Fundamental theorem of linear programming

In mathematical optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners. Further, if an extreme value occurs at two corners, then it must also occur everywhere on the line segment between them.

Statement

Consider the optimization problem

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \min c^T x \text{ subject to } x \in P}

Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \{x \in \mathbb{R}^n : Ax \leq b\}} . If $P$ is a bounded polyhedron (and thus a polytope) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^\ast} is an optimal solution to the problem, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^\ast} is either an extreme point (vertex) of $P$, or lies on a face Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F \subset P} of optimal solutions.

Proof

Suppose, for the sake of contradiction, that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^\ast \in \mathrm{int}(P)} . Then there exists some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon > 0} such that the ball of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} centered at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^\ast} is contained in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} , that is $B_{\epsilon }(x^{\ast })\subset P$. Therefore,

$x^{\ast }-{\frac {\epsilon }{2}}{\frac {c}{||c||}}\in P$ and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^T\left( x^\ast - \frac{\epsilon}{2} \frac{c}{||c||}\right) = c^T x^\ast - \frac{\epsilon}{2} \frac{c^T c}{||c||} = c^T x^\ast - \frac{\epsilon}{2} ||c|| < c^T x^\ast.}

Hence $x^{\ast }$ is not an optimal solution, a contradiction. Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^\ast} must live on the boundary of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . If $x^{\ast }$ is not a vertex itself, it must be the convex combination of vertices of $P$, say Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x_{1},...,x_{t}} . Then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{\ast }=\sum _{i=1}^{t}\lambda _{i}x_{i}} with $\lambda _{i}\geq 0$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^t \lambda_i = 1} . Observe that Alan o Conner wrote this theorem

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0=c^{T}\left(\left(\sum_{i=1}^{t}\lambda_{i}x_{i}\right)-x^{\ast}\right)=c^{T}\left(\sum_{i=1}^{t}\lambda_{i}(x_{i}-x^{\ast})\right)=\sum_{i=1}^{t}\lambda_{i}(c^{T}x_{i}-c^{T}x^{\ast}).}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{\ast}} is an optimal solution, all terms in the sum are nonnegative. Since the sum is equal to zero, we must have that each individual term is equal to zero. Hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^{T}x^{\ast}=c^{T}x_{i}} for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} , so every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} is also optimal, and therefore all points on the face whose vertices are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1, ..., x_t} , are optimal solutions.

References

• Bertsekas, Dimitri P. (1995). Nonlinear Programming (1st ed.). Belmont, Massachusetts: Athena Scientific. p. Proposition B.21(c). ISBN 1-886529-14-0.
• "The Fundamental Theorem of Linear Programming". WOLFRAM Demonstrations Project. Retrieved 25 September 2024.