## Linear equations and inequalities on finite dimensional, real or complex, vector spaces: a unified theory.(English)Zbl 0174.31502

This paper presents a unified theory of linear equations and inequalities, real or complex, on finite-dimensional vector spaces. Notations: Let $$S\subset \mathbb C^n$$ be a closed convex cone, $S^*= \{y\in\mathbb C^n : x\in S\Rightarrow \text{Re}(y^Hx)\ge 0\},\ A\in \mathbb C^{m\times m}, b\in \mathbb C^m.$ The system $$(*)$$ $$Ax = b$$, $$x\in S$$ is “consistent” [ “asymptotically consistent” ] if $$\exists\, x \in S \ni Ax = b$$ [ $$\exists\, \{x_k\}\subset S \ni \lim Ax_k = b$$ ]. Sample results:
Theorem: The following are equivalent (where $$A^T, A^H, A^+$$ denote the transpose, the conjugate transpose, the generalized inverse of $$A$$)::
(a) $$(*)$$ is asymptotically consistent,
(b) $$A^H y\in S^* \Rightarrow \text{Re}(b^H y)\ge 0$$,
(c) $$b\in R(A)$$ and $$A^+ b\in \text{cl}(N(A) + S)$$.
Corollary: Let $$N(A)+S$$ be closed. Then $$(*)$$ is consistent if, and only if, $$A^H y\in S^* \Rightarrow \text{Re}(b^H y) = 0$$.
Corollary (Solvability of linear equations): $$Ax = b$$ is consistent if, and only if, $$A^H y = 0 \Rightarrow b^H y = 0$$.
Corollary (Farkas): Let $$A\in \mathbb R^{n\times n}$$, $$b\in\mathbb R^n$$. Then $$Ax =b$$, $$x \ge 0$$ is consistent if, and only if, $$A^T y\ge 0 \Rightarrow b^T y\ge 0$$.
These results are used to prove a duality theorem for complex linear programming, following and slightly extending N. Levinson [J. Math Anal. Appl. 14, 44–62 (1966; Zbl 0136.13802)].

### MSC:

 15A06 Linear equations (linear algebraic aspects) 15A03 Vector spaces, linear dependence, rank, lineability 90C46 Optimality conditions and duality in mathematical programming 90C05 Linear programming

Zbl 0136.13802
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### References:

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