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In pure integer programming problems, the feasible set is the set of integers (or some subset thereof). In linear programming problems, the feasible set is a convex polytope: a region in multidimensional space whose boundaries are formed by hyperplanes and whose corners are vertices.
An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. [1] The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, [2] which runs in provably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...
In decision problem versions of the art gallery problem, one is given as input both a polygon and a number k, and must determine whether the polygon can be guarded with k or fewer guards. This problem is ∃ R {\displaystyle \exists \mathbb {R} } -complete , as is the version where the guards are restricted to the edges of the polygon. [ 10 ]
Cutting plane methods for MILP work by solving a non-integer linear program, the linear relaxation of the given integer program. The theory of Linear Programming dictates that under mild assumptions (if the linear program has an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a ...
In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm.The Big M method extends the simplex algorithm to problems that contain "greater-than" constraints.
Many other LP-type problems can also be expressed using quasiconvex functions in this way; for instance, the smallest enclosing circle problem is the problem of minimizing max i f i where each of the functions f i measures the Euclidean distance from one of the given points. [10] LP-type problems have also been used to determine the optimal ...
A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem.
By formulating MAX-2-SAT as a problem of finding a cut (that is, a partition of the vertices into two subsets) maximizing the number of edges that have one endpoint in the first subset and one endpoint in the second, in a graph related to the implication graph, and applying semidefinite programming methods to this cut problem, it is possible to ...