Dual Subgradient Method Research Articles

On Relay Selection and Subcarrier Assignment for Multiuser Cooperative OFDMA Networks With QoS Guarantees

This paper investigates quality of service (QoS)-aware relay selection and subcarrier assignment in multiuser OFDMA relay networks. In contrast to some existing works, which solve the sum-rate maximization problem directly, we first simplify it by exploiting the structure of the optimal solution and then consider the simplified problem from the perspectives of both primal and dual optimality. To characterize the optimal sum-rate performance, we solve the simplified problem optimally by the branch-and-cut method. In addition to the sum-rate maximization problem, we demonstrate that its two variants, namely, the worst-user rate maximization problem and the mixed rate maximization problem, can be also similarly simplified and then solved optimally by the branch-and-cut method. The optimal method serving as the benchmark is particularly suitable for problems of moderate scale. Furthermore, we study the simplified sum-rate maximization problem comprehensively from a dual perspective and propose three types of subgradient methods for the dual problem. The dual subgradient methods obtaining near-optimal solutions can be implemented in a distributed manner and thus are quite suitable for high-dimensional problems. To provide more in-depth analysis, we further consider the scenario in which subcarrier pairing is incorporated and suggest the natural generalization from the one-source-node case to the multiple-source-node case. Finally, two groups of simulations are conducted to validate the optimality of the branch-and-cut method and the effectiveness of the dual subgradient methods.

DISTRIBUTED PROXIMAL-GRADIENT METHOD FOR CONVEX OPTIMIZATION WITH INEQUALITY CONSTRAINTS

AbstractWe consider a distributed optimization problem over a multi-agent network, in which the sum of several local convex objective functions is minimized subject to global convex inequality constraints. We first transform the constrained optimization problem to an unconstrained one, using the exact penalty function method. Our transformed problem has a smaller number of variables and a simpler structure than the existing distributed primal–dual subgradient methods for constrained distributed optimization problems. Using the special structure of this problem, we then propose a distributed proximal-gradient algorithm over a time-changing connectivity network, and establish a convergence rate depending on the number of iterations, the network topology and the number of agents. Although the transformed problem is nonsmooth by nature, our method can still achieve a convergence rate, ${\mathcal{O}}(1/k)$, after $k$ iterations, which is faster than the rate, ${\mathcal{O}}(1/\sqrt{k})$, of existing distributed subgradient-based methods. Simulation experiments on a distributed state estimation problem illustrate the excellent performance of our proposed method.

Primal convergence from dual subgradient methods for convex optimization

When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favorably utilized, since they often find near-optimal dual solutions quickly. However, an optimal primal solution is generally not obtained directly through such a subgradient approach unless the Lagrangian dual function is differentiable at an optimal solution. We construct a sequence of convex combinations of primal subproblem solutions, a so called ergodic sequence, which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen. We generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions. In contrast to previously proposed rules, they exploit more information from later subproblem solutions than from earlier ones. We evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed.

Convergence Study of Decentralized Min-Cost Subgraph Algorithms for Multicast in Coded Networks

The problem of establishing minimum-cost multicast connections in coded networks can be viewed as an optimization problem, and decentralized algorithms were proposed by Lun to compute the optimal subgraph using the dual subgradient method. However, the convergence rate problem for these algorithms remains open. There are limited results in the literature, which bound the amount of infeasibility of the primal solution recovered after each iterations or the convergence rate. However, due to the special structure of the network coding problem, we have an algorithm that generates a feasible solution after each iterations. In addition, the convergence rate of the primal problem is O(1/n) to a neighborhood of the optimal solution. We also propose heuristics to further improve our algorithm and demonstrate through simulations that the distributed algorithm converges to the optimal subgraph quickly and is robust against network topology changes.

Relay Selection and Subcarrier-Pair Based Energy-Efficient Resource Allocation for Multirelay Cooperative OFDMA Networks

Energy-efficient resource allocation is investigated for a relay-based multiuser cooperation orthogonal frequency division multiple access (OFDMA) uplink system with amplify-and-forward (AF) protocol for all relays. The objective is to maximize the total energy efficiency (EE) of the uplink system with consideration of some practical limitations, such as the individual power constraint for the users and relays and the quality of service (QoS) for every user. We formulate an energy-efficient resource allocation problem that seeks joint optimization of subcarrier pairing, relay selection, subcarrier assignment, and power allocation. Unlike previous optimization throughput models, we transform the considered EE problem in fractional form into an equivalent optimal problem in subtractive form, which is solved by using dual decomposition and subgradient methods. To reduce computation costs, we propose two low-complexity suboptimal schemes. Numerical studies are conducted to evaluate the EE of the proposed algorithms.

Dual averaging method for solving multi-agent saddle-point problems with quantized information

In this paper we study the multi-agent saddle-point problems where multiple agents try to collectively optimize a sum of local convex–concave functions, each of which is available to one specific agent in the network. We propose a distributed primal–dual subgradient method under the constraint that agents can only communicate quantized information. The method can be implemented over a time-varying network while satisfying some standard connectivity conditions. We provide convergence results and convergence rate estimates for the proposed method. In particular, we provide an error bound on its convergence rate to highlight the dependence on the quantization resolution. We have also provided a numerical example to show the effectiveness of the proposed method.

Transmission Optimization of Wireless Visual Sensor Networks with Compressed Sensing Encoder

The design and optimization of wireless visual sensor network (WVSN) is challenging due to error-prone wireless link, limited resource and low computational capacity. Traditional video coding methods and layered transmission protocols are not efficient for visual information collection in WVSNs. In this paper, we adopted the compressed sensing (CS) paradigm to sense and encode the visual information, and proposed a cross-layer transmission algorithm to achieve optimal trade-off between total distortion and power consumption by simultaneously controlling the CS measurement rate and allocating the power and bandwidth for each wireless link. The algorithm was obtained by using convex dual theory and sub-gradient iterative method. The optimality condition of the proposed algorithm was also given. The algorithm can be implemented in a practical WVSN in a distributed way with only limited local information exchanges.

