Fundamental theorem of linear programming: Difference between revisions

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Hence <math>x^\ast</math> is not an optimal solution, a contradiction. Therefore, <math>x^\ast</math> must live on the boundary of <math>P</math>. If <math>x^\ast</math> is not a vertex itself, it must be the convex combination of vertices of <math>P</math>, say <math>x_1, ..., x_t</math>. Then <math>x^\ast = \sum_{i=1}^t \lambda_i x_i</math> with <math>\lambda_i \geq 0</math> and <math>\sum_{i=1}^t \lambda_i = 1</math>. Observe that

:<math>0=c^{T}\left(\left(\sum_{i=1}^{t}\lambda_{1i}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}).</math>

Since <math>x^{\ast}</math> 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, <math>c^{T}x^{\ast}=c^{T}x_{i}</math> for each <math>x_i</math>, so every <math>x_i</math> is also optimal, and therefore all points on the face whose vertices are <math>x_1, ..., x_t</math>, are optimal solutions.