Lagrangian Dual Approach Research Articles

Energy-efficient maximization algorithm for energy harvesting under imperfect channel information

In this paper, we propose a novel energy-efficient resource optimization scheme in non-orthogonal multiple access (NOMA) networks with simultaneous wireless information and power transfer (SWIPT). In this system, we consider practical imperfect channel information that accounts for random channel delays and channel estimation errors. Additionally, the small cell users (SCUs) in a heterogeneous network (HetNet) can harvest energy from the small base station (SBS) signal by energy harvesting. Based on the non-linear energy harvesting (NEH) model, the problem of maximizing energy efficiency with imperfect channel state information (CSI) is formulated. Since the formulated problem is a probabilistic mixed non-convex optimization problem, a two-stage algorithm is developed to jointly optimize sub-channel matching and power allocation. In particular, the problem is first transformed into a non-probabilistic problem through the relaxation method. For the sub-channels matching problem, a low-complexity many-to-many matching algorithm is designed to achieve dynamic matching based on the preference lists. For the power allocation problem, the closed-form transmission power of SCUs can be derived by the Dinkelbach method and Lagrangian dual approach. Simulation results show that the proposed scheme can achieve higher energy efficiency than existing linear energy harvesting (LEH)-NOMA and NEH-OFDMA schemes, with improvements of 37.99% and 84.69%, respectively.

A robust optimization approach to a repair shop network planning

Robust optimization model is a paradigm for decision-making under uncertainty, where parameters are given in the form of uncertainty sets. In this paper, we develop a robust optimization model for a repair shop network planning problem. The linear optimization model involves product of uncertain parameters in the constraints. We formulate its robust counterpart with the help of min-max regret and Lagrangian dual approach, considering the partial information of uncertain parameters is given in the form of ellipsoidal and polyhedral uncertainty sets. We also consider ellipsoidal+polyhedral uncertainty set, which is the intersection of ellipsoidal and polyhedral uncertainty sets. We apply the robust optimization model to a bi-objective multi-plant repair shop network planning problem where multiple pieces of equipment are repaired and overhauled using several resources over a multi-period planning horizon. We consider uncertainty in resource and demand parameters. Numerical examples are presented for illustrating the theoretical results.

Energy-Efficient Resource Allocation for NOMA-Based Heterogeneous 5G Mine Internet of Things

The Mine Internet of Things (MIoT), as an application paradigm of the IoT in mine environment, realizes the state perception and information interaction by connecting massive smart sensing devices deployed in mine, plays an important role in coal mine production. The fifth-generation (5G) is a key enabling technology to provide an efficient and reliable communication link guarantee for MIoT networks. To meet the increasing demand for huge amounts of data transmission, a NOMA-based heterogeneous MIoT communication system is proposed. In this paper, different types of mine smart devices can occupy the same subchannel resources for data transmission by the NOMA technology, which improves the device access and spectrum utilization of the MIoT system. We aim to maximize the energy efficiency (EE) of all small cell networks via power allocation and subchannel assignment. Under the imperfect channel state information (CSI), a joint power allocation and subchannel assignment iterative algorithm is proposed. Specifically, firstly, by considering the cross-layer interference power constraints, maximum power constraints and QoS constraints, the EE optimization problem is formulated as a mixed integer nonlinear fractional programming problem. Secondly, the uncertain CSI is modeled as an elliptical uncertainty set, and the original problem is transformed into an equivalent convex optimization form by using the Dinkelbach method. Finally, an approximate solution is obtained by using the Lagrangian dual approach. The numerical results validate the effectiveness of the proposed algorithm and significantly show its superior performance as compared with the baseline algorithms.

Differentially Private and Fair Deep Learning: A Lagrangian Dual Approach

A critical concern in data-driven decision making is to build models whose outcomes do not discriminate against some demographic groups, including gender, ethnicity, or age. To ensure non-discrimination in learning tasks, knowledge of the sensitive attributes is essential, while, in practice, these attributes may not be available due to legal and ethical requirements. To address this challenge, this paper studies a model that protects the privacy of the individuals’ sensitive information while also allowing it to learn non-discriminatory predictors. The method relies on the notion of differential privacy and the use of Lagrangian duality to design neural networks that can accommodate fairness constraints while guaranteeing the privacy of sensitive attributes. The paper analyses the tension between accuracy, privacy, and fairness and the experimental evaluation illustrates the benefits of the proposed model on several prediction tasks.

