Fractional programming (FP) refers to a family of optimization problems that involve ratio term(s). This two-part paper explores the use of FP in the design and optimization of communication systems. Part I of this paper focuses on FP theory and on solving continuous problems. The main theoretical contribution is a novel quadratic transform technique for tackling the multiple-ratio concave-convex FP problem-in contrast to conventional FP techniques that mostly can only deal with the single-ratio or the max-min-ratio case. Multiple-ratio FP problems are important for the optimization of communication networks, because system-level design often involves multiple signal-to-interference-plus-noise ratio terms. This paper considers the applications of FP to solving continuous problems in communication system design, particularly for power control, beamforming, and energy efficiency maximization. These application cases illustrate that the proposed quadratic transform can greatly facilitate the optimization involving ratios by recasting the original nonconvex problem as a sequence of convex problems. This FP-based problem reformulation gives rise to an efficient iterative optimization algorithm with provable convergence to a stationary point. The paper further demonstrates close connections between the proposed FP approach and other well-known algorithms in the literature, such as the fixed-point iteration and the weighted minimum mean-square-error beamforming. The optimization of discrete problems is discussed in Part II of this paper. Index Terms-Fractional programming (FP), quadratic transform, power control, beamforming, energy efficiency I. OVERVIEWO PTIMIZATION is a key aspect of communication system design [3], [4]. This two-part work explores the application of fractional programming (FP) in the design and optimization of communication systems. FP refers to a family of optimization problems containing ratio term(s). Its history can be traced back to an early paper on economic expansion [5] by von Neumann in 1937; it has since been studied extensively in broad areas in economics, management science, information theory, optics, graph theory, and computer science [6]- [8]. For example, FP has recently been applied in [9]-[12] to solve the energy efficiency maximization problem for wireless communication systems.

Abstract-This paper considers the optimization of the user and base-station (BS) association in a wireless downlink heterogeneous cellular network under the proportional fairness criterion. We first consider the case where each BS has a single antenna and transmits at fixed power, and propose a distributed price update strategy for a pricing-based user association scheme, in which the users are assigned to the BS based on the value of a utility function minus a price. The proposed price update algorithm is based on a coordinate descent method for solving the dual of the network utility maximization problem, and it has a rigorous performance guarantee. The main advantage of the proposed algorithm as compared to the existing subgradient method for price update is that the proposed algorithm is independent of parameter choices and can be implemented asynchronously. Further, this paper considers the joint user association and BS power control problem, and proposes an iterative dual coordinate descent and the power optimization algorithm that significantly outperforms existing approaches. Finally, this paper considers the joint user association and BS beamforming problem for the case where the BSs are equipped with multiple antennas and spatially multiplex multiple users. We incorporate dual coordinate descent with the weighted minimum mean-squared error (WMMSE) algorithm, and show that it achieves nearly the same performance as a computationally more complex benchmark algorithm (which applies the WMMSE algorithm on the entire network for BS association), while avoiding excessive BS handover.

This two-part paper develops novel methodologies for using fractional programming (FP) techniques to design and optimize communication systems. Part I of this paper proposes a new quadratic transform for FP and treats its application for continuous optimization problems. In this Part II of the paper, we study discrete problems, such as those involving user scheduling, which are considerably more difficult to solve. Unlike the continuous problems, discrete or mixed discrete-continuous problems normally cannot be recast as convex problems. In contrast to the common heuristic of relaxing the discrete variables, this work reformulates the original problem in an FP form amenable to distributed combinatorial optimization. The paper illustrates this methodology by tackling the important and challenging problem of uplink coordinated multi-cell user scheduling in wireless cellular systems. Uplink scheduling is more challenging than downlink scheduling, because uplink user scheduling decisions significantly affect the interference pattern in nearby cells. Further, the discrete scheduling variable needs to be optimized jointly with continuous variables such as transmit power levels and beamformers. The main idea of the proposed FP approach is to decouple the interaction among the interfering links, thereby permitting a distributed and joint optimization of the discrete and continuous variables with provable convergence. The paper shows that the well-known weighted minimum meansquare-error (WMMSE) algorithm can also be derived from a particular use of FP; but our proposed FP-based method significantly outperforms WMMSE when discrete user scheduling variables are involved, both in term of run-time efficiency and optimizing results. Index Terms-Fractional programming (FP), Lagrangian dual transform, user scheduling, discrete power control, discrete beamforming I. OVERVIEW F RACTIONAL programming (FP) is a valuable tool for the design and optimization of communication systems, because of the prominent role fractional terms-in particular the signal-to-interference-plus-noise (SINR) ratio-plays in the performance analysis of communication links. Part I of this paper [3] proposes a novel quadratic transform technique to tackle FP problems involving multiple ratios, which are frequently encountered in communication system design, but are typically beyond the capabilities of classic FP techniques, such as Schaible's transform [4] and Dinkelbach's method

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