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Globally Optimal Energy-Efficient Power Control and Receiver Design in Wireless Networks

Alessio Zappone, Emil Björnson, Luca Sanguinetti, Eduard Jorswieck

arXiv:1602.02923v2math.OCcs.IT

TL;DR

The paper develops optimization frameworks for characterizing energy-efficient performance in wireless networks. It combines monotonic or sequential optimization with fractional programming, achieving global optimality in several practical scenarios while reducing complexity for sequential solutions.

  • Problem

    The work addresses the need to characterize ultimate energy-efficient performance in interference-limited wireless networks.

  • Method

    The paper combines monotonic optimization or sequential optimization with fractional programming to develop globally or first-order optimal frameworks for energy-efficiency maximization.

  • Results

    Sequential fractional programming achieves global optimality in several practical communication scenarios, while its low-complexity solutions achieve near-optimal performance against monotonic fractional benchmarks.

  • Takeaways & Limitations

    The frameworks support multiple energy-efficiency metrics, constraints, transmit-power allocation, and joint transmit-power and receiver-design optimization.

Abstract

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The characterization of the global maximum of energy efficiency (EE) problems in wireless networks is a challenging problem due to the non-convex nature of investigated problems in interference channels. The aim of this work is to develop a new and general framework to achieve globally optimal solutions. First, the hidden monotonic structure of the most common EE maximization problems is exploited jointly with fractional programming theory to obtain globally optimal solutions with exponential complexity in the number of network links. To overcome this issue, we also propose a framework to compute suboptimal power control strategies characterized by affordable complexity. This is achieved by merging fractional programming and sequential optimization. The proposed monotonic framework is used to shed light on the ultimate performance of wireless networks in terms of EE and also to benchmark the performance of the lower-complexity framework based on sequential programming. Numerical evidence is provided to show that the sequential fractional programming framework achieves global optimality in several practical communication scenarios.

I. INTRODUCTION

The paper addresses energy-efficiency optimization in interference-limited wireless networks, where interference makes standard fractional-programming approaches insufficient for global optimization. It develops globally optimal and lower-complexity frameworks, using the former to characterize ultimate EE performance and benchmark the latter.

  • I. INTRODUCTION: ICT accounted for 5% of global CO2-equivalent emissions, while connected devices and data traffic were projected to grow rapidly.The cited projections were 50 billion connected devices by 2020 and 1000× higher data traffic over the following decade.
  • I. INTRODUCTION: Energy efficiency measures reliably transmitted information per Joule and mattered to operators' electricity costs and end-users' battery lifetimes.The paper motivates EE optimization as an alternative to restricting ICT usage.
  • I. INTRODUCTION: Interference makes EE numerators non-concave in transmit power, so conventional fractional programming cannot directly guarantee global optimization.Alternative interference-handling methods use alternating optimization, but the cited approaches do not ensure the same global guarantee.
  • B. Major contributions: The paper develops monotonic fractional programming for global optimality and sequential fractional programming for reduced-complexity solutions satisfying first-order optimality conditions.The frameworks target broad classes of EE metrics, constraints, power-control problems, and joint receiver-design problems.
  • B. Major contributions: Monotonic fractional programming characterizes the energy-efficient Pareto boundary and benchmarks lower-complexity resource-allocation algorithms.The sequential framework uses fractional programming with sequential optimization and requires solving convex subproblems.

II. SIGNAL MODEL AND PROBLEM FORMULATION

The paper formulates energy-efficiency optimization for mutually interfering wireless links, covering several SINR models, network metrics, constraints, and CSI settings. Because interference makes the fractional problems non-convex, the paper introduces monotonic and sequential optimization frameworks.

  • Signal model: The model considers K mutually interfering links with transmit powers p_k bounded by per-link maximum-power constraints.Each transmitter is single-antenna, while receivers may have multiple antennas.
  • Signal model: The framework accommodates general SINR and rate representations, including conventional interference channels, self-interference models, and LMMSE vector-channel reception.LMMSE is identified as the optimal linear receive structure from an energy-efficiency perspective.
  • Energy-efficiency metrics: The considered metrics include GEE, weighted minimum EE, weighted sum EE, and weighted product EE, with weights enabling prioritization of individual links.GEE measures network-wide reliably transmitted data per total consumed energy, whereas the other metrics support multi-objective trade-offs.
  • Constraints: The optimization can impose maximum-power, minimum-rate, total-power, and interference-temperature constraints through general constraint functions.The formulation permits per-user or network-level constraints and supports rate protection for other network nodes.
  • CSI assumptions: The formulation supports perfect, statistical, and imperfect CSI, including statistical or worst-case treatments of rate and SINR expressions.These options extend the model to fast-fading scenarios and channel-estimation or limited-feedback errors.
  • Optimization problem: Interference-limited EE optimization violates the concavity and convexity conditions required for polynomial-complexity fractional programming.The paper therefore combines fractional programming with monotonic optimization for global solutions and with sequential optimization for affordable-complexity local solutions.

