Fundamental theorem of linear programming: Difference between revisions

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:<math>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^Tx_i - c^Tx^\ast)</math>
 
Since all terms in the sum are negativenonpositive and the sum is equal to zero, we must have that each individual term is equal to zero. Hence, 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 all optimal solutions.
 
==References==