Linear Programming Lecture Notes - Penn State Personal Web Server
Linear Programming Lecture Notes - Penn State Personal Web Server
Linear Programming Lecture Notes - Penn State Personal Web Server
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8.1 System 2 has a solution if (and only if) the vector c is contained inside the<br />
positive cone constructed from the rows of A. 124<br />
8.2 System 1 has a solution if (and only if) the vector c is not contained inside the<br />
positive cone constructed from the rows of A. 124<br />
8.3 An example of Farkas’ Lemma: The vector c is inside the positive cone formed<br />
by the rows of A, but c ′ is not. 125<br />
8.4 The Gradient Cone: At optimality, the cost vector c is obtuse with respect to<br />
the directions formed by the binding constraints. It is also contained inside the<br />
cone of the gradients of the binding constraints, which we will discuss at length<br />
later. 130<br />
8.5 This figure illustrates the optimal point of the problem given in Example 8.13.<br />
Note that at optimality, the objective function gradient is in the dual cone of<br />
the binding constraint. That is, it is a positive combination of the gradients of<br />
the left-hand-sides of the binding constraints at optimality. The gradient of the<br />
objective function is shown in green. 136<br />
9.1 The dual feasible region in this problem is a mirror image (almost) of the primal<br />
feasible region. This occurs when the right-hand-side vector b is equal to the<br />
objective function coefficient column vector c T and the matrix A is symmetric. 146<br />
9.2 The simplex algorithm begins at a feasible point in the feasible region of the<br />
primal problem. In this case, this is also the same starting point in the dual<br />
problem, which is infeasible. The simplex algorithm moves through the feasible<br />
region of the primal problem towards a point in the dual feasible region. At the<br />
conclusion of the algorithm, the algorithm reaches the unique point that is both<br />
primal and dual feasible. 148<br />
9.3 Degeneracy in the primal problem causes alternative optimal solutions in the<br />
dual problem and destroys the direct relationship between the resource margin<br />
price that the dual variables represent in a non-degenerate problem. 153<br />
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