Feasible Regions

Optimization · Axiom Academy

Understanding the set of valid solutions in optimization 1 Definition of Feasible Region The feasible region (or feasible set) is the set of all points that satisfy all constraints of an optimization problem. It represents every possible solution that meets the problem's requirements. Watch as we build a feasible region by adding constraints one at a time. Each constraint eliminates part of the space, and the intersection of all valid areas forms our feasible region. 2 Bounded vs Unbounded Regions Feasible regions can be classified by whether they extend infinitely in any direction: This distinction is important because unbounded regions may allow the objective function to grow without limit, potentially leading to unbounded optimal values. 3 Convex vs Non-convex Regions A feasible region is convex if for any two points in the region, the entire line segment connecting them also lies within the region. Watch the line segment test: if any line segment between two feasible points leaves the region, it's non-convex. When constraints contradict each other, no point can satisfy all of them simultaneously. This creates an empty feasible region , making the problem infeasible . Watch as contradictory constraints leave no valid region where all are satisfied. 5 Finding Optimal Points on Boundaries For linear optimization problems with convex feasible regions, the optimal solution always occurs at a vertex (corner point) of the feasible region or along an edge.

This is the written version of the interactive lesson above. See the full Optimization course.