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For example, if the feasible region is defined by the constraint set {x ≥ 0, y ≥ 0}, then the problem of maximizing x + y has no optimum since any candidate solution can be improved upon by increasing x or y; yet if the problem is to minimize x + y, then there is an optimum (specifically at (x, y) = (0, 0)).
A closed feasible region of a problem with three variables is a convex polyhedron. The surfaces giving a fixed value of the objective function are planes (not shown). The linear programming problem is to find a point on the polyhedron that is on the plane with the highest possible value.
Linear programming feasible region farmer example: Image title: Graphical solution to the farmer example by CMG Lee. After shading regions violating the conditions, the vertex of the unshaded region with the dashed line farthest from the origin gives the optimal combination. Width: 100%: Height: 100%
For example, if is non-basic and its coefficient in is positive, then increasing it above 0 may make larger. If it is possible to do so without violating other constraints, then the increased variable becomes basic (it "enters the basis"), while some basic variable is decreased to 0 to keep the equality constraints and thus becomes non-basic ...
The blue region is the feasible region. The tangency of the line with the feasible region represents the solution. The line is the best achievable contour line (locus with a given value of the objective function).
Graphical solution to the farmer example – after shading regions violating the conditions, the vertex of the remaining feasible region with the dashed line farthest from the origin gives the optimal combination (its lying on the land and pesticide lines implies that revenue is limited by land and pesticide, not fertilizer)
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 ...
Informally, Slater's condition states that the feasible region must have an interior point (see technical details below). Slater's condition is a specific example of a constraint qualification. [2] In particular, if Slater's condition holds for the primal problem, then the duality gap is 0, and if the dual value is finite then it is attained.