Sizing Method Panel
The Sizing Method panel controls how the automatic sizing will determine the ideal sizes.
Sizing Level

Continuous  Allow the sized pipes to be any diameter

Discrete  Restrict the sized pipes to diameters directly specified in their Candidate Set.
Discrete and Continuous Sizing each have their own advantages. Discrete Sizing will present a set of pipes that are commercially available, as defined by the associated Candidate Sets. Continuous Sizing is generally much easier computationally, and hence faster.
It is good practice to begin with Continuous Sizing and only proceed to Discrete Sizing when the general design of the system has been determined.
Search Method
There are many ways to approach the search for a minimum or maximum.
A foolproof way to determine if a system is really the "best" is to compare every single possible combination allowed. Then, it will be guaranteed that the lowest cost option has been found. Unfortunately, attempting to brute force the problem is only viable with a very small number of variables. Instead, the solution must be searched for in an intelligent manner, without explicitly checking every possible option.
There are several such methods available in ANS:

Continuous Methods  These methods use gradients rates of change of the Objective (total system cost) with respect to one design variable (a pipe diameter). For the most basic system, the gradient will show an increasing cost with increasing pipe size. However, this quickly becomes less rigid when considering Design Requirements and the impacts a certain pipe size has on the system. A smaller pipe size may make the pipe itself cost less, but it may require a larger pump which costs more, effectively making the gradient show a decreasing cost with increasing size. Regardless, if the direction of decreasing cost is known, an intelligent choice to search in that direction can be made. By intelligently running sizing subproblems, it is possible to find a discrete solution with continuous methods. The Branch and Bound method searches for a discrete minimum by placing additional requirements on certain variables (branching) and analyzing the results of the continuous solution to determine if this section of the design space is worth further investigation (bounding).

Modified Method of Feasible Directions  Reliable and applicable to most problems.

Sequential Quadratic Programming  Frequently the most efficient, but more sensitive to system definition.

Sequential Unconstrained  Sensitive to initial sizes and control parameters, but more capable of handling large numbers of uniquely sized pipes.

Discrete Methods  Some methods are inherently discrete, and can only consider discrete possibilities during the sizing process. These methods are sometimes more adept at finding global minimums, but are also generally much slower than gradient based methods.

Genetic Algorithm  Crossbreeds a random "population" of designs, mimicking biological evolution. Less likely to become trapped in a local minimum, but very slow compared to other methods.
Note: Full understanding of the Search Methods is not required for effective use of ANS. However, it is important to realize that different methods are better suited to different systems. It is recommended to use multiple Search Methods on any system to ensure the best case sizes are located, and to determine if one method is more efficient than the others.
Related Topics
Related Examples
Related Blogs
What is the ANS Module Asking For? Introduction to the Sizing window in the ANS Addon Module