Augmented Lagrangian Penalty Function Research Articles

Research on Cooperative Planning of Distributed Generation Access to AC/DC Distribution (Micro) Grids Based on Analytical Target Cascading

With the wide application of distributed generation (DG) and the rapid development of alternating current/direct current (AC/DC) hybrid microgrids, the optimal planning of distributed generation connecting to AC/DC distribution networks/microgrids has become an urgent problem to resolve. This paper presents a collaborative planning method for distributed generation access to AC/DC distribution (micro) grids. Based on the grid structure of the AC/DC distribution network, the typical interconnection structure of the AC/DC hybrid microgrid and AC/DC distribution network is designed. The optimal allocation models of distributed power supply for the AC/DC distribution network and microgrid are established based on analytical target cascading. The power interaction between the distribution network and microgrid is used to establish a coupling relationship, and the augmented Lagrangian penalty function is used to solve the collaborative programming problem. The results of distributed power supply allocation are obtained, solving the problem so that distribution generation with different capacity levels is connected to the power grid system in a single form.

A New Augmented Lagrangian Objective Penalty Function for Constrained Optimization Problems

In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization problems are proved. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker (KKT) condition. Especially, when the KKT condition holds for convex programming its saddle point exists. Based on the augmented Lagrangian objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions.

Achievable rate maximization for decode-and-forward MIMO-OFDM networks with an energy harvesting relay.

This paper investigates the system achievable rate for the multiple-input multiple-output orthogonal frequency division multiplexing (MIMO–OFDM) system with an energy harvesting (EH) relay. Firstly we propose two protocols, time switching-based decode-and-forward relaying (TSDFR) and a flexible power splitting-based DF relaying (PSDFR) protocol by considering two practical receiver architectures, to enable the simultaneous information processing and energy harvesting at the relay. In PSDFR protocol, we introduce a temporal parameter to describe the time division pattern between the two phases which makes the protocol more flexible and general. In order to explore the system performance limit, we discuss the system achievable rate theoretically and formulate two optimization problems for the proposed protocols to maximize the system achievable rate. Since the problems are non-convex and difficult to solve, we first analyze them theoretically and get some explicit results, then design an augmented Lagrangian penalty function (ALPF) based algorithm for them. Numerical results are provided to validate the accuracy of our analytical results and the effectiveness of the proposed ALPF algorithm. It is shown that, PSDFR outperforms TSDFR to achieve higher achievable rate in such a MIMO–OFDM relaying system. Besides, we also investigate the impacts of the relay location, the number of antennas and the number of subcarriers on the system performance. Specifically, it is shown that, the relay position greatly affects the system performance of both protocols, and relatively worse achievable rate is achieved when the relay is placed in the middle of the source and the destination. This is different from the MIMO–OFDM DF relaying system without EH. Moreover, the optimal factor which indicates the time division pattern between the two phases in the PSDFR protocol is always above 0.8, which means that, the common division of the total transmission time into two equal phases in previous work applying PS-based receiver is not optimal.

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.

A particle swarm–BFGS algorithm for nonlinear programming problems

This article proposes a hybrid optimization algorithm based on a modified BFGS and particle swarm optimization to solve medium scale nonlinear programs. The hybrid algorithm integrates the modified BFGS into particle swarm optimization to solve augmented Lagrangian penalty function. In doing so, the algorithm launches into a global search over the solution space while keeping a detailed exploration into the neighborhoods. To shed light on the merit of the algorithm, we provide a test bed consisting of 30 test problems to compare our algorithm against two of its variations along with two state-of-the-art nonlinear optimization algorithms. The numerical experiments illustrate that the proposed algorithm makes an effective use of hybrid framework when dealing with nonlinear equality constraints although its convergence cannot be guaranteed.