III. MATHEMATICAL PRELIMINARIES

This section reviews the three optimization frameworks used later: fractional programming, monotonic optimization, and sequential optimization.

  • Mathematical preliminaries: The paper introduces fractional programming, monotonic optimization, and sequential optimization as mathematical foundations for the subsequent frameworks.The section positions these theories as the basis for complexity and algorithmic developments in later sections.

A. Fractional programming

Fractional programming transforms ratio optimization into auxiliary-function root finding, but global convergence depends on globally solving each non-convex subproblem. Convexity enables polynomial complexity, whereas non-convexity requires more expensive global optimization.

  • Fractional-programming formulation: A generalized fractional program optimizes multiple ratios f_k(x)/g_k(x), while K = 1 gives the single-ratio case.The denominator functions are positive on the domain.
  • Fractional-programming formulation: Because the fractional objective is generally non-concave, standard convex optimization algorithms are not guaranteed to solve the problem.Specific fractional-programming algorithms are therefore required.
  • Dinkelbach method: The optimum is characterized by the unique zero of an auxiliary function F(λ), which generalized Dinkelbach’s algorithm seeks iteratively.The algorithm updates the parameter λ while solving the associated auxiliary optimization problem.
  • Complexity: When the numerator is concave, the denominator convex, and constraints convex, each subproblem is a concave maximization with polynomial-time complexity.Under these conditions, the algorithm also has a super-linear convergence rate because its λ update follows Newton’s method.
  • Complexity: When the auxiliary problem is non-convex, global optimization methods are required, substantially increasing complexity and often requiring exploration of the whole feasible set.The same convexity requirements also apply to alternative methods such as the Charnes-Cooper transform.

B. Monotonic optimization

Monotonic optimization exploits monotonicity or hidden monotonicity to search only relevant boundaries rather than the complete feasible set. It guarantees global convergence but remains exponential in the number of variables.

  • Framework: Monotonic optimization exploits increasing or decreasing objective and constraint structures without requiring convexity.Its boundary-search principle avoids exploring the complete feasible set when the objective increases in all variables.
  • Framework: A monotonic optimization problem uses an increasing objective over a compact normal set and a closed co-normal constraint set.These set structures define the canonical form of the framework.
  • Global solution: The solution lies on the upper boundary of the feasible intersection, enabling polyblock and BRB algorithms to search only that boundary.This boundary restriction can drastically simplify the global search.
  • Complexity: The complexity of monotonic optimization remains exponential in the number of variables.It depends on the iterations needed to search the boundary and on objective-function evaluation complexity, although searching the frontier is less demanding than searching the whole feasible set.

C. Sequential optimization

Sequential optimization generates affordable-complexity candidate solutions for non-convex EE problems, with limit points satisfying KKT conditions under stated assumptions. The framework combines this approach with fractional programming to reduce the exponential complexity of global monotonic optimization.

  • C. Sequential optimization: Sequential optimization solves a sequence of easier approximate problems to generate feasible candidate solutions for non-convex optimization.Its practical value depends on finding approximations that are easier to solve than the original problem.
  • C. Sequential optimization: Under suitable constraint qualifications, every limit point achieving the limiting objective value satisfies the original problem’s KKT conditions.Proposition 3 establishes this result for compact feasible sets and differentiable objectives and constraints.
  • C. Sequential optimization: The sequence of approximate solutions monotonically increases the original objective, converging to the objective value attained at a KKT point.The result applies when the approximation sequence satisfies Proposition 3’s assumptions.
  • C. Sequential optimization: Extensions of sequential optimization show that stationarity can hold even when the original objective is not strictly concave, provided the feasible set is convex and only the objective is approximated.The supplied discussion distinguishes this case from results requiring strict concavity.
  • C. Sequential optimization: For the EE framework, fractional programming is combined with sequential optimization because the global monotonic framework still has exponential complexity.The required monotonicity assumption represents achievable rates and constraint functions as differences of monotonic functions, without requiring concavity or convexity of the positive parts.