Power Allocation for an Energy-Efficient Massive MIMO System With Imperfect CSI

Massive multiple-input and multiple-output (MIMO) has been recognized as a core technology for the fifth generation networks due to its huge spectral utilization brought by multiplexing gain and array gain. Most previous works on Massive MIMO systems have focused on the power optimization to maximize the system energy efficiency, in which the ergodic achievable sum-rate is replaced with a lower bound closed-form expression, which can result in suboptimal performance. In this paper, we aim to optimize the power assignment to achieve the maximum energy efficiency for a downlink system of single-cell Massive MIMO based on a tight approximate expression for the achievable sum-rate. Considering two linear precoding strategies at base station, i.e., maximum ratio transmission and zero-forcing, we first derive the corresponding approximate closed-form of the achievable sum-rate under imperfect channel state information. Furthermore, an energy-efficient power allocation scheme is formulated as a non-convex optimization problem. An iterative power allocation algorithm that maximizes the system energy efficiency is developed based on the Lagrangian dual approach. Simulation results demonstrate the tightness of the proposed approximate closed-form expressions, and show that our designed algorithm yields much better energy efficiency performance over the existing algorithms.

Joint Radio Communication, Caching, and Computing Design for Mobile Virtual Reality Delivery in Fog Radio Access Networks

The emerging virtual reality (VR) experience demands ultra-high-transmission-rate and ultra-low-latency deliveries, which is challenging for the current cellular networks. Since fog radio access networks (F-RANs) take full advantages of both edge fog computing and caching technologies and benefit different quality-of-service requirements, it is anticipated that high-quality VR experience could be well addressed in F-RANs. This paper presents an F-RAN-based mobile VR delivery framework, in which the core idea is to cache parts of the VR videos in advance and run a certain processing procedure at the edge of F-RANs. To optimize resource allocation at both mobile VR devices and fog access points (F-APs), a joint radio communication, caching and computing decision problem is formulated to maximize the average tolerant delay with meeting a given transmission rate constraint. This problem is formulated as a multiple choice multiple dimensional knapsack problem and solved with the Lagrangian dual decomposition approach. Furthermore, the optimal joint caching and computing decision is analyzed in a specific case with a closed-form expression of the average tolerant delay. The communications-caching-computing tradeoff at both mobile VR devices and F-APs is revealed, and the numerical results demonstrate that local caching and computing capabilities have significant impacts on the average tolerant delay. The proposed mobile VR delivery framework is promising in improving spectral efficiency by maximizing average tolerant delay while meeting high transmission rate requirements.

Optimal and Robust Interference Efficiency Maximization for Multicell Heterogeneous Networks

To improve energy efficiency and reduce the interference to macro users (MUs), this paper investigates the robust resource allocation (RA) problem for maximizing the interference efficiency of users (i.e., total rate/sum interference) in heterogeneous networks. Under perfect channel state information (CSI), the considered RA problem is formulated as a multivariate nonlinear programming problem with constraints on the maximum interference power of MUs, the minimum rate requirement of each femto user, and the maximum transmit power of the femto base station. The original fractional programming problem is converted into a convex optimization problem by using Dinkelbach's method, the logarithmic transformation method, and the successive convex approximation method, where the closed-form solution is obtained with the Lagrangian dual approach. Moreover, with imperfect CSI consideration, the original problem is reformulated as a robust RA problem that is transformed into a deterministic and convex optimization problem by using an inequality transformation approach. In addition, the computational complexity and the cost of robustness are also analyzed. The simulation results verify that the proposed algorithm has better interference efficiency and robustness by comparing with the existing algorithms.

Duality of nonconvex optimization with positively homogeneous functions

We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the p-norm function, where p is a positive real number. The problem, which is not necessarily convex, extends the absolute value optimization proposed in Mangasarian (Comput Optim Appl 36:43–53, 2007). In this work, we propose a dual formulation that, differently from the Lagrangian dual approach, has a closed-form and some interesting properties. In particular, we discuss the relation between the Lagrangian duality and the one proposed here, and give some sufficient conditions under which these dual problems coincide. Finally, we show that some well-known problems, e.g., sum of norms optimization and the group Lasso-type optimization problems, can be reformulated as positively homogeneous optimization problems.

A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. We adapt the augmented Lagrangian method framework to address the presence of nonconvexity in the non-relaxed constraint set and to enable efficient parallelization. The development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions. However, deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized. Pivotal in our modification to the augmented Lagrangian method is an integration of the simplicial decomposition method and the nonlinear block Gauss–Seidel method. An adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted. Under mild conditions optimal dual convergence is proven, and we report computational results on test instances from the stochastic optimization literature. We demonstrate improvement in parallel speedup over a baseline parallel approach.

