This chapter presents a class of distributionally robust optimization problems in which a decision-maker has to choose an action in an uncertain environment. The decision-maker has a continuous action space and aims to learn her optimal strategy. The true distribution of the uncertainty is unknown to the decision-maker. This chapter provides alternative ways to select a distribution based on empirical observations of the decision-maker. This leads to a distributionally robust optimization problem. Simple algorithms, whose dynamics are inspired from the gradient flows, are proposed to find local optima. The method is extended to a class of optimization problems with orthogonal constraints and coupled constraints over the simplex set and polytopes. The designed dynamics do not use the projection operator and are able to satisfy both upper-and lower-bound constraints. The convergence rate of the algorithm to generalized evolutionarily stable strategy is derived using a mean regret estimate. Illustrative examples are provided.

Maximizing frequency reuse is of high interest to multi-tier wireless cellular networks. The coexistence of macrocell and femtocell requires advanced cross-tier interference mitigation to satisfy the spectral efficiency sought for the evolution of cellular networks. In this paper, our proposal combines an opportunistic cancellation of the interference at the base station of the femtocell(HeNB) which mitigates dead zone occurrences at the femtocell premises in closed access regime, and a transmit power limitation on the femtocell mobile user for interference avoidance at the base station of the macro cell. Our proposed com bination profits from the network infrastructure where successive interference cancellation does not apply, and neighbor discovery enabled at the HeNBs. Through simulation results and analysis, our proposal improves the underlay network performance with implementation of a self organized-self optimized network. I1Idex Terms-Macrocell, femtocell, two-tier networks MMSE Cross-tier interference mitigation, power control, SON ' ,

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In a multi-tier cellular communication system, the interference from one tier to another, denoted as cross-tier interference, is a limiting factor for the system performance. In spectrum-sharing usage, we consider the uplink crosstier interference management of heterogeneous networks using femtocells overlaid onto the macrocells. We propose a variation of the cellular architecture and introduce a novel femtocell clustering based on interference cancellation to enhance the sum rate capacity. Our proposal is to use a distributed antenna system (DAS) as an interface to mitigate the cross-tier interference between the macrocell and femtocell tiers. By placing a DAS remote antenna unit (RAU) near a set of femtocells that experience interference from a macrocell user, the DAS can retrieve the interference symbols and feed them back to the femtocells, where each cell can perform interference cancellation when necessary. In addition, the DAS can forward the recovered data to the macrocell base station (MBS); thus, the macrocell user can reduce its transmit power to reach a RAU located closer than the MBS. By distributing the sensor nodes within the macrocell coverage, the proposed scheme can mitigate the cross-tier interference at different locations for several femtocell clusters. Our simulation results show substantial improvement in the network sum rate capacity.