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单词 FarkasLemma
释义

Farkas lemma


Given an m×n matrix A and an 1×n real row vectorMathworldPlanetmath c, both with real coefficients, one and only one of the following systems has a solution:

  1. 1.

    Ax0 and cx>0 for some n-column vector x;

  2. 2.

    wA=c and w0 for some m-row vector w.

Equivalently, one and only one of the following has a solution:

  1. 1.

    Ax0, x0 and cx>0 for some n-column vector x;

  2. 2.

    wAc and w0 for some m-row vector w.

Remark. Here, Ax0 means that every of Ax is nonnegative, and similarly with the other expressions.

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更新时间:2025/5/4 22:42:13