Optimization of functions of one variable Section 2

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 1

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 2

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 3

Golden section method xlo First iteration Second iteration lo’ x x 1’ x 1 x 2’ x 2 xhi’ xhi 4

Bounds on minimum Fl Fu 1. 618(xu-xl) xl First iteration Second iteration l’ x xu x 1’ x 2’ xu’ 5

Bounding minimum algorithm Given, xl, Fl, xmax Guess xu Y Fu>Fl N Expand Minimum in [xl, xu] STOP x 1=xu* xu=x 1+1/ (x 1 -xl) Y N * Stop if xu>xmax Fu>F 1 Expand xl=x 1 6

Example of minimizing function using second degree polynomial approximation obtained through regression. Four data points are used from minimum bounding solution 7

Example of minimizing function using second degree polynomial approximation obtained through regression. Five data points uniformly distributed between 10 and 30 are used 8

Example of minimizing function using second degree polynomial approximation obtained using three data points (exact fit) 9

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 10