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Multi-Objective Signal Processing Optimization: The Way to Balance Conflicting Metrics in 5G Systems

Emil Björnson, Eduard Jorswieck, Mérouane Debbah, Björn Ottersten

arXiv:1406.2871v2cs.ITcs.NI

TL;DR

5G design must handle multiple coupled objectives whose improvements can conflict, rather than optimizing a single metric in isolation. The paper reviews multi-objective optimization and its visualization and scalarization tools, then illustrates the framework with massive MIMO. It concludes that Pareto-boundary operating points provide the relevant choices for balancing conflicting objectives, while practical communication-network applications remain largely unexplored.

  • Problem

    5G networks combine multiple conflicting objectives, creating a need for optimization tools that represent their tradeoffs instead of selecting one objective alone.

  • Method

    The paper surveys MOO definitions, properties, algorithms, visualization, and scalarization, and applies the framework to massive MIMO network dimensioning.

  • Results

    Pareto-boundary analysis identifies attainable operating points for balancing conflicting objectives, with massive MIMO illustrating tradeoffs between user rate, area rate, and energy efficiency.

  • Takeaways & Limitations

    Network designers can use MOO to visualize tradeoffs and select subjectively preferred operating points rather than seek a generally nonexistent global optimum.

  • Takeaways & Limitations

    Applications of established MOO analytic tools to communication networks remain greatly unexplored, and practical models must capture propagation, hardware imperfections, and heterogeneous characteristics.

Abstract

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The evolution of cellular networks is driven by the dream of ubiquitous wireless connectivity: Any data service is instantly accessible everywhere. With each generation of cellular networks, we have moved closer to this wireless dream; first by delivering wireless access to voice communications, then by providing wireless data services, and recently by delivering a WiFi-like experience with wide-area coverage and user mobility management. The support for high data rates has been the main objective in recent years, as seen from the academic focus on sum-rate optimization and the efforts from standardization bodies to meet the peak rate requirements specified in IMT-Advanced. In contrast, a variety of metrics/objectives are put forward in the technological preparations for 5G networks: higher peak rates, improved coverage with uniform user experience, higher reliability and lower latency, better energy efficiency, lower-cost user devices and services, better scalability with number of devices, etc. These multiple objectives are coupled, often in a conflicting manner such that improvements in one objective lead to degradation in the other objectives. Hence, the design of future networks calls for new optimization tools that properly handle the existence and tradeoffs between multiple objectives. In this article, we provide a review of multi-objective optimization (MOO), which is a mathematical framework to solve design problems with multiple conflicting objectives. (...) We provide a survey of the basic definitions, properties, and algorithmic tools in MOO. This reveals how signal processing algorithms are used to visualize the inherent conflicts between 5G performance objectives, thereby allowing the network designer to understand the possible operating points and how to balance the objectives in an efficient and satisfactory way. For clarity, we provide a case study on massive MIMO.

INTRODUCTION

5G networks face heterogeneous requirements and multiple performance objectives that share resources and often conflict. This motivates optimization methods that can represent and balance these tradeoffs.

  • INTRODUCTION: 5G expectations include higher user and area rates, more connected devices, improved energy efficiency, and heterogeneous network operation.The requirements also vary across devices, services, network deployments, and user conditions.
  • INTRODUCTION: Heterogeneous devices impose different data-rate and energy requirements, from high-rate handhelds to low-rate sensors.
  • INTRODUCTION: Heterogeneous services require different latency, reliability, continuity, and delivery characteristics.
  • INTRODUCTION: Shared resources such as time, frequency, space, power, and hardware couple objectives, often making improvement in one objective degrade another.Higher rates may require more power, favor users with good channels, or use intricate signal processing.
  • INTRODUCTION: Because 5G objectives conflict and cannot be treated separately, network design requires a framework that handles multiple objectives and searches for attainable operating points.

CONVENTIONAL SINGLE-OBJECTIVE OPTIMIZATION

Conventional wireless optimization selects one scalar objective and converts other objectives into constraints. The paper argues that this approach is no longer viable for heterogeneous, long-term 5G network design.