Fair dynamic resource allocation in transit-based evacuation planning

Resource allocation in transit-based emergency evacuation is studied in this paper. The goal is to find a method for allocation of resources to communities in an evacuation process which is (1) fair, (2) reasonably efficient, and (3) able to dynamically adapt to the changes to the emergency situation. Four variations of the resource allocation problem, namely maximum rate, minimum clearance time, maximum social welfare, and proportional fair resource allocation, are modeled and compared. It is shown that the optimal answer to each problem can be found efficiently. Additionally, a distributed and dynamic algorithm based on the Lagrangian dual approach, called PFD2A, is developed to find the proportional fair allocation of resources and update the evacuation process in real time whenever new information becomes available. Numerical results for a sample scenario are presented.

Fair Dynamic Resource Allocation in Transit-based Evacuation Planning

Abstract Resource allocation in transit-based emergency evacuation is studied in this paper. The goal is to find a method for allocation of resources to communities in an evacuation process which is (1) fair, (2) reasonably efficient, and (3) able to dynamically adapt to the changes to the emergency situation. Four variations of the resource allocation problem, namely maximum rate, minimum clearance time, maximum social welfare, and proportional fair resource allocation, are modeled and compared. It is shown that the optimal answer to each problem can be found efficiently. Additionally, a distributed and dynamic algorithm based on the Lagrangian dual approach, called PFD2A, is developed to find the proportional fair allocation of resources and update the evacuation process in real time whenever new information becomes available. Numerical results for a sample scenario are presented.

Cross-Layer Optimization and Receiver Localization for Cognitive Networks Using Interference Tweets

A cross-layer resource allocation scheme for underlay multi-hop cognitive radio networks is formulated, in the presence of uncertain propagation gains and locations of primary users (PUs). Secondary network design variables are optimized under long-term probability-of-interference constraints, by exploiting channel statistics and maps that pinpoint areas where PU receivers are likely to reside. These maps are tracked using a Bayesian approach, based on 1-bit messages - here refereed to as "interference tweet" - broadcasted by the PU system whenever a communication disruption occurs due to interference. Although nonconvex, the problem has zero duality gap, and it is optimally solved using a Lagrangian dual approach. Numerical experiments demonstrate the ability of the proposed scheme to localize PU receivers, as well as the performance gains enabled by this minimal primary-secondary interplay.

Four proofs of Gittins’ multiarmed bandit theorem

We study four proofs that the Gittins index priority rule is optimal for alternative bandit processes. These include Gittins’ original exchange argument, Weber’s prevailing charge argument, Whittle’s Lagrangian dual approach, and Bertsimas and Nino-Mora’s proof based on the achievable region approach and generalized conservation laws. We extend the achievable region proof to infinite countable state spaces, by using infinite dimensional linear programming theory.

A Lagrangian Dual Approach to the Single-Source Localization Problem

The single-source localization problem (SSLP), which is nonconvex by its nature, appears in several important multidisciplinary fields such as signal processing and the global positioning system. In this paper, we cast SSLP as a Euclidean distance embedding problem and study a Lagrangian dual approach. It is proved that the Lagrangian dual problem must have an optimal solution under the generalized Slater condition. We provide a sufficient condition for the zero-duality gap and establish the equivalence between the Lagrangian dual approach and the existing Generalized Trust-Region Subproblem (GTRS) approach studied by Beck et al. [“Exact and Approximate Solutions of Source Localization Problems,” IEEE Trans. Signal Process., vol. 56, pp. 1770-1778, 2008]. We also reveal new implications of the assumptions made by the GTRS approach. Moreover, the Lagrangian dual approach has a straightforward extension to the multiple-source localization problem. Numerical simulations demonstrate that the Lagrangian dual approach can produce localization of similar quality as the GTRS and can significantly outperform the well-known semidefinite programming solver for the multiple source localization problem on the tested cases.

A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization

In this paper, we propose a unifiednonlinear augmented Lagrangian dual approach for a nonconvex vectoroptimization problem by applying a class of vector-valued nonlinearaugmented Lagrangian penalty functions. We establish weak and strongduality results, necessary and sufficient conditions for uniformlyexact penalization and exact penalization in the framework ofnonlinear augmented Lagrangian.

