Optimization of functions of one variable (Section 2) Find minimum of function of one variable Occurs directly Part of iterative algorithm (line search) Unimodal function, single optimum -- step toward optimum results in reduction of objective function along the path

Two methods Golden section search Polynomial approximation Golden section search; known convergence rate, guaranteed to find interval bounding optimum (tolerance interval). Provides information about confidence in solution. Expensive Polynomial approximation. Efficient but not as robust as Golden section search

Golden section search Starts with interval known to contain minimum (tolerance interval) Proceeds by narrowing tolerance interval Uses four data points for which objective function is evaluated. In each iteration -- one additional function evaluation Tolerance interval reduces to 61.8% of interval from previous iteration

Golden section method xlo xhi x1 x2 xlo’ xhi’ x2’ x1’ Second iteration First iteration

Bounds on minimum Fl Fu 1.618(xu-xl) xu First iteration xl Second iteration xl’ x1’ x2’ xu’

Bounding minimum algorithm Given, xl, Fl, xmax Guess xu Y Fu>Fl Minimum in [xl,xu] STOP N Expand x1=xu* xu=x1+1/(x1-xl) Y Fu>F1 N Expand xl=x1 * Stop if xu>xmax

Example of minimizing function using second degree polynomial approximation obtained through regression. Four data points are used from minimum bounding solution

Example of minimizing function using second degree polynomial approximation obtained through regression. Five data points uniformly distributed between 10 and 30 are used

Example of minimizing function using second degree polynomial approximation obtained using three data points (exact fit)

Minimizing constrained functions of one variable Direct approach Deal with each function (objective, constraint) individually Indirect approach Develop and use pseudo objective function that includes both the objective function and the constraint functions