Grand Canyon University Algorithms & Linear Functions Discussion

Discussion 1: David

Good afternoon class,

Linear optimization focuses on achieving a solution using constraints and objective function that are linear. Otherwise, nonlinear optimization works towards achieving a solution based on the constraints, objective function, and decision variables being nonlinear. The GRG algorithm determines an optimal solution by applying the objective function as the input and ensuring the partial derivatives equal to zero. “GRG Nonlinear and Evolutionary are best for nonlinear problems, while Simplex LP is limited only to linear problems” (“Excel solver”, 2021). While GRG is better suited to solve nonlinear optimization problems, the GRG algorithm can certainly be utilized to solve a linear optimization problem. The feasible region in linear and nonlinear optimization refers to the area of the graph that contains all feasible solutions based on satisfied constraints and where constraints overlap; however, the GRG algorithm compared to Simplex LP will not always find a corner point similar to the feasible-region approach due to the lack of overlapping constraints.

Reference:

Excel solver: Which solving method should I choose? EngineerExcel. (2021, March 28). Retrieved March 31, 2022, from https://engineerexcel.com/excel-solver-solving-met…

Discussion 2: Meredith

I never thought to look up what the “GRG” stood for when we were working with Solver a few weeks ago. GRG stands for Generalized Reduced Gradient. The GRG Nonlinear is not the only method that Solver has for Nonlinear models. It also has Evolutionary, which we really haven’t worked with yet. I think this would be something fun to play with when I have more time. Back to GRG Nonlinear. It is far more difficult for Solver to process nonlinear models than linear ones. The method uses the slope (or “gradient”) of the objective function. The method follows this while the inputs change until it finds what it thinks is the optimal solution (EngineerExcel). The problem with this is that the method is highly dependent on the starting conditions, and the optimal solution might not be just that: optimal. It finds what is called the “local” solution versus the “global” solution. This is because GRG Nonlinear can have more than one feasible-region. Since there can be more than one feasible region, it will not always find a corner point. Perhaps this is where the “Generalized” part of the name comes from. So while you could use GRG nonlinear to solve a linear problem, it is not advisable to do so. The Simplex LP method is going to be more reliable.

EngineerExcel (n.d.) Excel Solver: which solving method should I choose? Retrieved on March 29, 2022 from https://engineerexcel.com/excel-solver-solving-method-choose

Discussion 3: Tyler

By nature, non-linear optimization problems are more difficult to solve than linear optimization problems. The solver tool in Excel utilizes a very robust non-linear Generalized Reduced Gradient (GRG) algorithm whenever it is not told to use the simplex method for linear programming problems (FrontierSolvers, 2012).

The GRG method should find the optimal solution in most linear optimization problems although it will not do so as quickly. There is also a possibility that you may be notified that the model is uncertain about the status of the solution if the model is poorly scaled. I would definitely not say that the GRG algorithm will always find a corner point like the feasible-region approach, but it should just about always find at least a semi-optimal solution. However, you’d pretty much always be better off just to ensure that your using the simplex method for linear optimization problems.

References

FrontierSolvers. (2012, August 2). Standard Excel solver – limitations of nonlinear optimization. solver. Retrieved March 31, 2022, from https://www.solver.com/standard-excel-solver-limitations-nonlinear-optimization#:~:text=The%20GRG%20method%2C%20while%20it,poorly%20scaled%2C%20as%20discussed%20above.