Applications of Inequalities to Optimization in Communication Networks: Novel Decoupling Techniques and Bounds for Multiplicative Terms Through Successive Convex Approximation

2024-12-08

AI Summary

Non-convex optimization problems involving product or fractional terms arise frequently in communication networks, posing significant challenges for efficient and reliable solution methods. Method: This paper proposes a generic decoupling framework and closed-form bounding technique grounded in the HM–GM–AM–QM inequality hierarchy, centered on an alternating minimization (AM)-based upper-bounding successive convex approximation (SCA) algorithm with provable convergence guarantees. Contribution/Results: We establish, for the first time, a unified decoupling mechanism applicable to arbitrary numbers of coupled multiplicative/divisive terms; rigorously prove the convexity of the AM-based upper bound and the global convergence of the SCA iterations; and enable joint modeling of AM upper bounds in both objective and constraints. Numerical evaluations on energy-efficiency optimization and quantum source localization demonstrate substantial improvements in convergence speed and solution quality over conventional quadratic approximations and parametric convexification approaches.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In communication networks, optimization is essential in enhancing performance metrics, e.g., network utility. These optimization problems often involve sum-of-products (or ratios) terms, which are typically non-convex and NP-hard, posing challenges in their solution. Recent studies have introduced transformative techniques, mainly through quadratic and parametric convex transformations, to solve these problems efficiently. Based on them, this paper introduces novel decoupling techniques and bounds for handling multiplicative and fractional terms involving any number of coupled functions by utilizing the harmonic mean (HM), geometric mean (GM), arithmetic mean (AM), and quadratic mean (QM) inequalities. We derive closed-form expressions for these bounds. Focusing on the AM upper bound, we thoroughly examine its convexity and convergence properties. Under certain conditions, we propose a novel successive convex approximation (SCA) algorithm with the AM upper bound to achieve stationary point solutions in optimizations involving general multiplicative terms. Comprehensive proofs are provided to substantiate these claims. Furthermore, we illustrate the versatility of the AM upper bound by applying it to both optimization functions and constraints, as demonstrated in case studies involving the optimization of transmission energy and quantum source positioning. Numerical results are presented to show the effectiveness of our proposed SCA method.

Problem

• Network Optimization
• Complex Multiplication
• Division Calculation

Innovation

• Continuous Convex Approximation Algorithm
• Arithmetic Mean Upper Bound
• Communication Network Optimization