Find necessary and sufficient conditions on the reals a and...

Find necessary and sufficient conditions on the reals $a$ and $b$ under which the linear program
\[
\begin{array}{c}
\max x+y \\
a x+b y \leq 1 \\
x, y \geq 0
\end{array}
\]
(a) Is infeasible.
(b) Is unbounded.
(c) Has a unique optimal solution.

Key Concepts

Feasibility in Linear Programming

Feasibility refers to whether there exists at least one point that satisfies all of the constraints of a linear program. Analyzing feasibility involves determining if the intersection of all constraint regions is non-empty. A linear program is called infeasible when no such point exists, meaning there are conflicting or overly restrictive constraints that do not allow any simultaneous solution.

Unboundedness in Linear Programming

Unboundedness occurs in a linear programming problem when the objective function can be improved without bound, typically due to a direction of increase that the constraints do not restrict. In essence, if one can move infinitely in some direction while still remaining within the feasible region, the optimal value is not finite. This characteristic is a major consideration when formulating models to ensure that a maximum or minimum value is attainable.

Unique Optimal Solution in Linear Programming

A unique optimal solution exists when there is exactly one feasible point that provides the best value for the objective function, rather than a continuum of points yielding the same optimum. This uniqueness is typically determined by the geometry of the feasible region and the orientation of the objective function; for example, if the objective function is not parallel to any edge of the feasible region, then the optimal solution is likely to occur at a single vertex, guaranteeing its uniqueness.

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