parameter leads to sequence of unconstrained minimization technique (SUMT). Why is it important? Slide 7 Example 5.7.1 Slide 8 Contours for r=1. Slide 9 Contours for r=1000. For non-derivative methods can avoid this by having penalty proportional to absolute value of violation instead of its square! Slide 10 Problems Penalty Find how many function evaluatons fminunc and fminsearch solve Example 5.7.1 when you use r=1000 starting from x0=[2,2]. Compare to going through r=1,10,100,1000, and starting the solution for the next r value where the solution for the previous one stopped. Slide 11 5.9: Projected Lagrangian methods Sequential quadratic programming Find direction by solving Find alpha by minimizing Slide 12 Matlab function fmincon FMINCON attempts to solve problems of the form: min F(X) subject to: A*X