Optimization: Finding the maximum or minimum of a function f
Function f can be a function of a single variable, e.g., f = f(x) in one-dimensional optimization problems. In multidimensional problems f = f(x1,x2,...,xn).
Some optimization problems are posed with constraints, such as
All algorithms for optimization begin with a starting guess and the guess is updated/improved to obtain the final solution. The most frequently used algorithms are- (i)Conjugate gradient method, (ii) Newton's method, and (iii) Marquadt's method.
In the figure below, a typical one-dimensional problem is shown, schematically.
The distinction between global min/max from local min/max can be a very difficult problem. One method of distinction is to find the max/min based on a widely varying starting guesses and from the set of solutions that are obtained, select the largest/smallest of these as the global solution.
Warning: For large multi-dimensional problems, there may be no practical way to ensure that you have located a global optimum.
The procedure for setting up an optimization problem on Excel is shown in the two diagrams, below. The built-in "Solver" program is used. You will find the "Solver" in the tools menu of Excel.
Run the Excel Solver with a number of trial starting guesses and ensure, as best as you can, that you have reached the global optimum.
