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The secret weapon of linear programming: What is the core concept of the simplex method?
In the field of mathematical optimization, Dantzig's simplex algorithm (or simplex method) is undoubtedly one of the most popular algorithms for linear programming. The method is named after the concept of "simplex", although these geometric shapes are not used directly in practice. The simplex method operates on a geometric object, the polyhedron, and the development of the algorithm has evolved in line with the needs of the twentieth century.
History of the Simplex Method
George Danziger, the creator of the simplex method, designed plans for the U.S. Army Air Force during World War II, and his original purpose was to allocate resources efficiently. In 1946, Danziger's colleagues challenged him to mechanize the planning process, a challenge that led him to transform the problem into linear inequalities. Although he did not include an objective function at the time, this move led to the subsequent development of methods.
Eventually, he realized that most of the military's "ground rules" could be transformed into a linear objective function that needed to be maximized.
How the Simplex Method Works
The simplex method operates on linear programs of standard form, with the goal of maximizing an objective function while satisfying a set of constraints. In geometry, all feasible solutions form a convex polyhedron, and every vertex of this polyhedron (or "basic feasible solution") is the best possible solution.
The core of this method is to visit the edges of the polyhedron and gradually find the maximum value of the objective function. The simplex method starts from the initial extreme point. If the current extreme point is not the optimal solution, it moves along the edge to other extreme points with higher objective function values until it reaches the optimal solution or finds that the objective function has no upper limit.
This process not only ensures that the algorithm will terminate, but also expects that the number of poles that need to be visited is relatively small.
Standard Form of Linear Programming
To convert a linear program into standard form, we first need to process each constraint. New variables are introduced to represent the difference between the original variables and their boundaries, thereby simplifying all constraints to non-negative conditions. An important step in this conversion process is to convert the inequality constraints into equality constraints.
For example, given the following inequality constraints:
x1 ≥ 5
We can introduce a new variable y1 so that:
y1 = x1 - 5
This converts the constraint into equation form.
Simple form
Once a linear program has been converted to standard form, it can be evaluated using simplex tables. Each row of the table represents a constraint, and the first row defines the objective function. If the table can be rearranged to a form that contains the identity matrix, then the table is called a standard table.
In this table, the distinction between basic variables and non-basic variables is particularly important. By setting the non-basic variables to zero, the values of the basic variables can be easily calculated, which is also a core step in the solution process.
ConclusionThus, through repeated row operations, the simplex method can iteratively find the optimal solution.
With the development of mathematics and computing technology, the simplex method has played an important role in various practical applications. In areas such as business, transportation, and network optimization, the implications of this approach are not limited to finding mathematical solutions. Today, with the continuous evolution of technology, whether this classic algorithm can be used to achieve the best results in a changing environment will be an important direction for future research. What innovative strategies do you think will emerge in future optimization problems to replace or assist the traditional application of the simplex method?
