Functions of Several Variables The Gradient and Hessian

Optimizing a univariate function is rarely seen in pharmacokinetics. Multivariate optimization is more the norm. For example, in pharmacokinetics one often wishes to identify many different rate constants and volume terms. One solution to a multivariate problem can be done either directly using direct search (Khora-sheh, Ahmadi, and Gerayeli, 1999) or random search algorithms (Schrack and Borowski, 1972), both of which are basically brute force algorithms that repeatedly evaluate the function at selected values under the 

Direct search methods have a disadvantage in that as the number of variables increases, so does the length of time to solve the problem, sometimes leading to exceedingly long search times. They also fail to take into account the information contained by evaluating the derivatives at the point under evaluation. For example if the first derivative is positive, the function is increasing and viceversa. This is useful information when one wants to find the maximum or minimum. 

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