A. GEE maximization

GEE maximization initially lacks the monotonic objective and canonical feasible-set structure required by monotonic optimization. Fractional programming and auxiliary-variable reformulations expose hidden monotonicity and yield a canonical monotonic problem under Assumption 3.

  • A. GEE maximization: GEE maximization is handled through Dinkelbach’s algorithm by solving an auxiliary problem for a given parameter λj.The auxiliary problem introduces a transformed objective involving the achievable-rate numerator and λj-dependent terms.
  • A. GEE maximization: The original auxiliary formulation is not directly monotonic because achievable rates and interference-related negative terms make its objective non-monotonic in power.Its constraint set also need not be the intersection of normal and co-normal sets because differences of increasing functions are not generally increasing.
  • A. GEE maximization: Under Assumption 3, GEE auxiliary problem (25) can be expressed as a monotonic optimization problem in canonical form.The assumption requires relevant rate and constraint functions to be differences of monotonic functions.
  • A. GEE maximization: Introducing t = q−(pmax, λj) − q−(p, λj) transforms the objective into q+(p) + t, with q+(p) and q−(p, λj) increasing in p.This reformulation supplies the monotonic objective structure used in the canonical problem.
  • A. GEE maximization: Replacing multiple constraints with one equivalent difference-of-increasing-functions constraint and introducing s produces a feasible set that is the intersection of normal and co-normal sets.Monotonicity and continuity of the reformulated constraints complete the canonical-form proof.

B. WMEE maximization

WMEE maximization is treated as a generalized fractional program whose auxiliary objective is initially non-monotonic. Under Assumption 3, reformulation with auxiliary variables converts it into a canonical monotonic problem, with broader model support than affine power consumption.

  • B. WMEE maximization: WMEE maximization belongs to generalized fractional programming and requires solving an auxiliary problem at each iteration j.The auxiliary formulation uses λj-dependent terms to transform the generalized fractional objective.
  • B. WMEE maximization: Although the WMEE auxiliary objective is not monotonic, Proposition 5 states that it becomes a canonical monotonic problem when Assumption 3 holds.The proposition provides the WMEE counterpart to the GEE reformulation.
  • B. WMEE maximization: The reformulation introduces t using λj-dependent numerator terms, then expresses the objective through increasing components.The displayed fragments specify bounds for t based on differences evaluated at pmax and 0K.
  • B. WMEE maximization: Rewriting all constraints ck(p) ≥ 0 as one constraint and applying the GEE proof arguments establishes canonical monotonicity for WMEE.The same normal/co-normal feasible-set reasoning is reused.
  • B. WMEE maximization: The results extend beyond affine power consumption to any model for which the relevant λj-dependent functions are monotonic in p, and to multiple constraint functions per user.Multiple constraints remain admissible when each can be expressed as a difference of two monotonic functions.

C. WSEE and WPEE maximization

The framework also addresses WSEE and WPEE, whose sum-of-ratios and product-of-ratios forms are difficult even under favorable convexity assumptions. It supports broad interference-limited scenarios and offers a lower-complexity sequential alternative with KKT guarantees and reported global-optimality agreement.

  • C. WSEE and WPEE maximization: WSEE and WPEE are difficult sum-of-ratios and product-of-ratios problems, even with concave numerators, convex denominators, and convex feasible sets.The proposed monotonic fractional programming framework is nevertheless applied to both objectives.
  • C. WSEE and WPEE maximization: WSEE can be expressed as a single ratio whose numerator is a difference of increasing functions and whose denominator is increasing.This permits use of the GEE maximization method for global optimization.
  • C. WSEE and WPEE maximization: WPEE admits the same framework because expanding its numerator produces a difference of increasing functions while its denominator is increasing.The product of increasing functions remains increasing, supporting the reformulation.
  • C. WSEE and WPEE maximization: Beamforming extensions are formally possible through domain changes, but the transformed feasible set generally lacks a closed form and increases polyblock-method complexity.This is the stated practical boundary for extending the framework beyond power control and linear receiver design.
  • C. WSEE and WPEE maximization: The monotonic fractional programming framework has exponential complexity, motivating sequential fractional programming for affordable-complexity EE solutions satisfying KKT conditions.The sequential approach is reported to attain global optimality in the supplied result passage.
  • C. WSEE and WPEE maximization: The framework handles SINRs arising in joint power control and receiver design, multiple constraint types including interference-temperature constraints, and broader communication-system formulations.The paper contrasts this scope with earlier sequential approaches limited to more specific SINRs or constraint settings.