  • CONVENTIONAL SINGLE-OBJECTIVE OPTIMIZATION: Conventional formulations maximize a scalar utility, such as weighted user rates, or minimize power subject to rate constraints.Energy efficiency has also become a scalar optimization objective.
  • CONVENTIONAL SINGLE-OBJECTIVE OPTIMIZATION: The approach assumes one objective dominates and that suitable constraint values for the remaining objectives are known beforehand.
  • CONVENTIONAL SINGLE-OBJECTIVE OPTIMIZATION: Conventional network-utility problems usually emphasize short-term objective values rather than the long-term values important for network design.
  • CONVENTIONAL SINGLE-OBJECTIVE OPTIMIZATION: Given increased heterogeneity, long-term optimization needs, and diverse 5G expectations, the conventional approach is no longer viable.

NEW PARADIGM: MULTI-OBJECTIVE OPTIMIZATION

Multi-objective optimization models network design as simultaneous maximization of multiple objective functions over feasible resource utilizations. It provides a rigorous framework for examining attainable objective combinations and their conflicts.

  • NEW PARADIGM: MULTI-OBJECTIVE OPTIMIZATION: MOO treats area throughput, guaranteed rates, simultaneous users, and energy efficiency as multiple objectives rather than reducing them to one scalar goal.
  • NEW PARADIGM: MULTI-OBJECTIVE OPTIMIZATION: The resource bundle X contains feasible utilizations of time, frequency, space, power, and hardware, while each objective function measures satisfaction.Objectives are assumed bounded, continuous, and non-negative, with an all-zero dissatisfaction operating point.
  • NEW PARADIGM: MULTI-OBJECTIVE OPTIMIZATION: The network designer seeks to maximize all M objectives simultaneously without imposing a prior ordering among them.
  • NEW PARADIGM: MULTI-OBJECTIVE OPTIMIZATION: A MOOP maximizes the vector-valued function g(x) over x ∈X, making the objective explicitly multi-dimensional.
  • NEW PARADIGM: MULTI-OBJECTIVE OPTIMIZATION: Conflicting objectives generally prevent a global optimum, so the framework shifts attention toward attainable combinations and subjective operating-point selection.
  • NEW PARADIGM: MULTI-OBJECTIVE OPTIMIZATION: The attainable objective set G contains all objective-value combinations generated by feasible resource utilizations.

PARETO OPTIMAL OPERATING POINTS

The Pareto boundary contains attainable operating points that cannot improve one objective without degrading another. For conflicting objectives, selecting among these points requires subjective preference because no global optimum exists.

  • PARETO OPTIMAL OPERATING POINTS: Interior points of the attainable objective set are strictly suboptimal, leaving the Pareto boundary as the candidate efficient operating region.
  • PARETO OPTIMAL OPERATING POINTS: Strong Pareto-boundary points cannot improve any objective without degrading another and are mutually unordered.
  • PARETO OPTIMAL OPERATING POINTS: The strong Pareto boundary is a subset of the upper boundary, while the weak Pareto boundary is the complete upper boundary and may include further-improvable points.
  • PARETO OPTIMAL OPERATING POINTS: The utopia point simultaneously maximizes all objectives; when objectives conflict, it is unattainable and no global optimum exists.
  • PARETO OPTIMAL OPERATING POINTS: A resource point x* is Pareto optimal when its objective vector g(x*) lies on the Pareto boundary.
  • PARETO OPTIMAL OPERATING POINTS: Mapping a Pareto-optimal resource point to the boundary is given by g(x*), whereas the inverse mapping is hard to derive in most cases.Multiple resource points can produce the same objective point, including through common phase rotations in multi-antenna transmission.

SOLVING A MOOP BY VISUALIZATION

Visualization-based methods sample the attainable objective set or its Pareto boundary so designers can inspect tradeoffs and iteratively select an operating point.