A distributed power control and mode selection algorithm for D2D communications

Device-to-device (D2D) communications underlaying a cellular infrastructure has recently been proposed as a means of increasing the resource utilization, improving the user throughput and extending the battery lifetime of user equipments. In this article we propose a new distributed power control algorithm that iteratively determines the signal-to-noise-and-interference-ratio (SINR) targets in a mixed cellular and D2D environment and allocates transmit powers such that the overall power consumption is minimized subject to a sum-rate constraint. The performance of the distributed power control algorithm is benchmarked with respect to the optimal SINR target setting that we obtain using the Augmented Lagrangian Penalty Function method. The proposed scheme shows consistently near optimum performance both in a single-input-multiple-output and a multiple-input-multiple-output setting. We also propose a joint power control and mode selection algorithm that requires single cell information only and clearly outperforms the classical cellular mode operation.

On Opportunistic Power Control for Alamouti and SM MIMO Systems

Transmit antenna diversity and single user spatial multiplexing have become attractive in practical systems, because they achieve performance gains without requiring sophisticated channel state information (CSI) feedback mechanisms. On the other hand, when fast and accurate CSI at the transmitter is available, opportunistic power control (OPC) is an attractive alternative to signal-to-interference-and-noise ratio (SINR) target following approaches, because it maximizes throughput by taking advantage of fast channel variations. In this paper we examine the question whether OPC is worth the pain of obtaining fast CSI by evaluating the gains of OPC for the downlink of a system employing multiple input multiple output (MIMO) systems with Alamouti and open loop spatial multiplexing (SM). We formulate the OPC problem as a throughput maximization task subject to power budget and fairness constraints. We solve this task by the Augmented Lagrangian Penalty Function and find that without fairness constraints, OPC in concert with SM provides superior throughput. With increasingly tight fairness constraints, Alamouti along with equal power allocation becomes a viable alternative to the SM OPC scheme. Both from fairness and throughput perspectives, Alamouti along with OPC is particularly efficient when adaptive MCS is employed and users with large differences in channel qualities have to share the total transmit power.

Near Optimum Power Control and Precoding under Fairness Constraints in Network MIMO Systems

We consider the problem of setting the uplink signal-to-noise-and-interference (SINR) target and allocating transmit powers for mobile stations in multicell spatial multiplexing wireless systems. Our aim is twofold: to evaluate the potential of such mechanisms in network multiple input multiple output (MIMO) systems, and to develop scalable numerical schemes that allow real-time near-optimal resource allocation across multiple sites. We formulate two versions of the SINR target and power allocation problem: one for maximizing the sum rate subject to power constraints, and one for minimizing the total power needed to meet a sum-rate target. To evaluate the potential of our approach, we perform a semianalytical study inMathematicausing the augmented Lagrangian penalty function method. We find that the gain of the joint optimum SINR setting and power allocation may be significant depending on the degree of fairness that we impose. We develop a numerical technique, based on successive convexification, for real-time optimization of SINR targets and transmit powers. We benchmark our procedure against the globally optimal solution and demonstrate consistently strong performance in realistic network MIMO scenarios. Finally, we study the impact of near optimal precoding in a multicell MIMO environment and find that precoding helps to reduce the sum transmit power while meeting a capacity target.

A Comparative Study of Three Different Mathematical Methods for Solving the Unit Commitment Problem

The unit commitment (UC) problem which is an important subject in power system engineering is solved by using Lagragian relaxation (LR), penalty function (PF), and augmented Lagrangian penalty function (ALPF) methods due to their higher solution quality and faster computational time than metaheuristic approaches. This problem is considered to be a nonlinear programming-(NP-) hard problem because it is nonlinear, mixed-integer, and nonconvex. These three methods used for solving the problem are based on dual optimization techniques. ALPF method which combines the algorithmic aspects of both LR and PF methods is firstly used for solving the UC problem. These methods are compared to each other based on feasible schedule for each stage, feasible cost, dual cost, duality gap, duration time, and number of iterations. The numerical results show that the ALPF method gives the best duality gap, feasible and dual cost instead of worse duration time and the number of iterations. The four-unit Tuncbilek thermal plant which is located in Kutahya region in Turkey is chosen as a test system in this study. The programs used for all the analyses are coded and implemented using general algebraic modeling system (GAMS).

