| Simplex Algorithm |
Article Index for Simplex |
Shopping Simplex |
Articles about Simplex Algorithm |
Website Links For Simplex |
Information AboutSimplex Algorithm |
| CATEGORIES ABOUT SIMPLEX ALGORITHM | |
| optimization algorithms | |
| operations research | |
|
In both cases, the method uses the concept of a in one dimension, a Triangle in two dimensions, a Tetrahedron in three-dimensional space and so forth. DESCRIPTION See Also: Linear programming as a feasible region. The simplex algorithm begins at a starting Vertex and moves along the edges of the polytope until it reaches the vertex of the optimum solution.]] A linear programming problem consists of a collection of Linear inequalities on a number of Real Variable s and a given Linear Functional (on these real variables) which is to be maximized or minimized. Further details are given in the linear programming article. In geometric terms we are considering a Closed Convex Polytope , ''P'', defined by intersecting a number of Half-space s in ''n''-dimensional Euclidean Space ; each half-space is the area which lies on one side of a Hyperplane . If the objective is to maximise a linear functional ''L''(''x''), consider the hyperplanes ''H''(''c'') defined by ; as ''c'' increases, these form a parallel family. If the problem is well-posed, we want to find the largest value of ''c'' such that ''H''(''c'') intersects ''P'' (if there is no such largest value of ''c'', this isn't a reasonable question for optimization as it stands). In this case we can show that the optimum value of ''c'' is attained on the boundary of ''P''. Methods for finding this optimum point on P work in several ways: some attempt to improve a possible point by moving through the interior of ''P'' (so-called ''interior point methods''); others start and remain on the boundary searching for an optimum. The simplex algorithm follows this latter methodology. The idea is to move along the facets of ''P'' in search of the optimum, from point to point. Note that, unless the optimum occurs on an edge or face that is parallel to ''H'', the optimum will be unique and occur at a vertex of the polytope. If an optimum is found on an edge or face that is parallel to ''H'', the optimum is not unique and can be obtained at any point on the edge or face. Since the simplex algorithm is concerned only with finding a single optimal point (even if other equally-optimal points exist), it is possible to look solely at moves skirting the edge of a Simplex , ignoring the interior. The algorithm specifies how this is to be done. We start at some vertex of the polytope, and at every iteration we choose an adjacent vertex such that the value of the objective function does not decrease. If no such vertex exists, we have found a solution to the problem. But usually, such an adjacent vertex is nonunique, and a ''pivot rule'' must be specified to determine which vertex to pick. Various pivot rules exist. In ''P'' is a distortion of an ''n''-dimensional cube. They showed that the simplex method as formulated by Dantzig visits all 2''n'' vertices before arriving at the optimal vertex. This shows that the worst-case complexity of the algorithm is Exponential Time . Similar examples have been found for other pivot rules. It is an open question if there is a pivot rule with Polynomial Time worst-case complexity. Nevertheless, the simplex method is remarkably efficient in practice. Attempts to explain this employ the notion of average complexity or (recently) smoothed complexity. Other algorithms for solving linear programming problems are described in the Linear Programming article. PROBLEM INPUT Consider a linear programming problem, : maximize : subject to The simplex algorithm requires the linear programming problem to be in Augmented Form . The problem can then be written as follows in matrix form: : Maximize ''Z'' in: : : where x are the variables from the ''standard form'', xs are the introduced slack variables from the augmentation process, '''c''' contains the optimization coefficients, '''A''' and '''b''' describe the system of constraint equations, and ''Z'' is the variable to be maximized. The system is typically ''underdetermined'', since the number of variables exceed the number of equations. The difference between the number of variables and the number of equations gives us the ''degrees of freedom'' associated with the problem. Any solution, optimal or not, will therefore include a number of variables of arbitrary value. The simplex algorithm uses zero as this arbitrary value, and the number of variables with value zero equals the degrees of freedom. Variables of nonzero value are called ''basic variables'', and variables of zero value are called ''nonbasic variables'' in the simplex algorithm. definition is problematic, since basic variables can also take zero value. This form simplifies finding the initial ''basic feasible solution'' (BF), since all variables from the ''standard form'' can be chosen to be nonbasic (zero), while all new variables introduced in the ''augmented form'' are basic (nonzero), since their value can be trivially calculated ( for them, since the augmented problem matrix is diagonal on its right half). REVISED SIMPLEX ALGORITHM Matrix form of the simplex algorithm At any iteration of the simplex algorithm, the tableau will be of this form: : where are the coefficients of basic variables in the c-matrix; and '''B''' is the columns of corresponding to the basic variables. It is worth noting that B and are the only variables in this equation (except ''Z'' and '''x''' of course). Everything else is constant throughout the algorithm. Algorithm
: This implies that B is the identity matrix, and is a zero-vector.
REFERENCES
NOTE SEE ALSO EXTERNAL LINKS
|
|
|