A. GEE maximization

The GEE problem is reformulated through approximations that expose a tractable single-ratio structure, enabling sequential optimization with convergence and first-order optimality guarantees.

  • Problem: Difference-of-concave numerators and constraints prevent direct fractional-programming solutions for the original GEE problem.The framework addresses this non-concavity by exploiting hidden structure.
  • Guarantees: The sequential GEE objective values increase monotonically and converge from any feasible initialization.This property follows from tight lower-bound approximations at the current iterate.
  • Guarantees: Any limit point attaining the converged GEE value satisfies the original problem’s KKT conditions under suitable constraint qualifications.Slater’s condition is identified as a suitable qualification for the relevant pseudo-concave maximization problems.
  • Method: For each iterate, concave numerators and affine denominators yield a single-ratio problem solvable globally by fractional programming.The equivalent convex Charnes–Cooper reformulation is also available.

B. WMEE maximization

The WMEE framework extends sequential fractional optimization to weighted metrics, retaining affordable subproblem complexity and monotonic convergence while supporting broader receiver-design settings.

  • Method: WMEE numerators and constraint functions can likewise be represented as differences of concave functions, motivating a sequential approximation.The approach mirrors the GEE construction while targeting weighted network energy efficiency.
  • Guarantees: The WMEE sequence of objective values is monotonically increasing and convergent.The proof follows the same reasoning used for the GEE proposition.
  • Method: Each WMEE approximation has a concave numerator, convex denominator, and concave or affine constraints, so fractional programming solves it globally.This provides affordable-complexity subproblems.
  • Caveat: Unlike GEE, WMEE is non-differentiable, so KKT conditions are not available for the original WMEE problem.The framework instead establishes convergence of the objective sequence.
  • Applications: The proposed frameworks cover two practical case studies: multi-antenna LTE and massive MIMO networks.These scenarios represent leading 4G and candidate 5G technologies, respectively.

A. MIMO LTE network with LMMSE detection

The numerical studies apply the frameworks to multi-cell LTE and massive-MIMO uplinks with linear receivers, comparing global monotonic optimization against lower-complexity sequential optimization.

  • System model: The LTE model uses multi-antenna UEs and base stations with LMMSE detection, producing SINRs formally equivalent to the framework’s established form.The receiver structure is treated as the optimal linear receive structure in the cited model.
  • Simulation setup: In the LTE simulations, three base stations serve randomly placed two-antenna UEs over a 2 km × 2 km area with uncorrelated Rayleigh fading.Users associate with their nearest base station.
  • LTE results: For K = 2, monotonic fractional programming characterizes the complete energy-efficient Pareto region, whereas 40000 grid points miss portions of it.The region is sampled using 200 weighted WMEE optimizations.
  • LTE results: The K = 2 maximum-GEE point coincides across sequential and monotonic fractional programming and lies inside the Pareto region, while full-power allocation is also interior and inefficient.The result shows that GEE maximization need not select a Pareto-boundary point.
  • LTE results: For K = 3, the energy-efficient Pareto region is non-convex, so WMEE maximization is generally required to characterize it.This non-convexity is directly visible in the monotonic-programming result.
  • Massive-MIMO results: In massive-MIMO experiments, sequential fractional programming matches the monotonic framework’s global GEE maximum across tested settings, including K = 8.The setup includes hardware impairments, channel-estimation errors, and comparisons over maximum transmit power.

VII. CONCLUSION

The paper develops two fractional-programming frameworks for energy-efficient wireless-network optimization: one globally optimal but exponentially complex, and one lower-complexity framework that achieves first-order optimality and global optimality in several practical scenarios.

  • VII. CONCLUSION: The monotonic fractional programming framework guarantees global optimality while its complexity still grows exponentially with the number of network links.It is used to characterize the ultimate energy-efficiency performance of wireless networks.
  • VII. CONCLUSION: The sequential fractional programming framework combines sequential optimization with fractional programming to provide affordable-complexity solutions satisfying first-order optimality conditions.The framework solves a sequence of easier approximate fractional problems.
  • VII. CONCLUSION: Figures 5–7 benchmark the two proposed frameworks against sequential sum-rate maximization and full power allocation using achieved GEE versus Pmax.The comparisons cover K = 2, K = 3, and K = 8 mobiles.
  • VII. CONCLUSION: Numerical results show that sequential fractional programming achieves global optimality in several practical communication scenarios.Figure 4 illustrates convergence to the global GEE optimum for K = 3, while Figures 5–7 compare GEE against Pmax for K = 2, K = 3, and K = 8.
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