  • A posteriori visualization: The a posteriori method visualizes Pareto-boundary tradeoffs before the designer selects a preferred operating point.The Pareto boundary is not known beforehand, so visualization supports informed preference decisions.
  • Grid sampling: Grid traversal evaluates g(x) over finitely many resource vectors, producing 6^D samples when each of D variables has six values.This approach is efficient when g(x) is easy to evaluate, but its sample count grows exponentially with D.
  • Boundary sampling: Directional traversal searches for outermost attainable points along nonnegative directions v, using a weighted Chebyshev problem.The resulting samples lie on the weak Pareto boundary; strong-boundary attainment requires a slight modification.
  • Comparison: Grid sampling reveals G’s shape and objective conflicts but has non-uniform density and may leave the Pareto boundary unsampled.Most grid samples can lie in the interior and require postprocessing or discard operations.
  • Comparison: Directional sampling gives sparse Pareto-boundary points with high resolution, but requires tractable solution of the search problem.The method is practical when membership in the objective set can be tested efficiently.
  • Interactive refinement: Interactive visualization can iteratively shrink the resource bundle as the designer specifies preferred minimum objective levels.The process continues until the designer is satisfied, described as psychological convergence.

FINDING THE PARETO BOUNDARY BY BISECTION

Bisection finds a weak Pareto-boundary point by repeatedly testing objective-set membership along a chosen direction and narrowing the feasible scaling range.

  • Membership testing: The search problem checks whether candidate objective vectors μ belong to the attainable set G through a membership test.The test imposes gm(x*) = μm for each objective.
  • Computational tractability: Membership-test complexity provides a baseline for related optimization problems using the same resources and objective functions.The test is efficiently solvable in many cellular beamforming designs because it is often convex.
  • Bisection procedure: The algorithm initializes λminv inside G and λmaxv outside G, then repeatedly halves the interval [λmin, λmax].The lower bound can be zero because the origin is attainable; the upper bound depends on the MOOP.
  • Accuracy: Bisection converges quickly, keeping the distance from the resulting point a to the Pareto boundary below ϵ∥v∥.The guarantee holds for accuracy parameter ϵ > 0.

SOLVING A MOOP BY SCALARIZATION

Scalarization converts multiple objectives into a single subjective goal function, allowing conventional optimization to select Pareto-boundary operating points while exposing computational tradeoffs.

  • Scalarization: Scalarization combines M objectives into one scalar goal function while retaining x ∈ X as the feasibility constraint.Unlike the conventional sole-objective formulation, other objectives need not be converted into constraints.
  • Goal functions: Weighted sums, weighted products, Chebyshev functions, and distance functions represent different preference structures over operating points.Weights can encode objective priorities, while a preferable point can define a distance-based preference.
  • Goal functions: The weighted geometric mean is less affected by relative scaling than the weighted arithmetic mean when objectives have different numerical ranges.This comparison concerns numerical-range sensitivity, not overall optimality.
  • Complexity: Scalarization choice affects computational complexity: some formulations are convex or quasi-convex, whereas others are non-convex with exponential or worse complexity.The weighted Chebyshev function is identified as the safest computational choice when membership testing is tractable.
  • Pragmatic approach: Because goal functions encode subjective preferences, the pragmatic approach selects weighted Chebyshev scalarization for tractability and adapts weights to designer needs.No goal function is universally better in optimality because the preference specification is subjective.
  • Illustration: In the two-objective illustration, sum, product, Chebyshev, and distance scalarizations touch G at different optimal operating points.For equal Chebyshev weights, the search follows the line from the origin where the two objectives have equal values.
  • Weight selection: Different weight selections produce different Pareto-optimal points, and some function classes can attain the whole Pareto boundary or only a subset.Weights are nonnegative and sum to one in the stated set W.

CASE STUDY: DESIGNING MASSIVE MIMO SYSTEMS

The case study models massive MIMO network design as balancing average user rate, average area rate, and energy efficiency through resource choices for antennas, users, and transmit power. It also describes the signal-processing and power-consumption models used to evaluate these objectives.

