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A basis B of the LP is called dual-optimal if the solution = is an optimal solution to the dual linear program, that is, it minimizes . In general, a primal-optimal basis is not necessarily dual-optimal, and a dual-optimal basis is not necessarily primal-optimal (in fact, the solution of a primal-optimal basis may even be unfeasible for the ...
The strong duality theorem states that if the primal has an optimal solution, x *, then the dual also has an optimal solution, y *, and c T x * =b T y *. A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem.
In linear programming, a discipline within applied mathematics, a basic solution is any solution of a linear programming problem satisfying certain specified technical conditions. For a polyhedron P {\displaystyle P} and a vector x ∗ ∈ R n {\displaystyle \mathbf {x} ^{*}\in \mathbb {R} ^{n}} , x ∗ {\displaystyle \mathbf {x} ^{*}} is a ...
In the latter case the linear program is called infeasible. In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point. The possible results from Phase II are either an optimum basic feasible solution or an infinite edge on which the objective function is unbounded above.
A LP can also be unbounded or infeasible. Duality theory tells us that: If the primal is unbounded, then the dual is infeasible; If the dual is unbounded, then the primal is infeasible. However, it is possible for both the dual and the primal to be infeasible. Here is an example:
The space of all candidate solutions, before any feasible points have been excluded, is called the feasible region, feasible set, search space, or solution space. [2] This is the set of all possible solutions that satisfy the problem's constraints. Constraint satisfaction is the process of finding a point in the feasible set.
In the study of algorithms, an LP-type problem (also called a generalized linear program) is an optimization problem that shares certain properties with low-dimensional linear programs and that may be solved by similar algorithms.
If Ax=b is infeasible, then of course the original problem is infeasible. Otherwise, it has some solution x 0, and the set of all solutions can be presented as: Fz+x 0, where z is in R k, k=n-rank(A), and F is an n-by-k matrix. Substituting x = Fz+x 0 in the original problem gives: