Semidefinite Relaxation for Downlink OFDMA Resource Allocation Using Adaptive Modulation Pablo Adasme * Abdel Lisser * Ismael Soto ⋆ * Laboratoire de Recherche en Informatique, Universite Paris-Sud XI Batiment 490, 91405, Orsay Cedex, France ⋆ Departamento de Ingenieria Industrial, Universidad de Santiago de Chile Av. Ecuador 3769, Santiago, Chile Abstract This paper proposes a new binary quadratic formulation for the problem of minimizing power sub- ject to rate and sub-carrier allocation constraints over wireless downlink (DL) Orthogonal Frequency Division Multiple Access (OFDMA) networks when using adaptive modulation. The model allows to decide what modulation and what sub-carriers are going to be used by a particular user in the system depending on its bits rate requirements. Thus, we derive a semidefinite relaxation whereby simulation results, we get a total tightness approximation average gain of 18.65 % to the optimal solution of the problem when compared to the linear programming relaxation obtained by applying Fortet linearization method to the quadratic model. Keywords: Orthogonal Division Multiple Access, downlink allocation, adaptive modulation, semidef- inite programming. 1 Introduction When many users are connected to a Base Station (BS) and a large number of signals are using a wireless channel, greater complexity is generated by the negative phenomena of Multiple Access Interfer- ence (MAI) and Multi-path distortions. OFDMA is a suitable technology for combating these negative phenomena and it is currently the type of modulation used in wireless multi-user systems such as IEEE 802.11a/g WLAN, in networks of fixed access such as IEEE 802.16a and also for mobile WiMax deploy- ments networks ensuring high quality of service (QoS) requirements [1]. OFDMA divides the channel into several orthogonal narrow band frequencies forming sub-carriers (sub-channels) giving access to several users simultaneously. In order to solve the resource allocation problem of DL sub-carrier and power over OFDMA systems, several schemes and algorithms have been proposed [2]. In [3] for example, the prob- lem of minimizing total power consumption with constraints on bit error rate (BER) for users requiring different services is formulated while in [4], a sub-carrier allocation algorithm is proposed to increase the total user data rates subject to BER, total transmission power and proportional rate constraints. In this work we deal with the problem of minimizing power subject to assignment sub-carrier constraints depending on the bits required by each user using adaptive modulation, which means varying the number of bits to be sent in the different sub-channels. We state a quadratic formulation of this problem and derive a Semidefinite programming (SDP) relaxation, besides we get an equivalent linear program which is obtained by linearizing the quadratic terms using Fortet linearization [5]. Then, we compare gaps obtained by SDP relaxation and the relaxed solution of the equivalent linear formulation. We use SDP to get tighter bounds due to its proven efficiency in combinatorial optimization [6]. Moreover, the solvers uses interior point algorithms with polynomial time complexity [7]. This paper is organized as follows: 1