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We present and prove a theorem which gives the optimal dual vector for which a Lagrangian dual problem in the Single Period Design Problem (SPDP) is maximized.
Furthermore we give a straightforward generalization to the Multi-Period Design Problem (MPDP). Based on the optimal dual values derived we compute the solution of the Lagrangian relaxation and compare it
with the linear relaxation and optimal IP values.

The current work considers random access communications within a single cell and provides
an algorithmic suggestion for the almost optimal coordination of user actions.
In the scenario considered, an access effort is successful if (a) the signal is
detected at the receiver and (b) no collision occurs. The first event is controlled by the user transmission power and the second by the
choice of access (back-off) probability. These constitute the user action pair.
The algorithm aims at exploiting information from measurements and user reports, in order to estimate current
values of the system situation. Based on these, two optimization problems can be formulated and solved
for the so called contention level and transmission power level at the
base station side. The methodology to find the optimal values is based on minimization of a drift function.
The two values are broadcast in order to help the users update their actions ``almost optimally``.
In this way the wireless cell can achieve self-optimization, without outside coordination, by relying on such intelligent information exchange
and parameter estimation. Numerical results illustrate the great benefits of the
suggested algorithm, compared to scenarios where the actions remain fixed, at a very low or even zero cost in power expenditure and delay.

Base station cooperation in the downlink of cellular systems has been recently suggested as a promising concept towards a
better exploitation of the communication system physical resources. It may offer a high gain in capacity through interference mitigation.
This however,
comes at a cost of high information exchange between cooperating entities and a high computational burden.
Clustering of base stations into subgroups is an alternative to
guarantee such cooperation benefits in a lower scale. The optimal definition of clusters, however, and a systematic way to find a solution to
such problem is not yet available. In this work, we highlight the combinatorial nature of the problem, exploit this to describe the
system of users and base stations as a graph and formulate a pure 0-1 program. Its solution
suggests a cost optimal way to form clusters and assign user subsets to them.