Lagrangian Duality Theory Research Articles

Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability

While the literature on dynamic portfolio selection with stochastic interest rates only confines its investigation to the continuous-time setting up to now, this paper studies a multi-period mean-variance portfolio selection problem with a stochastic interest rate, where the movement of the interest rate is governed by the discrete-time Vasicek model. Invoking dynamic programming approach and the Lagrange duality theory, we derive the analytical expressions for both the efficient investment strategy and the efficient mean-variance frontier of the model formulation. We then extend our model to the situation with an uncontrollable liability.

Power management in adjacent cognitive femtocells with distance-dependent interference in full coverage area

In cognitive heterogeneous networks, when multiple adjacent femtocells are deployed, uncontrolled transmission power can lead to severe mutual interference. In order to reduce interference while satisfying capacity requirements, we propose a power management scheme for an adjacent femtocell network. Firstly, we build an interference model for adjacent femtocells, in which distance-dependent interference to terminals within the whole femtocell coverage area is considered. Then, we calculate the total capacity of adjacent femtocells based on the mutual interference and formulate a non-convex optimization problem to maximize the capacity under the transmission power constraints. To solve the non-convex problem, we divide it into two convex subproblems and solve them with the Lagrange dual theory and linear programming method. Finally, we derive the closed-form expression of the optimal power configuration to maximize capacity while minimizing energy consumption simultaneously. The simulation results indicate that the proposed power scheme demonstrates obvious improvement in terms of capacity and power economization, compared with the maximal power configuration method.

Resource optimisation using bandwidth‐power product for multiple‐input multiple‐output orthogonal frequency‐division multiplexing access system in cognitive radio networks

This study investigates resource allocation problems for a point-to-point multi-carrier multiple-input multiple-output cognitive radio network. Different from conventional resource optimisation problems, a joint power and bandwidth resource optimisation framework using the novel optimisation metric, namely bandwidth-power product, is developed. Besides, rate requirement of the secondary system is satisfied, and the interferences introduced to the primary users (PUs) are below threshold of tolerance. The optimal source precoding matrix is designed and two methods, namely the project-channel singular value decomposition (SVD) and direct-channel SVD methods, are applied to satisfy interference power constraints for PUs. Then the unified power and channel allocation problem is derived and found to be a mixed-integer programming problem. Hence, a sub-optimal and tractable algorithm with low complexity is proposed. The innovative idea is to determine the channel resource budget by selecting the best channels, where the criterion for evaluating the quality of channel is detailed discussed. Then the power optimisation subproblem and channel allocation subproblem can be performed independently using the Lagrange-duality theory and Gauss–Newton method, respectively. The simulation results show significant improvement in spectral efficiency by using this framework compared to classical power optimisation framework using the waterfilling scheme.

Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems

In this paper, we mainly consider the augmented Lagrangian duality theory and explore second-order conditions for the existence of augmented Lagrange multipliers for eigenvalue composite optimizati...

Optimization of Metro Train Schedules With a Dwell Time Model Using the Lagrangian Duality Theory

This paper proposes an optimization method of train scheduling for metro lines with a train dwell time model according to passenger demand. An optimization problem of train scheduling is established with constraints of a headway equation, passenger equation, and train dwell time equation, where the train dwell time is modeled as a function of boarding and alighting passenger volumes. The aim of the optimization problem is to minimize the waiting time of passengers and train operation cost. Lagrangian duality theory is adopted to solve this optimization problem with high dimensionality. Finally, simulation results illustrate that this method is efficient to generate the train schedule, which meets the passengers' exchanging requirements between trains and platforms. The contribution of this paper is that a dwell time model is introduced in train schedule optimization, which provides the possibility of reducing the operation cost in the precondition that the exchanging time of passengers between platforms and trains is assured.

Transmit Power Optimisation in Wireless Network

Transmit power optimisation in wireless networks based on beamforming have emerged as a promising technique to enhance the spectrum efficiency of present and future wireless communication systems. The aim of this study is to minimise the access point power consumption in cellular networks while maintaining a targeted quality of service (QoS) for the mobile terminals. In this study, the targeted quality of service is delivered to a mobile station by providing a desired level of Signal to Interference and Noise Ratio (SINR). Base-stations are coordinated across multiple cells in a multi-antenna beamforming system. This study focuses on a multi-cell multi-antenna downlink scenario where each mobile user is equipped with a single antenna, but where multiple mobile users may be active simultaneously in each cell and are separated via spatial multiplexing using beamforming. The design criteria is to minimize the total weighted transmitted power across the base-stations subject to SINR constraints at the mobile users. The main contribution of this study is to define an iterative algorithm that is capable of finding the joint optimal beamformers for all basestations, based on a correlation-based channel model, the full-correlation model. Among all correlated channel models, the correlated channel model used in this study is the most accurate, giving the best performance in terms of power consumption. The environment here in this study is chosen to be Non-Light of- Sight (NLOS) condition, where a signal from a wireless transmitter passes several obstructions before arriving at a wireless receiver. Moreover there are many scatterers local to the mobile, and multiple reflections can occur among them before energy arrives at the mobile. The proposed algorithm is based on uplink-downlink duality using the Lagrangian duality theory. Time-Division Duplex (TDD) is chosen as the platform for this study since it has been adopted to the latest technologies in Fourth Generation (4G) wireless communication systems. Monte Carlo simulation results and discussions are also provided to complement the analysis.

Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems

In this paper, we mainly consider the augmented Lagrangian duality theory and explore second-order conditions for the existence of augmented Lagrange multipliers for eigenvalue composite optimization problems. In the approach, we reformulate the augmented Lagrangian introduced by Rockafellar into a new form in terms of the Moreau envelope function and characterize second-order conditions via the epi-derivatives of the augmented Lagrangian.

Mean Energy Efficiency Maximization in Cognitive Radio Channels With PU Outage Constraint

Energy efficiency (EE) is very crucial for future green wireless communication systems, particularly for cognitive radio networks (CRNs). However, the EE design for CRNs in fast-fading scenarios has not been fully studied. This letter investigates the mean EE maximization problem for the secondary user (SU) in the fading cognitive radio channels consisting of both SUs and primary users (PUs). We adopt PU outage probability constraint to ensure the quality of service (QoS) of PUs and consider both the peak and mean transmit power constraints for SUs. However, this problem is nontrivial since the mean EE is nonconvex and PU outage probability constraint belongs to chance constraints. With the aid of the fractional programming and Lagrangian duality theory, we propose an efficient algorithm to derive the optimal power allocation strategy for SU to maximize its mean EE while guaranteeing the QoS of PUs. Simulation results assess the performance of our proposed scheme and show the tradeoff between SUs' mean EE and PU outage probability threshold only in some certain range.

A Semidistributed Approach for the Feasible Min-Max Fair Agent-Assignment Problem With Privacy Guarantees

In cyberphysical systems, a relevant problem is assigning agents to slots by distributed decisions capable of preserving an agent's privacy. For example, in future intelligent transportation systems, city-level coordinators may optimally assign cars (the agents) to parking slots depending on the cars' distance to final destinations in order to ensure social fairness and without disclosing or even using the car's destination information. Unfortunately, these assignment problems are combinatorial, whereas traditional solvers are exponentially complex, are not scalable, and do not ensure privacy of the agents' intended destinations. Moreover, no emphasis is placed to optimize the agents' social benefit. In this paper, the aggregate social benefit of the agents is considered by an agent-slot assignment optimization problem whose objective function is the fairness among the agents. Due to the problem's complexity, the problem is solved by an approximate approach based on Lagrange duality theory that enables the development of an iterative semidistributed algorithm. It is shown that the proposed algorithm is gracefully scalable compared to centralized methods, and that it preserves privacy in the sense that an eavesdropper will not be able to discover the destination of any agent during the algorithm iterations. Numerical results illustrate the performance and tradeoff of the proposed algorithm compared to the ideal optimal assignment and a greedy method.

On the Existence of a Saddle Value

In this work, we achieve a complete characterization of the existence of a saddle value, for bifunctions which are convex, proper, and lower semi continuous in their first argument, by considering new suitably defined notions of special directions of recession. As special cases, we obtain some recent results of Lagrangian duality theory on zero duality gap for convex programs.

Convex Optimization of Wireless Power Transfer Systems With Multiple Transmitters

Wireless power transfer systems with multiple transmitters promise advantages of higher transfer efficiencies and focusing effects over single-transmitter systems. From the standard formulation, straightforward maximization of the power transfer efficiency is not trivial. By reformulating the problem, a convex optimization problem emerges, which can be solved efficiently. Further, using Lagrangian duality theory, analytical results are found for the achievable maximum power transfer efficiency and all parameters involved. With these closed-form results, planar and coaxial wireless power transfer setups are investigated.

Decentralized Throughput Maximization in Cognitive Radio Wireless Mesh Networks

Scheduling and spectrum allocation are tasks affecting the performance of cognitive radio wireless networks, where heterogeneity in channel availability limits the performance and poses a great challenge on protocol design. In this paper, we present a distributed algorithm for scheduling and spectrum allocation with the objective of maximizing the network’s throughout subject to a delay constraint. During each time slot, the scheduling and spectrum allocation problems involve selecting a subset of links to be activated, and based on spectrum sensing outcomes, allocate the available resources to these links. This problem is addressed as an aggregate utility maximization problem. Since the throughput of any data flow is limited by the throughput of the weakest link along its end-to-end path, the utility of each flow is chosen as a function of this weakest link’s throughput. The throughput and delay performance of the network are characterized using a queueing theoretic analysis, and throughput is maximized via the application of Lagrangian duality theory. The dual decomposition framework decouples the problem into a set of subproblems that can be solved locally, hence, it allows us to develop a scalable distributed algorithm. Numerical results demonstrate the fast convergence rates of the proposed algorithm, as well as significant performance gains compared to conventional design methods.