An augmented Lagrangian decomposition method for quasi-separable problems in MDO

Several decomposition methods have been proposed for the distributed optimal design of quasi-separable problems encountered in Multidisciplinary Design Optimization (MDO). Some of these methods are known to have numerical convergence difficulties that can be explained theoretically. We propose a new decomposition algorithm for quasi-separable MDO problems. In particular, we propose a decomposed problem formulation based on the augmented Lagrangian penalty function and the block coordinate descent algorithm. The proposed solution algorithm consists of inner and outer loops. In the outer loop, the augmented Lagrangian penalty parameters are updated. In the inner loop, our method alternates between solving an optimization master problem, and solving disciplinary optimization subproblems. The coordinating master problem can be solved analytically; the disciplinary subproblems can be solved using commonly available gradient-based optimization algorithms. The augmented Lagrangian decomposition method is derived such that existing proofs can be used to show convergence of the decomposition algorithm to KKT points of the original problem under mild assumptions. We investigate the numerical performance of the proposed method on two example problems.

A Boundary Point Method to Solve Semidefinite Programs

We investigate the augmented Lagrangian penalty function approach to solve semidefinite programs. It turns out that this method generates iterates which lie on the boundary of the cone of semidefinite matrices which are driven to the affine subspace described by the linear equations defining the semidefinite program. We provide some computational experience with this method and show in particular, that it allows to compute the theta number of a graph to reasonably high accuracy for instances which are beyond reach by other methods.

An Equality Constrained RQP Algorithm Based on the Augmented Lagrangian Penalty Function

In comparative studies of constrained optimization methods the equality constrained recursive quadratic programming procedure has performed very favorably, particularly in terms of required computer time for execution. Biggs has formulated a strategy based on a quadratic penalty function and proved the global convergence of the method. This paper reformulates the procedure based on an augmented Lagrangian penalty function leading to improved performance and reduced sensitivity to the algorithm parameters. The formulation and algorithm are described herein and global convergence is demonstrated. Evaluation results on several test problems are presented to allow comparisons with other algorithms of the same type.

Simultaneous Optimization of Dynamic Reactions of a Four-Bar Linkage With Prescribed Maximum Shaking Force

A three-counterweight technique for simultaneously minimizing the maximum values of such dynamic reactions as the bearing force, the input moment, and the shaking moment of a constant input speed four-bar linkage, while additionally obtaining a prescribed maximum value of the shaking force, is introduced. The chosen optimization formulation is new to the field of mechanism design; a certain minimization parameter is made the objective function. The maximum values of the individual dynamic reactions are then minimized by inequality constraints which force the differences between these reactions (due to the presence of counterweights) and the maximum value of the same reactions in the unbalanced mechanism to be smaller than the minimization parameter at as many mechanism positions as required. The prescribed maximum shaking force is attained by an equality constraint which has been called the general equipollent circle constraint equation. The method, which employs an augmented Lagrangian penalty function code, produced good results in the optimization of the dynamic reactions of partially and fully force balanced example mechanisms.

A New Concept for Force Balancing Machines for Planar Linkages. Part 2: Application to Four-Bar Linkage and Experiment

This paper represents Part 2 of a two-part investigation on balancing devices for planar linkages. The theory shown in Part 1 is applied to the development of an experimental machine for theoretically fully force balanced four-bar linkages. To accomodate the inaccuracies in the various experimental quantities, it became necessary to obtain the correction masses from overdetermined sets of shaking force equations. Minimax and least squares approaches were formulated and adapted for solution by an existing augmented Lagrangian penalty function code. Both methods reduced the magnitudes of the actual shaking force components by more than fifty percent.