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# 数学代写|线性规划代写Linear Programming代考|MATH417 Fundamental Theorem of Linear Programming

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## 数学代写|线性规划代写Linear Programming代考|Fundamental Theorem of Linear Programming

Now that we have a Phase I algorithm and a variant of the simplex method that is guaranteed to terminate, we can summarize the main gist of this chapter in the following theorem:

THEOREM 3.4. For an arbitrary linear program in standard form, the following statements are true:
(1) If there is no optimal solution, then the problem is either infeasible or unbounded.
(2) If a feasible solution exists, then a basic feasible solution exists.
(3) If an optimal solution exists, then a basic optimal solution exists.

PRoOf. The Phase I algorithm either proves that the problem is infeasible or produces a basic feasible solution. The Phase II algorithm either discovers that the problem is unbounded or finds a basic optimal solution. These statements depend, of course, on applying a variant of the simplex method that does not cycle, which we now know to exist.

## 数学代写|线性规划代写Linear Programming代考|Geometry

The set of points satisfying $x_1+2 x_3=3$ is a plane. The inequality $x_1+2 x_3 \leq 3$ therefore consists of all points on one side of this plane; that is, it is a halfspace. The same is true for each of the other four inequalities. The feasible set consists of those points in space that satisfy all five inequalities, i.e., those points lying in the intersection of these halfspaces. This set is the polyhedron shown in Figure 3.1. This polyhedron is bordered by five facets, each facet being a portion of one of the planes that was defined by replacing a constraint inequality with an equation. For example, the “front” facet in the figure is a portion of the plane $x_1+2 x_3=3$. The facets acquire a particularly simple description if we introduce slack variables into the problem:
$$\begin{array}{lr} w_1=3-x_1 & -2 x_3 \ w_2=2 & -x_2-2 x_3 \end{array}$$

## 数学代写|线性规划代写Linear Programming代考|Fundamental Theorem of Linear Programming

(1) 如果没有最优解，则问题不可行或无界。
(2) 若存在可行解，则存在基本可行解。
(3) 若存在最优解，则存在基本最优解。

## 数学代写|线性规划代写Linear Programming代考|Geometry

$$w_1=3-x_1 \quad-2 x_3 w_2=2 \quad-x_2-2 x_3$$

## MATLAB代写

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