论文标题
产品约束下的总和最小化
Minimization of the sum under product constraints
论文作者
论文摘要
我们系统地探索了一类带有线性目标函数的约束优化问题，并且是优化变量对数的线性组合的约束。这些问题可以看作是算术和几何方式之间不平等的概括。在一般情况下，在自然的假设下证明了最小化器的存在和独特性。我们详细研究特殊的子类，其中用组合术语（定向图，根树）描述了一组约束。特别是，如果有一个定向，连接的图形，我们试图最大程度地减少在循环产物约束下的所有ARC值的总数。我们在给定（大）变量数量的问题中获得了一些最小值的估计和渐近学。同样，在这种情况下，我们重新审视了称为J. Shallit最小化问题的渐近结果。该材料以问题书的形式呈现。除了构成主要理论线程的问题外，还有许多练习，一些迷你偏氧化物和一些数值方法。
We systematically explore a class of constrained optimization problems with linear objective function and constraints that are linear combinations of logarithms of the optimization variables. Such problems can be viewed as a generalization of the inequality between the arithmetic and geometric means. The existence and uniqueness of the minimizer is proved under natural assumptions in the general case. We study in detail special subclasses where the set of constraints is described in combinatorial terms (oriented graphs, rooted trees). In particular, given a directed, strongly connected graph, we seek to minimize the total of all arc values under cyclic product constraints. We obtain some estimates and asymptotics for the minimum in problems with given (large) number of variables. Also in this context we revisit an asymptotical result known as J. Shallit's minimization problem. The material is presented in the form of a problem book. Along with problems that constitute main theoretical threads, there are many exercises, some mini-paradoxes, and a touch of numerical methods.
