Minimization of the sum under product constraints

Abstract

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.