Load-balance cell association for improving network capacity in heterogeneous cellular networks

Heterogeneous cellular networks improve the spectrum efficiency and coverage of wireless communication networks by deploying low power base station (BS) overlapping the conventional macro cell. But due to the disparity between the transmit powers of the macro BS and the low power BS, cell association strategy developed for the conventional homogeneous networks may lead to a highly unbalanced traffic loading with most of the traffic concentrated on the macro BS. In this paper, we propose a load-balance cell association scheme for heterogeneous cellular network aiming to maximize the network capacity. By relaxing the association constraints, we can get the upper bound of optimal solution and convert the primal problem into a convex optimization problem. Furthermore we propose a Lagrange multipliers based distributed algorithm by using Lagrange dual theory to solve the convex optimization, which converges to an optimal solution with a theoretical performance guarantee. With the proposed algorithm, mobile terminals (MTs) need to jointly consider their traffic type, received signal-to-interference-noise-ratios (SINRs) from BSs, and the load of BSs when they choose server BS. Simulation results show that the load balance between macro and pico BS is achieved and network capacity is improved significantly by our proposed cell association algorithm.

Equilibrium Model of Discrete Dynamic Supply Chain Network with Random Demand and Advertisement Strategy

The advertisement can increase the consumers demand; therefore it is one of the most important marketing strategies in the operations management of enterprises. This paper aims to analyze the impact of advertising investment on a discrete dynamic supply chain network which consists of suppliers, manufactures, retailers, and demand markets associated at different tiers under random demand. The impact of advertising investment will last several planning periods besides the current period due to delay effect. Based on noncooperative game theory, variational inequality, and Lagrange dual theory, the optimal economic behaviors of the suppliers, the manufactures, the retailers, and the consumers in the demand markets are modeled. In turn, the supply chain network equilibrium model is proposed and computed by modified project contraction algorithm with fixed step. The effectiveness of the model is illustrated by numerical examples, and managerial insights are obtained through the analysis of advertising investment in multiple periods and advertising delay effect among different periods.

Throughput maximization in distributed cognitive radio wireless mesh networks using lagrangian duality theory

Throughput maximization in distributed cognitive radio wireless mesh networks using lagrangian duality theory

Asset allocation for a DC pension fund with stochastic income and mortality risk: A multi-period mean–variance framework

This paper investigates an asset allocation problem for defined contribution pension funds with stochastic income and mortality risk under a multi-period mean–variance framework. Different from most studies in the literature where the expected utility is maximized or the risk measured by the quadratic mean deviation is minimized, we consider synthetically both to enhance the return and to control the risk by the mean–variance criterion. First, we obtain the analytical expressions for the efficient investment strategy and the efficient frontier by adopting the Lagrange dual theory, the state variable transformation technique and the stochastic optimal control method. Then, we discuss some special cases under our model. Finally, a numerical example is presented to illustrate the results obtained in this paper.

Markowitz’s mean–variance defined contribution pension fund management under inflation: A continuous-time model

In defined contribution (DC) pension schemes, the financial risk borne by the member occurs during the accumulation phase. To build up sufficient funds for retirement, scheme members invest their wealth in a portfolio of assets. This paper considers an optimal investment problem of a scheme member facing stochastic inflation under the Markowitz mean–variance criterion. Besides, we consider a more general market with multiple assets that can all be risky. By applying the Lagrange method and stochastic dynamic programming techniques, we derive the associated Hamilton–Jacobi–Bellman (HJB) equation, which can be converted into six correlated but relatively simple partial differential equations (PDEs). The explicit solutions for these six PDEs are derived by using the homogenization approach and the variable transformation technique. Then the closed-form expressions for the optimal strategy and the efficient frontier can be obtained through the Lagrange dual theory. In addition, we illustrate the results by some numerical examples.

Lagrange Duality in Set Optimization

Based on the complete-lattice approach, a new Lagrangian type duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. “Saddle sets” replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand, and the existence of a saddle set for the Lagrangian on the other hand. Applications to set-valued risk measures are indicated.

Optimizing Client Association for Load Balancing and Fairness in Millimeter-Wave Wireless Networks

Millimeter-wave communications in the 60-GHz band are considered one of the key technologies for enabling multigigabit wireless access. However, the special characteristics of such a band pose major obstacles to the optimal utilization of the wireless resources, where the problem of efficient client association to access points (APs) is of vital importance. In this paper, the client association in 60-GHz wireless access networks is investigated. The AP utilization and the quality of the rapidly vanishing communication links are the control parameters. Because of the tricky nonconvex and combinatorial nature of the client association optimization problem, a novel solution method is developed to guarantee balanced and fair resource allocation. A new distributed, lightweight, and easy-to-implement association algorithm, based on Lagrangian duality theory and subgradient methods, is proposed. It is shown that the algorithm is asymptotically optimal, that is, the relative duality gap diminishes to zero as the number of clients increases .

A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming

A simplicial branch and bound duality-bounds algorithm is presented to globally solving the linear multiplicative programming (LMP). We firstly convert the problem (LMP) into an equivalent programming one by introducingpauxiliary variables. During the branch and bound search, the required lower bounds are computed by solving ordinary linear programming problems derived by using a Lagrangian duality theory. The proposed algorithm proves that it is convergent to a global minimum through the solutions to a series of linear programming problems. Some examples are given to illustrate the feasibility of the present algorithm.