最优化问题句法逻辑表达式的下界与对偶性--用Lagrange对偶刻画最优性条件
Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions
摘要：
我们表明，简单的语法表达式，如存在二阶（ESO）通用霍恩公式可以表示NP-难优化问题。决策问题和优化问题的表达方式有很大的不同。这类似于两类问题的计算时间差异;例如，2SAT Horn公式可以在多项式时间内满足，而优化版本则是NP难的。已知所有多项式可解的决策问题在后继关系存在的情况下都可以表示为ESO泛（$Π_1$）Horn语句。我们在这里表明，另一方面，如果$P \neq NP$，优化问题无视这样的特征，通过证明，即使是$Π_0$（量词免费）霍恩公式是无法保证多项式时间可解性。最后，通过连接的概念，在优化对偶与描述的复杂性，我们将展示一种方法，通过该方法，优化问题可以解决一个单一的调用“决策”图灵机，而不是多个调用使用经典的二分搜索设置。
英文摘要：
We show that simple syntactic expressions such as existential second order (ESO) universal Horn formulae can express NP-hard optimisation problems. There is a significant difference between the expressibilities of decision problems and optimisation problems. This is similar to the difference in computation times for the two classes of problems; for example, a 2SAT Horn formula can be satisfied in polynomial time, whereas the optimisation version in NP-hard. It is known that all polynomially solvable decision problems can be expressed as ESO universal ($Π_1$) Horn sentences in the presence of a successor relation. We show here that, on the other hand, if $P \neq NP$, optimisation problems defy such a characterisation, by demonstrating that even a $Π_0$ (quantifier free) Horn formula is unable to guarantee polynomial time solvability. Finally, by connecting concepts in optimisation duality with those in descriptive complexity, we will show a method by which optimisation problems can be solved by a single call to a "decision" Turing machine, as opposed to multiple calls using a classical binary search setting.
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