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Chapter 7 Optimization

Chapter 7 Optimization. Content. Introduction One dimensional unconstrained Multidimensional unconstrained Example. Introduction (1). Root finding & optimization are closely related !!. Optimization. Root finding. Introduction (2). An optimization problem

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Chapter 7 Optimization

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  1. Chapter 7 Optimization

  2. Content • Introduction • One dimensional unconstrained • Multidimensional unconstrained • Example

  3. Introduction (1) Root finding & optimization are closely related !! Optimization Root finding

  4. Introduction (2) An optimizationproblem “Find x, which minimizes or maximizes f(x) subject to… x is an n-dimensional design vector f(x) is the objective function di(x) are inequality constraints ei(x) are equality constraints ai and bi are constants ”

  5. Introduction (3) Classification • Constrained optimization problem 1)Linear programming : f(x) & constraints are linear 2) Quadratic programming : f(x) is quadratic and constraints are linear 3) Nonlinear programming : f(x) is either quadratic or nonlinear and/or constraints are nonlinear • Unconstrainedoptimization problem

  6. One-dimensional (1) interested in finding the absolute highest or lowest value of a function. Multimodal function

  7. One-dimensional (2) To distinguish global minimum (or maximum) from local ones we can try • Graphing to gain insight into the behavior of the function. • Using randomly generated starting guesses and picking the largest of the optima as global. • Perturbing the starting point to see if the routine returns a better point or the same local minimum.

  8. One-dimensional (3) Golden section search (for a unimodal function) • A unimodal function has a single maximum or a minimum in the a given interval. • Step of GSS • Pick two points that will bracket your extremum [xl, xu]. • Pick an additional third point within this interval to determine whether a maximum occurred. • Then pick a fourth point to determine whether the maximum has occurred within the first three or last three points • The key is making this approach efficient by choosing intermediate points wisely thus minimizing the function evaluations by replacing the old values with new values.

  9. One-dimensional (4) GSS principle is always keep the ratio of section length equal @ each iteration

  10. One-dimensional (5) Step I: Starts with two initial guesses (xl ,xu) Step II: Calculate points x1,x2 according to the golden ratio where Step III: Evaluate function at x1 and x2

  11. One-dimensional (6) Step IV: if f(x1)> f(x2) xU = x1 if f(x1)< f(x2) xL= x2 Step V: Calculate new x1 x Try with the following example !!

  12. One-dimensional (7) Ex Use the GSS to find the minimum of assume that xl = 0 and xu = 4 Answer x = 1.4276

  13. Multidimensional… (1) • Techniques to find minimum and maximum of a function of several variables. • Classified as: • Requires derivative evaluation • Gradient or descent (or ascent) methods • Not require derivative evaluation • Non-gradient or direct methods.

  14. Multidimensional… (2)

  15. Multidimensional… (3) DIRECT METHODS :Random Search • Based on evaluation of the function randomly at selected values of the independent variables. • If a sufficient number of samples are conducted, the optimum will be eventually located. • Ex: Locate the maximum of a function f (x, y)=y-x-2x2-2xy-y2

  16. Multidimensional… (4)

  17. Multidimensional… (5) • Advantages/ • Works even for discontinuous and nondifferentiable functions. • Always finds the global optimum rather than the global minimum. • Disadvantages/ • As the number of independent variables grows, the task can become onerous. • Not efficient, it does not account for the behavior of underlying function.

  18. Multidimensional… (6) Univariate and Pattern Searches • More efficient than random search and still doesn’t require derivative evaluation. • The basic strategy is: • Change one variable at a time while the other variables are held constant. • Thus problem is reduced to a sequence of one-dimensional searches that can be solved by variety of methods. • The search becomes less efficient as you approach the maximum.

  19. Multidimensional… (7) Univariate and Pattern Searches

  20. Multidimensional… (8) Gradient method • If f(x,y) is a two dimensional function, the gradient vector tells us • What direction is the steepest ascend? • How much we will gain by taking that step? Directional derivative of f(x,y) at point x=a and y=b

  21. Multidimensional… (9) Gradient method (cont’d) • For n dimensions

  22. Multidimensional… (10) Steepest ascent method Use the gradient vector to change x Says that the we should change with along h-axis The gradient vector represents a direction of maximum rate of increase for the function f(x)at x .

  23. Multidimensional… (11) Example Minimize the function where x = 0.6, y = 4

  24. Multidimensional… (11) Example Maximize the function where x = -1, y = 1 Answer x = 2, y = 1

  25. Multidimensional… (11) Example Maximize the function where x = 0.6, y = 4

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