Summary of simplex algorithm and duality theory

has always been on the shadow price, dual solutions, such as the concept is not particularly clear, recently work spare, on the re-review of the Downline planning, summed up the following points, since for other small partners reference;

First of all, the definition of the research object to put it, convenient for everyone to find a familiar feeling

simplex method definition

a general method for solving linear programming problems by simplex method. Simplex isUnited Statesthe mathematician g.b Danzic first came out in 1947. Its theoretical basis is that the feasible domain of linear programming problem is multi-faceted in n-dimensional vector space RnConvex Set, and its optimal value must be reached at a vertex of the convex set if it exists. The vertices correspond to theFeasible SolutionIs called the basic feasible solution. The basic idea of simplex method is: To find out a basic feasible solution, to identify it, to see if it is the optimal solution; if not, then change to another improved basic feasible solution according to a certain law, then identify it; if it is not, then convert it, repeat it. Because the basicFeasible Solutionis limited, so the problem can be obtained by finite conversion .Optimal Solution. If the problem is notOptimal SolutionThis method can also be used to discriminate.

original problem and duality problem definition:

Now the simplex table structure is decomposed according to different purposes, and the mystery of simple table is found.

1, according to base and non-base division:

2, according to the original variable and relaxation variable division:

Summarize:

1, the test number, can also be understood as marginal contribution, or reduce cost in commercial software, for the corresponding x to the objective function of the marginal contribution to the maximization of the problem, the optimal solution of the judging condition for the test number is not positive, and the increase of all the variables of the target value of the sense of no positive contribution;

2, for the value of the decision variable, the simplex multiplier, is also the problem of the dual problem of the decision-making variable value of the decision-making variable, also known as the Shadow Price;

3, according to the second structure division can more clearly express the shadow price and the optimal solution of the base variable structure. At the same time, the relationship between the original problem and duality problem is found by the combination of this structure and duality problem:

1), assuming that the original problem is a resource actual ability problem, the duality problem is the resource pricing problem, and the test number of the relaxation variable of the original problem is negative for the marginal contribution of the relaxation variable to the objective function.

2), relaxation variables and is the representative of the remaining resources, relaxation variables of the original problem of the marginal contribution, that is, the resource in the system resource pricing, which is the dual problem of the resource pricing problem, it is obvious that this kind of simple structure of the original problem and the dual problem tightly linked together.

Summary of simplex algorithm and duality theory