MAPEL: Achieving Global Optimality for a Non-convex Wireless Power Control Problem (0805.2675v2)

Abstract: Achieving weighted throughput maximization (WTM) through power control has been a long standing open problem in interference-limited wireless networks. The complicated coupling between the mutual interferences of links gives rise to a non-convex optimization problem. Previous work has considered the WTM problem in the high signal to interference-and-noise ratio (SINR) regime, where the problem can be approximated and transformed into a convex optimization problem through proper change of variables. In the general SINR regime, however, the approximation and transformation approach does not work. This paper proposes an algorithm, MAPEL, which globally converges to a global optimal solution of the WTM problem in the general SINR regime. The MAPEL algorithm is designed based on three key observations of the WTM problem: (1) the objective function is monotonically increasing in SINR, (2) the objective function can be transformed into a product of exponentiated linear fraction functions, and (3) the feasible set of the equivalent transformed problem is always normal although not necessarily convex. The MAPLE algorithm finds the desired optimal power control solution by constructing a series of polyblocks that approximate the feasible SINR region in increasing precision. Furthermore, by tuning the approximation factor in MAPEL, we could engineer a desirable tradeoff between optimality and convergence time. MAPEL provides an important benchmark for performance evaluation of other heuristic algorithms targeting the same problem. With the help of MAPEL, we evaluate the performance of several respective algorithms through extensive simulations.

• The paper introduces the MAPEL algorithm to find the global optimal solution for the non-convex Weighted Throughput Maximization (WTM) problem in wireless networks under general SINR conditions.
• MAPEL reformulates the problem using exponentiated linear fractional functions and approximates the feasible SINR region boundary via successive polyblock refinement.
• Simulation results show MAPEL accurately finds the global optimum, serving as a benchmark to evaluate the performance gaps of other algorithms like SPC and ADP.

An Overview of MAPEL: Achieving Global Optimality for Non-convex Wireless Power Control

The paper by Liping Qian, Ying Jun (Angela) Zhang, and Jianwei Huang presents the MAPEL algorithm, a method designed to tackle the non-convex Weighted Throughput Maximization (WTM) problem in wireless networks, particularly when interference is a limiting factor. Unlike earlier methods that perform efficiently only under high SINR assumptions, MAPEL is capable of achieving global optimal solutions in the general SINR regime, presenting itself as a robust solution in practical scenarios.

Problem Context

The WTM problem involves optimizing the power allocation across wireless links to maximize the weighted sum of logarithmic functions of SINR. This is emblematic of real-world applications where data communications dominate and require optimization of throughput in the face of mutual interferences. While previous work in high-SINR contexts allowed reformulation into convex problems solvable via geometric programming, the general SINR scenario retained its non-convexity, thus evoking a need for MAPEL.

Key Insights and Methodology

MAPEL derives its efficiency from several observations:

1. The monotonicity of the objective in terms of SINR implies that the optimal solution occurs at the boundary of the feasible SINR region.
2. Reformulating the objective as exponentiated linear fractional functions allows mapping the WTM problem to a Multiplicative Linear Fractional Programming (MLFP) framework.
3. The feasible set, though non-convex, maintains "normal" properties that enable constructing polyblocks to enclose the feasible SINR region.

The innovation in MAPEL lies in progressively approximating the SINR region's boundary through successive refinement of these polyblocks, eventually pinpointing the global optima. This is facilitated by a fine-tuning mechanism—the approximation factor—which balances precision and computational time. This tuning enables desirable tradeoffs between convergence speed and solution optimality.

Numerical Results and Comparisons

Simulation studies demonstrate MAPEL’s capability to serve as a benchmark against other heuristic and algorithmic strategies. For instance, in testing scenarios, MAPEL accurately established the global optimum, showcasing the performance gaps and room for improvement in existing methods like the Signomial Programming Condensation (SPC) and Asynchronous Distributed Pricing (ADP) algorithms. Notably, these comparisons highlight SPC’s ability to achieve near-optimal performance in a majority of cases, albeit without guarantees of global optimality, unlike MAPEL.

Theoretical and Practical Implications

Practically, the development of MAPEL offers a significant step forward in planning and managing interference-limited wireless networks, a core challenge for modern communication infrastructures. Theoretically, MAPEL illustrates the power of reformulating non-convex problems into forms amenable to intelligent geometric search strategies while ensuring global convergence.

Future Directions

This paper also sets the stage for future exploration in both research and application-oriented aspects of non-convex power control problems. Prospects include expanding MAPEL or similar methodologies to utility functions with broader definitions including non-concave scenarios or incorporating dynamics of time-varying channels for more adaptive power control mechanisms.

In conclusion, the MAPEL algorithm advances the frontier of solving non-convex wireless network problems by ensuring global optimality in varying SINR conditions, thus providing an important benchmark in the evaluation of other strategies that cannot inherently guarantee such outcomes.