Lectures 3 ( part ) , 4 and 5
نویسنده
چکیده
Recall that a linear program defines a polyhedron. For simplicity, let us assume here that this polyhedron is nonempty (i.e., the LP is feasible) and bounded (namely, it is a polytope). Then we know that the optimal value of the LP is attained at a vertex of the polytope (equivalently, at a basic feasible solution to the LP). We say that two vertices of a polytope are adjacent if they are connected by an edge of the polytope. The basic idea of the simplex algorithm is as follows. One starts at an arbitrary vertex of the polytope. (The question of how to find a starting vertex will be addressed shortly.) Thereafter, at every iteration, the algorithm moves to a neighboring vertex of better value (of the objective function). The algorithm ends when such a move is no longer possible, meaning that the current vertex is a local optimum compared to all adjacent vertices. The above description gives only the basic idea. A more serious treatment invloves proving that the solution found is optimal, showing how one can implement a single iteration, deciding which of several improving adjacent vertices to move to, how to find a starting feasible vertex, deal with polyhedrons that are unbounded, with degeneracies, analysing the number of iterations required (and showing that this number is finite) and so on. An even more serious treatment involves the many ideas that come into improved implementations of the simplex algorithm.
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تاریخ انتشار 2015