Distributed Primal–Dual Subgradient Method for Multiagent Optimization via Consensus Algorithms

This paper studies the problem of optimizing the sum of multiple agents' local convex objective functions, subject to global convex inequality constraints and a convex state constraint set over a network. Through characterizing the primal and dual optimal solutions as the saddle points of the Lagrangian function associated with the problem, we propose a distributed algorithm, named the distributed primal-dual subgradient method, to provide approximate saddle points of the Lagrangian function, based on the distributed average consensus algorithms. Under Slater's condition, we obtain bounds on the convergence properties of the proposed method for a constant step size. Simulation examples are provided to demonstrate the effectiveness of the proposed method.

Cross-layer optimization of wireless multihop networks with one-hop two-way network coding

In this paper, we investigate optimal cross-layer design of congestion control, routing, one-hop two-way inter-commodity (OTIC) network coding and scheduling in wireless multi-hop networks utilizing the broadcast advantage of wireless medium. We first present an achievable rate region with OTIC network coding by introducing virtual flow rates in a node. Then we formulate the network utility maximization problem subject to constraints on this achievable rate region, and analyze the complexities of both node- and path-based formulation with no network coding, OTIC network coding, and overhearing network coding. After that, we solve it using dual decomposition and subgradient method. Based on the solution, we present a new queue model, which is able to facilitate the coding operation between two-way commodities, and then propose a backpressure-based cross-layer optimization algorithm with low coding complexity and overhead. Afterwards, we analyze the stability and asymptotical optimality of the proposed cross-layer algorithm by Lyapunov drift technique, and evaluate its performance through extensive simulation. By comparing with the pure routing scheme under both primary and two-hop interference models in an illustrative topology, it is shown that with the proposed joint optimization algorithms, the OTIC network coding can interact adaptively and optimally with other components in different layers and achieve high throughput gain. Simulation of the proposed algorithm in a mesh network with base station and a random ad hoc network justifies that OTIC network coding can obtain considerable throughput gain with low complexity in various kinds of networks.

Barrier subgradient method

In this paper we develop a new affine-invariant primal–dual subgradient method for nonsmooth convex optimization problems. This scheme is based on a self-concordant barrier for the basic feasible set. It is suitable for finding approximate solutions with certain relative accuracy. We discuss some applications of this technique including fractional covering problem, maximal concurrent flow problem, semidefinite relaxations and nonlinear online optimization. For all these problems, the rate of convergence of our method does not depend on the problem’s data.

Optimization decomposition in energy-constrained wireless networks

This study considers a joint congestion control, routing and power control for energy-constrained wireless networks. A mathematical framework is introduced which includes maximization of network utility, maximization of network lifetime and trade-off between network utility and network lifetime. The framework would maximize the overall throughput of the network where the overall throughput depends on the data flow rates which in turn is dependent on the link capacities. The link capacity, on the other hand, is a function of transmit power levels and link Signal to Interference plus Noise Ratio (SINR) which makes the power allocation problem inherently difficult to solve. Using dual decomposition techniques, subgradient method and logarithmic transformations, a joint algorithm for rate and power allocation problems was formulated. Numerical examples for each optimization problem were also provided.

Subgradient Methods for Saddle-Point Problems

We study subgradient methods for computing the saddle points of a convex-concave function. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. We first present a subgradient algorithm for generating approximate saddle points and provide per-iteration convergence rate estimates on the constructed solutions. We then focus on Lagrangian duality, where we consider a convex primal optimization problem and its Lagrangian dual problem, and generate approximate primal-dual optimal solutions as approximate saddle points of the Lagrangian function. We present a variation of our subgradient method under the Slater constraint qualification and provide stronger estimates on the convergence rate of the generated primal sequences. In particular, we provide bounds on the amount of feasibility violation and on the primal objective function values at the approximate solutions. Our algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily (equivalently, the minimum of the Lagrangian function at a dual solution cannot be computed efficiently), thus impeding the use of dual subgradient methods.

Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods

In this paper, we study methods for generating approximate primal solutions as a byproduct of subgradient methods applied to the Lagrangian dual of a primal convex (possibly nondifferentiable) constrained optimization problem. Our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods, such as the recent resource allocation problems in large-scale networks. For such problems, we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions. These algorithms use a constant stepsize in view of its simplicity and practical significance. We provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions. These estimates are given per iteration, thus providing a basis for analyzing the trade-offs between the desired level of error and the selection of the stepsize value. Our analysis relies on the Slater condition and the inherited boundedness properties of the dual problem under this condition. It also relies on the boundedness of subgradients, which is ensured by assuming the compactness of the constraint set.

Distributed utility maximization for network coding based multicasting: a shortest path approach

One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions

Experiments with primal - dual decomposition and subgradient methods for the uncapacitatied facility location problem

For optimization problems that are structured both with respect to the constraints and with respect to the variables, it is possible to use primal–dual solution approaches, based on decomposition principles. One can construct a primal subproblem, by fixing some variables, and a dual subproblem, by relaxing some constraints and king their Lagrange multipliers, so that both these problems are much easier to solve than the original problem. We study methods based on these subproblems, that do not include the difficult Benders or Dantzig-Wolfe master problems, namely primal–dual subgradient optimization methods, mean value cross decomposition, and several comtbinations of the different techniques. In this paper, these solution approaches are applied to the well-known uncapacitated facility location problem. Computational tests show that some combination methods yield near-optimal solutions quicker than the classical dual ascent method of Erlenkotter