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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.
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.
If it is not, there is guaranteed to exist a linear inequality that separates the optimum from the convex hull of the true feasible set. Finding such an inequality is the separation problem, and such an inequality is a cut. A cut can be added to the relaxed linear program. Then, the current non-integer solution is no longer feasible to the ...
English: A diagram showing an example of a linear programming problem. No specific problem is computed, just the way in which the feasible region is bounded by straight lines. No specific problem is computed, just the way in which the feasible region is bounded by straight lines.
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).
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%
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.
The feasible set of the optimization problem consists of all points satisfying the inequality and the equality constraints. This set is convex because D {\displaystyle {\mathcal {D}}} is convex, the sublevel sets of convex functions are convex, affine sets are convex, and the intersection of convex sets is convex.