48 research outputs found

    Network Utility Maximization With Nonconcave Utilities Using Sum-of-Squares Method

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    The Network Utility Maximization problem has recently been used extensively to analyze and design distributed rate allocation in networks such as the Internet. A major limitation in the state-of-the-art is that user utility functions are assumed to be strictly concave functions, modeling elastic flows. Many applications require inelastic flow models where nonconcave utility functions need to be maximized. It has been an open problem to find the globally optimal rate allocation that solves nonconcave network utility maximization, which is a difficult nonconvex optimization problem. We provide a centralized algorithm for off-line analysis and establishment of a performance benchmark for nonconcave utility maximization. Based on the semialgebraic approach to polynomial optimization, we employ convex sum-of-squares relaxations solved by a sequence of semidefinite programs, to obtain increasingly tighter upper bounds on total achievable utility for polynomial utilities. Surprisingly, in all our experiments, a very low order and often a minimal order relaxation yields not just a bound on attainable network utility, but the globally maximized network utility. When the bound is exact, which can be proved using a sufficient test, we can also recover a globally optimal rate allocation. In addition to polynomial utilities, sigmoidal utilities can be transformed into polynomials and are handled. Furthermore, using two alternative representation theorems for positive polynomials, we present price interpretations in economics terms for these relaxations, extending the classical interpretation of independent congestion pricing on each link to pricing for the simultaneous usage of multiple links

    Layering as Optimization Decomposition: Questions and Answers

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    Network protocols in layered architectures have historically been obtained on an ad-hoc basis, and much of the recent cross-layer designs are conducted through piecemeal approaches. Network protocols may instead be holistically analyzed and systematically designed as distributed solutions to some global optimization problems in the form of generalized Network Utility Maximization (NUM), providing insight on what they optimize and on the structures of network protocol stacks. In the form of 10 Questions and Answers, this paper presents a short survey of the recent efforts towards a systematic understanding of "layering" as "optimization decomposition". The overall communication network is modeled by a generalized NUM problem, each layer corresponds to a decomposed subproblem, and the interfaces among layers are quantified as functions of the optimization variables coordinating the subproblems. Furthermore, there are many alternative decompositions, each leading to a different layering architecture. Industry adoption of this unifying framework has also started. Here we summarize the current status of horizontal decomposition into distributed computation and vertical decomposition into functional modules such as congestion control, routing, scheduling, random access, power control, and coding. We also discuss under-explored future research directions in this area. More importantly than proposing any particular crosslayer design, this framework is working towards a mathematical foundation of network architectures and the design process of modularization

    Recovery of primal solution in dual subgradient schemes

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    Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.Includes bibliographical references (p. 97-99).In this thesis, we study primal solutions for general optimization problems. In particular, we employ the subgradient method to solve the Lagrangian dual of a convex constrained problem, and use a primal-averaging scheme to obtain near-optimal and near-feasible primal solutions. We numerically evaluate the performance of the scheme in the framework of Network Utility Maximization (NUM), which has recently drawn great research interest. Specifically for the NUM problems, which can have concave or nonconcave utility functions and linear constraints, we apply the dual-based decentralized subgradient method with averaging to estimate the rate allocation for individual users in a distributed manner, due to its decomposability structure. Unlike the existing literature on primal recovery schemes, we use a constant step-size rule in view of its simplicity and practical significance. Under the Slater condition, we develop a way to effectively reduce the amount of feasibility violation at the approximate primal solutions, namely, by increasing the value initial dual iterate; moreover, we extend the established convergence results in the convex case to the more general and realistic situation where the objective function is convex. In particular, we explore the asymptotical convergence properties of the averaging sequence, the tradeoffs involved in the selection of parameter values, the estimation of duality gap for particular functions, and the bounds for the amount of constraint violation and value of primal cost per iteration. Numerical experiments performed on NUM problems with both concave and nonconcave utility functions show that, the averaging scheme is more robust in providing near-optimal and near-feasible primal solutions, and it has consistently better performance than other schemes in most of the test instances.by Jing Ma.S.M

    Optimal Network Utility

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    ABSTRACT: The problem of scheduling for maximum throughput-utility in a network with random packet arrivals and time varying channel reliability, the network controller assesses the condition of its channels and selects a set of links for transmission. The success of each transmission depends on the collection of links selected and their corresponding reliabilities. The goal is to maximize a concave, non-decreasing function of the time, average through-put on each link. Such a function represents a utility function that acts as a measure of fairness for the achieved throughput vector

    Scalable Primal-Dual Actor-Critic Method for Safe Multi-Agent RL with General Utilities

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    We investigate safe multi-agent reinforcement learning, where agents seek to collectively maximize an aggregate sum of local objectives while satisfying their own safety constraints. The objective and constraints are described by {\it general utilities}, i.e., nonlinear functions of the long-term state-action occupancy measure, which encompass broader decision-making goals such as risk, exploration, or imitations. The exponential growth of the state-action space size with the number of agents presents challenges for global observability, further exacerbated by the global coupling arising from agents' safety constraints. To tackle this issue, we propose a primal-dual method utilizing shadow reward and κ\kappa-hop neighbor truncation under a form of correlation decay property, where κ\kappa is the communication radius. In the exact setting, our algorithm converges to a first-order stationary point (FOSP) at the rate of O(T−2/3)\mathcal{O}\left(T^{-2/3}\right). In the sample-based setting, we demonstrate that, with high probability, our algorithm requires O~(ϵ−3.5)\widetilde{\mathcal{O}}\left(\epsilon^{-3.5}\right) samples to achieve an ϵ\epsilon-FOSP with an approximation error of O(ϕ02κ)\mathcal{O}(\phi_0^{2\kappa}), where ϕ0∈(0,1)\phi_0\in (0,1). Finally, we demonstrate the effectiveness of our model through extensive numerical experiments.Comment: 50 page
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