An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem

Higham (2002, IMA J. Numer. Anal., 22, 329–343) considered two types of nearest correlation matrix problems, namely the W-weighted case and the H-weighted case. While the W-weighted case has since been well studied to make several Lagrangian dual-based efficient numerical methods available, the H-weighted case remains numerically challenging. The difficulty of extending those methods from the W-weighted case to the H-weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H-weight, unlike the case under the W-weight, is not available. In this paper we introduce an augmented Lagrangian dual-based approach that avoids the explicit computation of the metric projection under the H-weight. This method solves a sequence of unconstrained convex optimization problems, each of which can be efficiently solved by an inexact semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust.

On the implementation of a log-barrier progressive hedging method for multistage stochastic programs

A progressive hedging method incorporated with self-concordant barrier for solving multistage stochastic programs is proposed recently by Zhao [G. Zhao, A Lagrangian dual method with self-concordant barrier for multistage stochastic convex nonlinear programming, Math. Program. 102 (2005) 1–24]. The method relaxes the nonanticipativity constraints by the Lagrangian dual approach and smoothes the Lagrangian dual function by self-concordant barrier functions. The convergence and polynomial-time complexity of the method have been established. Although the analysis is done on stochastic convex programming, the method can be applied to the nonconvex situation. We discuss some details on the implementation of this method in this paper, including when to terminate the solution of unconstrained subproblems with special structure and how to perform a line search procedure for a new dual estimate effectively. In particular, the method is used to solve some multistage stochastic nonlinear test problems. The collection of test problems also contains two practical examples from the literature. We report the results of our preliminary numerical experiments. As a comparison, we also solve all test problems by the well-known progressive hedging method.

Equivalent primal and dual differentiable reformulations of the Euclidean multifacility location problem

In this paper, we consider two equivalent differentiable reformulations of the nondifferentiable Euclidean multifacility location problem (EMFLP). The first of these is derived via a Lagrangian dual approach based on the optimum of a linear function over a unit ball (circle). The resulting formulation turns out to be identical to the known dual problem proposed by Francis and Cabot [1]. Hence, besides providing an easy direct derivation of the dual problem, this approach lends insights into its connections with classical Lagrangian duality and related results. In particular, it characterizes a straightforward recovery of primal location decisions. The second equivalent differentiable formulation is constructed directly in the primal space. Although the individual constraints of the resulting problem are generally nonconvex, we show that their intersection represents a convex feasible region. We then establish the relationship between the Karush-Kuhn-Tucker (KKT) conditions for this problem and the necessary and sufficient optimality conditions for EMFLP. This lends insights into the possible performance of standard differentiable nonlinear programming algorithms when applied to solve this reformulated problem. Some computational results on test problems from the literature, and other randomly generated problems, are also provided.

Equivalent primal and dual differentiable reformulations of the Euclidean multifacility location problem

In this paper, we consider two equivalent differentiable reformulations of the nondifferentiable Euclidean multifacility location problem (EMFLP). The first of these is derived via a Lagrangian dual approach based on the optimum of a linear function over a unit ball (circle). The resulting formulation turns out to be identical to the known dual problem proposed by Francis and Cabot [1]. Hence, besides providing an easy direct derivation of the dual problem, this approach lends insights into its connections with classical Lagrangian duality and related results. In particular, it characterizes a straightforward recovery of primal location decisions. The second equivalent differentiable formulation is constructed directly in the primal space. Although the individual constraints of the resulting problem are generally nonconvex, we show that their intersection represents a convex feasible region. We then establish the relationship between the Karush–Kuhn–Tucker (KKT) conditions for this problem and the necessary and sufficient optimality conditions for EMFLP. This lends insights into the possible performance of standard differentiable nonlinear programming algorithms when applied to solve this reformulated problem. Some computational results on test problems from the literature, and other randomly generated problems, are also provided.

Computational development of a lagrangian dual approach for quadratic networks

A Lagrangian dual approach for large-scale quadratic programs due to Lin and Pang is adapted to the quadratic cost network flow problem. The unconstrained dual problem is solved iteratively by combining conjugate gradient directions with finite line-search methods. By exploiting the graph structure, matrix and vector operations are streamlined so that the technique can be used for large problems. A computational study of randomly generated networks with up to 500 nodes and 20,000 arcs gives a comparison between combinations of different conjugate direction formulas with three line-search techniques. The results demonstrate the clear superiority of the Polak-Ribiere direction formula combined with a variant of a line-search technique originally developed by Bitran and Hax for convex knapsack problems.