Microsoft Excel 2007 Help wrote:
Algorithms and methods used by Solver
The Microsoft Office Excel Solver tool uses the Generalized Reduced Gradient (GRG2) nonlinear optimization code, which was developed by Leon Lasdon, University of Texas at Austin, and Alan Waren, Cleveland State University.

Linear and integer problems use the simplex method with bounds on the variables and the branch-and-bound method, implemented by John Watson and Daniel Fylstra, Frontline Systems, Inc. For more information about the internal solution process that is used by Solver, contact:

Frontline Systems, Inc.
P.O. Box 4288
Incline Village, NV 89450-4288

Microsoft Help and Support KB100782 wrote:
The Goal Seek command uses a simple linear search beginning with guesses on the positive or negative side of the value in the source cell (By Changing Cell). Excel uses the initial guesses and recalculates the formula. Whichever guess brings the formula result closer to the targeted result (To Value) is the direction (positive or negative) in which Goal Seek heads. If neither direction appears to approach the target value, Goal Seek makes additional guesses that are further away from the source cell. After the direction is determined, Goal Seek uses an iterative process in which the source cell is incremented or decremented at varying rates until the target value is reached.
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The Goal Seek command uses a simple algorithm and, as a result, may have problems converging on a solution if the function is not linear. Because the iteration process begins with guesses around the source cell, if Goal Seek is having problems converging on a solution, changing the value in the source cell may help. Alternatively, you can use Microsoft Excel Solver.