A Unified Successive Approximation Framework for General Coupled Multiplicative and Fractional Optimization: HM-GM-AM-QM Transforms and Applications

Abstract

Optimization problems in communication networks and information systems often contain coupled multiplicative or fractional terms, such as sum-of-products, sum-of-ratios, and logarithmic product-ratio structures. These problems are generally non-convex and difficult to solve, which motivates the development of tractable transformation and approximation techniques. In this paper, we propose an inequality-based transform framework for handling multiplicative and fractional terms involving an arbitrary number of coupled functions. The proposed framework is built upon the harmonic-mean, geometric-mean, arithmetic-mean, and quadratic-mean inequalities, and yields lower-bound and upper-bound surrogates for product-type terms. We derive the corresponding auxiliary-variable updates in closed form and show that the constructed surrogates are tight and first-order consistent at the current iterate. Based on these properties, we develop a class of successive approximation (SA) methods for sum-of-products/ratios minimization and maximization problems. When the transformed surrogate is convex for minimization or concave for maximization, the proposed method reduces to a standard successive convex approximation (SCA) method. When such convexity or concavity is not guaranteed, we further develop gradient-based SA variants and establish their sublinear convergence to an $\epsilon$-stationary point under standard smoothness and boundedness assumptions. We also discuss extensions to logarithmic product-ratio objectives and non-convex constraints. Numerical studies and application examples, including transmit-energy minimization, age-of-information minimization, semantic utility maximization, reliability-aware routing, cooperative edge caching, and product-loss learning, demonstrate the versatility and effectiveness of the proposed transform framework.