  • Study purpose: The case study uses MOO to visualize tradeoffs and derive design insights about coordinated transmission and massive MIMO deployment.CoMP can improve area rates but faces signaling, complexity, and imperfect-CSI challenges that motivate the massive MIMO example.
  • System rationale: Massive MIMO uses large BS antenna arrays to serve fewer users, improving robustness to imperfect CSI and enabling low-complexity processing.It also supports simple implicit intercell coordination and robustness to hardware distortions.
  • System model: The 16-cell scenario optimizes three conflicting objectives: average user rate, average area rate, and energy efficiency.Each BS has N antennas and K single-antenna users, with 10 MHz bandwidth and per-BS transmit power P.
  • Resource variables: The optimization variables are the BS antenna count N, users per cell K, and transmit power P.The resource bundle limits N to 500 antennas, P to 20 W per BS antenna, and imposes K ≤ N/2.
  • Signal processing: The model assumes perfect CSI and zero-forcing precoding, which cancels intracell interference through beamforming and adapts power allocation for equal user rates.The rate model accounts for channel-acquisition overhead, noise, and intercell interference.
  • Power model: Total cell power includes transmit-power amplification, antenna and user hardware, static hardware, and zero-forcing precoding computation.The computational term is based on required floating-point operations and computational efficiency.

DESIGNING MASSIVE MIMO BY MOO FRAMEWORK

The MOO framework maps attainable operating points and Pareto tradeoffs among user rate, area rate, and energy efficiency. In the massive MIMO case study, the resulting tradeoffs show when objectives align, conflict, or require flexible resource adaptation.

  • MOO framework: The framework formulates the three-objective network design problem and analyzes its attainable objective set to study tradeoffs and operating points.The analysis uses Pareto-boundary exploration and scalarized goal functions.
  • User rate–EE tradeoff: Average user rate and energy efficiency align until 20.4 Mbit/s/user and 11.1 Mbit/J, after which further rate increases require drastic EE sacrifices.This boundary is visualized in Fig. 6.
  • Area rate–EE tradeoff: Average area rate and energy efficiency align until maximum EE, after which area rate can increase with only minor EE losses.This tradeoff is visualized in Fig. 7.
  • Area-rate mechanism: Area rate improves mainly by serving more users in parallel, rather than by increasing the rate per user.The three-dimensional objective set supports this conclusion.
  • Three-objective tradeoff: High area rates require low per-user rates and many active users, whereas high per-user rates require fewer active users.High energy efficiency is possible when the per-user rate is small.
  • A priori design: Different scalarization weights produce different operating points, while the utopia point is far outside the attainable set for strongly conflicting user rate and EE.For the milder area-rate–EE conflict, the utopia point lies closer to the attainable objective set.
  • Design implication: Flexible network architecture and real-time adaptation can support different Pareto-boundary operating points as traffic load and service requirements change.The proposed adaptation may involve switching off antennas and changing precoding.

CONCLUSIONS AND FUTURE DIRECTIONS

5G network design requires multi-objective optimization because performance expectations involve conflicting objectives rather than a single metric. The article surveys Pareto-based and goal-function methods and illustrates their use in massive MIMO network dimensioning.

  • Multi-objective optimization is needed because 5G design involves conflicting objectives including peak user rates, average area rates, and energy efficiency.
  • The a posterior method samples the Pareto boundary, where improving one objective necessarily degrades another, to support informed design decisions.
  • Alternatively, a network designer can specify acceptable tradeoffs through a goal function and optimize that function using conventional optimization.
  • A cellular network-dimensioning case study demonstrates how MOO can balance conflicting performance objectives when designing for massive MIMO deployment.
  • Applications of established MOO analytic tools to communication networks remain greatly unexplored, including the challenge of choosing an appropriate modeling granularity.

AUTHORS

The supplied passage provides author-biographical information, including a Swedish Research Council postdoctoral grant and affiliation with Linköping University.

  • The author received a Swedish Research Council postdoctoral grant and is a tenure-track research fellow at Linköping University, Sweden.
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