The mathematical theory behind linear programming states that an optimal solution to any problem will lie at a(n) ________ of the feasible region.

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BlackTom AI · Accepted Answer

The question asks where the optimal solution to a linear programming problem lies within the feasible region. Since the mathematical theory of LP states that an optimal solution occurs at a vertex of the feasible region, you should look for language that names a corner or vertex there.
Option notes: In this instance, the provided answer is 'Corner point or extreme point,' which aligns with the standard result that optima in LP occur at a corner (vertex) of the feasible set and correspond to a basic feasible solution in many cases.
Why this logic is true: Linear programs have linear objective functions and linear constraints, which yield a convex feasible region. A convex polyhedron’s optimum for a linear objective is achieved at extreme points—points that cannot be written as a strict convex combination of other feasible points. Intuitively, moving in any direction away from a corner along a feasible edge cannot improve the objective without leaving feasibility.
Why other potential statements (if present) would be misleading:
- If someone suggested the optimum is always in the interior, that would contradict the fundamental LP property that gradients guide toward the boundary; interior points can’t be optimal unless the objective is flat along a direction, which is a degenerate case often handled by basics but not the general rule.
- If a statement claimed the optimum lies on the boundary but not at a corner, you’d need to recognize that along an edge with a linear objective, every point on that edge between the endpoints yields the same objective value only in degenerate cases; typically, a pure optimum occurs at a vertex.
- If a claim asserted that the optimum could be anywhere in the feasible region, that would ignore the convexity and linearity that push the optimum to an extreme point.
Note: Because the provided options list is empty in this dataset, we cannot compare alternate phrasings directly, but the canonical correct idea remains that linear programming optima occur at corner (extreme) points of the feasible region.

MSIS 212 P 1 05 Mngrl Decision Makng Spring 2025 212.05.25.Exam 2

The mathematical theory behind linear programming states that an optimal solution to any problem will lie at a(n) ________ of the feasible region.

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