6,572 research outputs found
Extremal Problems for Subset Divisors
Let be a set of positive integers. We say that a subset of is
a divisor of , if the sum of the elements in divides the sum of the
elements in . We are interested in the following extremal problem. For each
, what is the maximum number of divisors a set of positive integers can
have? We determine this function exactly for all values of . Moreover, for
each we characterize all sets that achieve the maximum. We also prove
results for the -subset analogue of our problem. For this variant, we
determine the function exactly in the special case that . We also
characterize all sets that achieve this bound when .Comment: 10 pages, 0 figures. This is essentially the journal version of the
paper, which appeared in the Electronic Journal of Combinatoric
Average Relaxations of Extremal Problems
In this paper extremal problems that include averaging operation in constraints and objective are considered. The relaxation caused by a replacement of a problem without averaging with a problem that includes averaging is formally defined and investigated. Canonical form for nolinear programming problem with averaging is constructed and its conditions for optimality are derived. It is shown how optimality conditions for optimal control problems with various types of objectives and constraints can be derived using its averaged relaxation.averaging; constraint relaxation; nonlinear programming; optimal control problem; optimality conditions
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