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The Quadratic Approximations optimization method is one of two methods used in EES to find a minimum or maximum when there is one degree of freedom.  The alternative method is the Golden Section method.

EES requires that you specify the possible range for the independent variable, X, before the search begins.  EES evaluates the objective function, f, at the bounds of this range and at one point within the range.  The objective function is then assumed to depend on X in a quadratic manner, i.e.,

f = a X^2+b X+c

Knowing the values of X and f at three points (the bounds and one point within the range) allows the coefficients, a, b, and c to be determined by solving 3 linear equations and 3 unknowns.  Next, the optimum value of the objective function is estimated by setting the derivative of f with respect to X to zero to yield Xopt=-b/(2*a).  The value of the objective function at this new value of Xopt is determined.  Ideally, this point has a smaller value of the objective function (for minimization) than any of the original trial points.  The point having the largest value of the objective function is eliminated and the process is repeated until convergence is achieved.