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Linear Programming Computation
This monograph represents a historic breakthrough in the field of linear programming (LP)since George Dantzig first discovered the simplex method in...
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Simplex Interior-Point Method
To cope with degeneracy, the face (or dual face) method introduced previously does not seek the optimal solution, vertex by vertex, but instead, face...
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D-Reduced Simplex Method
Consider the following special form of the standard LP problem.
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Generalized Reduced Simplex Method
Although we always consider the standard LP problem, the LP problems from practice are various. The latter can be transformed into a more general...
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Implementation of Simplex Method
All algorithms formulated in this book, such as the simplex algorithm and the dual simplex algorithm, are theoretical or conceptual and cannot be put...
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Generalized Simplex Method
There are various problems from practice, and all can be put into the following general form.
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Deficient-Basis Method
As was mentioned, the assumption of nondegeneracy is entirely contrary to reality. When the simplex method is used to solve large-scale sparse...
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Dual Pivot Rule
A pivot rule used in the dual simplex method is termed dual pivot rule. Like in the primal simplex context, a dual pivot rule plays an important role...
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Dual Face Method with LU Factorization
In the next section, we first put forward the key to this method, which is important for understanding the subsequent derivation. In the following...
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Integer Linear Programming (ILP)
The feasible region of the LP problem is continuous since each variable is restricted to a continuous interval. If variables (or a part of variables)...
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Primal-Dual Simplex Method
Methods perform very differently when solving the same problem. It is a common case that a problem that is solved slowly by the simplex method would...
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Simplex Method
The simplex method is the most famous and widely used method for solving LP problems. Since created by George B. Dantzig in 1947, it has been...
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Dual Simplex Phase-l Method
The mission of a dual Phase-I procedure is to provide an initial dual feasible basis to get the dual simplex method started.
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Interior-Point Method
As it is known, the simplex method moves on the underlying polyhedron, from vertex to adjacent vertex along edges, until attaining an optimal vertex...
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Geometry of Feasible Region
The feasible region P, defined by Definition 1.4.1 , is of great importance to the LP problem. Theories and...
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Decomposition Method
Solving large-scale LP problems is a challenging task, putting forward high requirements on the algorithms’ efficiency, storage, and numerical...
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Duality Principle and Dual Simplex Method
Duality is an essential part of LP theory. It is related to the special relationship between one LP problem and another, both of which involve the...
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Pivot Rule
Consider the following standard LP problem.
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Introduction
As intelligent creatures, human beings plan and carry out activities with pre-set objectives. Early human ancestors relied on their experience only,...
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Reduced Simplex Method
Consider the standard LP problem ( 12.1 ) with the additional assumption that the cost c is not a linear...