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However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
In large linear-programming problems A is typically a sparse matrix and, when the resulting sparsity of B is exploited when maintaining its invertible representation, the revised simplex algorithm is much more efficient than the standard simplex method. Commercial simplex solvers are based on the revised simplex algorithm.
The dual of a given linear program (LP) is another LP that is derived from the original (the primal) LP in the following schematic way: The objective direction is inversed – maximum in the primal becomes minimum in the dual and vice versa. The weak duality theorem states that the objective value of the dual LP at any feasible solution is ...
Integer programming. An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear .
In the theory of linear programming, a basic feasible solution ( BFS) is a solution with a minimal set of non-zero variables. Geometrically, each BFS corresponds to a vertex of the polyhedron of feasible solutions. If there exists an optimal solution, then there exists an optimal BFS. Hence, to find an optimal solution, it is sufficient to ...
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
Linear programming relaxation. In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form. The relaxation of the original integer program instead uses a collection of linear ...
The Rainbow covering problem is the problem of finding a rainbow set Q that is a covering of P. The problem is NP-hard (by reduction from linear SAT). Conflict-free covering. A more general notion is conflict-free covering. In this problem: There is a set O of m objects, and a conflict-graph G O on O.