Linear programming and Simplex algorithm description

The problem of solving (1) Maximum value when satisfying (2) and (3) constraint conditions is called linear programming problem

Linear programming in various forms, but can be converted into the above form, the above form is called linear programming standard type

The matrix form of the standard type :

also solves (1) The maximum value when all other constraints except (1) are satisfied (similar expressions are understood, no longer described)

The following refers to the introduction of the algorithm:

Consider a linear plan with two variables below:

The geometry is as follows:

This gray convex area is called the feasible area , and the function that wants to maximize is called the objective function . Conceptually, we can evaluate the objective function at each point in the feasible region; We refer to the value of the target function at a specific point as the target value . We can find a point with the maximum target value as the optimal solution.

In two dimensions, we can find the optimal solution by a graphical step. For any given in Z, the set of points above is a straight line with a slope of-1. Z as a variable, let the feasible area in the coordinate system move to obtain the maximum value of z, such as:

Get.

While it is not easy to graphically represent a linear plan with more than two variables, the same intuition is still true. As in two-dimensional space, because the feasible area is convex (why?) ), the point set that obtains the optimal target value must contain a vertex of the feasible region. Similarly, if there are n variables, each constraint defines a half space in the n-dimensional space. We call the possible regions of these half-space intersections to be simplex . The objective function is now a super-plane, and because of its convexity, an optimal solution is still obtained at a vertex of simplicity.

The feasible area is a convex proof:

Set the feasible area to S, take two points in S, there

Also by the definition of convex set, we know that s is convex set.

PS: The visual interpretation of a convex region is that there is a line between any two points in the area, and the points on the segment are all in that area.

　　The simplex algorithm takes a linear programming as input and outputs an optimal solution. It starts with a single vertex of a simplex and performs sequential iterations. In each iteration, it moves along an edge of the simplex from the current vertex to an adjacent vertex whose target value is not less than (usually greater than) the current vertex. When a local maximum is reached, where there is a vertex, the target value of all adjacent vertices is less than the target value of the vertex, and the simplex algorithm terminates. Because the feasible area is convex, and the objective function is linear, the local optimality is actually the global optimal.

Although we use the geometrical method to describe the simplex algorithm very intuitively, we first write the given linear programming in the form of relaxation, that is, the set of linear equations. Then the arithmetic is performed from the algebraic point of view.

　

Blogging is a time-taking thing .... The actual working process and implementation method of Simplex algorithm will be supplemented when there is time.

Linear programming and Simplex algorithm description