论文标题
二元性理论和一类圆锥线性问题的最佳解决方案的特征
Duality theory and characterizations of optimal solutions for a class of conic linear problems
论文作者
论文摘要
For a primal-dual pair of conic linear problems that are described by convex cones $S\subset X$, $T\subset Y$, bilinear symmetric objective functions $\langle\cdot,\cdot\rangle_X$, $\langle\cdot,\cdot\rangle_Y$ and a linear operator $A:X\rightarrow Y$, we show that满足$ ax^*= b $和$ a^ty^*= c $的最佳解决方案的存在$ x^*\在t $中的$ y^*\最终的存在，最终取决于问题的一致性和解决性，$ min \ min \ langle z，z \ rangle_y， w，w \ rangle_x，\; w \ in \ {a^ty-c：y \ in t \} $。假设这两个问题是一致且可解决的，强大的二元性定理以及最佳溶液的几何和代数表征是通过Farkas的引理的自然概括而没有闭合条件来获得的。在复杂空间中连续线性编程和线性编程的情况下，讨论了主要理论的某些应用。
For a primal-dual pair of conic linear problems that are described by convex cones $S\subset X$, $T\subset Y$, bilinear symmetric objective functions $\langle\cdot,\cdot\rangle_X$, $\langle\cdot,\cdot\rangle_Y$ and a linear operator $A:X\rightarrow Y$, we show that the existence of optimal solutions $x^*\in S$, $y^*\in T$ that satisfy $Ax^*=b$ and $A^Ty^*=c$ eventually comes down to the consistency and solvability of the problems $min\langle z,z\rangle_Y,\;z\in\{Ax-b:x\in S\}$ and $ min\langle w,w\rangle_X,\; w\in\{A^Ty-c:y\in T\}$. Assuming that these two problems are consistent and solvable, strong duality theorems as well as geometric and algebraic characterizations of optimal solutions are obtained via natural generalizations of the Farkas' Lemma without a closure condition. Some applications of the main theory are discussed in the cases of continuous linear programming and linear programming in complex space.
