10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 29 Global Optimization Multiple Minima Convex vs Non Convex Convex only one minimum Non convex multiple relative minima minx f x Global Optimization Deterministic f x x Figure 1 A function with relative minima Convex Underestimator 2 3 xopt By looking at 2nd derivatives find parabola that is lower than other minimum Figure 2 Convex underestimator Choose a f x upper bound Divide domain Underestimate the lower bound with a parabola Find minimum of parabola Bound again If new upper bound is lower than lower bound in other region can stop considering that section Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY To converge lower bound rises at a certain rate upper bound decreases at a certain rate Going several zones deep creates many divisions 2Ndivisions 1 D Proteins 100 dimensional space or more 100N or more Current papers can solve 4 5 dimensions Method guarantees global optimum if you care about the global optimum If you have 20 variables use heuristics that often find the global optimum but there is no guarantee Multi start 8 8 8 x2 8 8 8 mesh of start points weighted Monte Carlo Figure 3 Begin in multiple locations and then run minimization In low dimension draw map One method do different starts Run a local minimization on each then compare values With enough points can make a space Can use mesh Can use Monte Carlo random guess If there are 100 points and 6 variables 1006 calculations Simulated Annealing Can use when there are lots of global minimum f x kT Figure 4 The molecule is heated and then cooled slowly so that conformational changes taking place will lead to a local minimum This process is repeated many times until several closely related low energy conformations are obtained f x z mixed integer hybrid Genetic Algorithms discretize everything DNA z1 z2 z3 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 29 Page 2 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY mutate zn zn reproduction exchange of DNA fragments replication death give everything probabilities to make it mirror evolution Non determinate methods do not exactly know when you are done 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 29 Page 3 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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