Lemma Farkasha

Lemma Farkas - statement about the properties of linear inequalities. It was formulated and proved by Gyula Farkas. Used in geometric programming .

Formulation

Let be $f_{1}(x),f_{2}(x),...,f_{r}(x)$ ${\ displaystyle f_ {1} (x), f_ {2} (x), ..., f_ {r} (x)}$ $f_{1}(x), f_{2}(x), ..., f_{r}(x)$ and $g(x)$ ${\ displaystyle g (x)}$ $g(x)$ - homogeneous linear functions $m$ ${\ displaystyle m}$ $m$ variables $x_{1},x_{2},...,x_{m}$ ${\ displaystyle x_ {1}, x_ {2}, ..., x_ {m}}$ $x_{1}, x_{2}, ..., x_{m}$ .

Suppose ratios $f_{1}(x)\geqslant 0,f_{2}(x)\geqslant 0,...,f_{r}(x)\geqslant 0,$ ${\ displaystyle f_ {1} (x) \ geqslant 0, f_ {2} (x) \ geqslant 0, ..., f_ {r} (x) \ geqslant 0,}$ $f_{1}(x) \geqslant 0, f_{2}(x) \geqslant 0, ..., f_{r}(x) \geqslant 0,$ entail inequality $g(x)\geqslant 0$ ${\ displaystyle g (x) \ geqslant 0}$ $g(x) \geqslant 0$ .

Then there are non-negative constants. $y_{1},y_{2},...,y_{r}$ ${\ displaystyle y_ {1}, y_ {2}, ..., y_ {r}}$ $y_{1}, y_{2}, ..., y_{r}$ such that $y_{1}f_{1}(x)+y_{2}f_{2}(x)+...+y_{r}f_{r}(x)\equiv g(x)$ ${\ displaystyle y_ {1} f_ {1} (x) + y_ {2} f_ {2} (x) + ... + y_ {r} f_ {r} (x) \ equiv g (x)}$ $y_{1}f_{1}(x)+y_{2} f_{2}(x)+ ... +y_{r}f_{r}(x) \equiv g(x)$ is an identity.

It is assumed that all constants and variables are real.

Proof

The proof is in the book ^[1] .

Notes

↑ Geometric programming, 1972 , p. 263.

Literature

R. Duffin, E. Peterson, K. Zener. Geometrical programming. - M .: Mir, 1972. - 311 p.

[_5ddc9557143af6ef-1] Geometric programming, 1972 , p. 263.

Lemma Farkasha

Formulation

Proof

Notes

Literature

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