Penalty and Augmented Lagrangian Methods for Constrained DC Programming

In this paper, we consider a class of structured nonsmooth difference-of-convex (DC) constrained DC programs in which the first convex component of the objective and constraints is the sum of a smooth and a nonsmooth function, and their second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have a weak convergence guarantee or require a feasible initial point. Inspired by the recent work by Pang et al. [Pang J-S, Razaviyayn M, Alvarado A (2017) Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. 42(1):95–118.], in this paper, we propose two infeasible methods with a strong convergence guarantee for the considered problem. The first one is a penalty method that consists of finding an approximate D-stationary point of a sequence of penalty subproblems. We show that any feasible accumulation point of the solution sequence generated by such a penalty method is a B-stationary point of the problem under a weakest possible assumption that it satisfies a pointwise Slater constraint qualification (PSCQ). The second one is an augmented Lagrangian (AL) method that consists of finding an approximate D-stationary point of a sequence of AL subproblems. Under the same PSCQ condition as for the penalty method, we show that any feasible accumulation point of the solution sequence generated by such an AL method is a B-stationary point of the problem, and moreover, it satisfies a Karush–Kuhn–Tucker type of optimality condition for the problem, together with any accumulation point of the sequence of a set of auxiliary Lagrangian multipliers. We also propose an efficient successive convex approximation method for computing an approximate D-stationary point of the penalty and AL subproblems. Finally, some numerical experiments are conducted to demonstrate the efficiency of our proposed methods.

Joint 3-D Positioning and Power Allocation for UAV Relay Aided by Geographic Information

<div>In this paper, we study to employ geographic information to address the blockage problem of air-to-ground links between UAV and terrestrial nodes. In particular, a UAV relay is deployed to establish communication links from a ground base station to multiple ground users. To improve communication capacity, we fifirst model the blockage effect caused by buildings according to the three-dimensional (3-D) geographic information. Then, an optimization problem is formulated to maximize the minimum capacity among users by jointly optimizing the 3-D position and power allocation of the UAV relay, under the constraints of link capacity, maximum transmit power, and blockage. To solve this complex non-convex problem, a two-loop optimization framework is developed based on Lagrangian relaxation. The outer-loop aims to obtain proper Lagrangian multipliers to ensure the solution of the Lagrangian problem converge to the tightest upper bound on the original problem. The inner-loop solves the Lagrangian problem by applying the block coordinate descent (BCD) and successive convex approximation (SCA) techniques, where UAV 3-D positioning and power allocation are alternately optimized in each iteration. Simulation results confifirm that the proposed solution signifificantly outperforms two benchmark schemes and achieves a performance close to the upper bound on the UAV relay system.</div>

Controller Design for a Delta Robot Using Lagrangian Multipliers

The novelty of this research is that the study utilizes Lagrangian multipliers for an articulated close-chain robot mechanism as time-varying states to augment an energy-based Lagrangian dynamic model, and the strategy facilitates the system computations while satisfying the forward/inverse dynamics and mechanism constraints simultaneously for the close-chained robot. Then, this article develops a controller based upon the augmented dynamics model for the Delta robot mechanism and drives its moving platform to track prescribed trajectories in space. As a result, the simulation results validate the effectiveness of the augmented system model in undertaking complex dynamics of close-chained robot mechanisms.

Improved CP-Based Lagrangian Relaxation Approach with an Application to the TSP

CP-based Lagrangian relaxation (CP-LR) is an efficient optimization technique that combines cost-based filtering with Lagrangian relaxation in a constraint programming context. The state-of-the-art filtering algorithms for the WeightedCircuit constraint that encodes the traveling salesman problem (TSP) are based on this approach. In this paper, we propose an improved CP-LR approach that locally modifies the Lagrangian multipliers in order to increase the number of filtered values. We also introduce two new algorithms based on the latter to filter WeightedCircuit. The experimental results on TSP instances show that our algorithms allow significant gains on the resolution time and the size of the search space when compared to the state-of-the-art implementation.

Joint 3-D Positioning and Power Allocation for UAV Relay Aided by Geographic Information

<div>In this paper, we study to employ geographic information to address the blockage problem of air-to-ground links between UAV and terrestrial nodes. In particular, a UAV relay is deployed to establish communication links from a ground base station to multiple ground users. To improve communication capacity, we fifirst model the blockage effect caused by buildings according to the three-dimensional (3-D) geographic information. Then, an optimization problem is formulated to maximize the minimum capacity among users by jointly optimizing the 3-D position and power allocation of the UAV relay, under the constraints of link capacity, maximum transmit power, and blockage. To solve this complex non-convex problem, a two-loop optimization framework is developed based on Lagrangian relaxation. The outer-loop aims to obtain proper Lagrangian multipliers to ensure the solution of the Lagrangian problem converge to the tightest upper bound on the original problem. The inner-loop solves the Lagrangian problem by applying the block coordinate descent (BCD) and successive convex approximation (SCA) techniques, where UAV 3-D positioning and power allocation are alternately optimized in each iteration. Simulation results confifirm that the proposed solution signifificantly outperforms two benchmark schemes and achieves a performance close to the upper bound on the UAV relay system.</div>

Joint 3-D Positioning and Power Allocation for UAV Relay Aided by Geographic Information

<div>In this paper, we study to employ geographic information to address the blockage problem of air-to-ground links between UAV and terrestrial nodes. In particular, a UAV relay is deployed to establish communication links from a ground base station to multiple ground users. To improve communication capacity, we fifirst model the blockage effect caused by buildings according to the three-dimensional (3-D) geographic information. Then, an optimization problem is formulated to maximize the minimum capacity among users by jointly optimizing the 3-D position and power allocation of the UAV relay, under the constraints of link capacity, maximum transmit power, and blockage. To solve this complex non-convex problem, a two-loop optimization framework is developed based on Lagrangian relaxation. The outer-loop aims to obtain proper Lagrangian multipliers to ensure the solution of the Lagrangian problem converge to the tightest upper bound on the original problem. The inner-loop solves the Lagrangian problem by applying the block coordinate descent (BCD) and successive convex approximation (SCA) techniques, where UAV 3-D positioning and power allocation are alternately optimized in each iteration. Simulation results confifirm that the proposed solution signifificantly outperforms two benchmark schemes and achieves a performance close to the upper bound on the UAV relay system.</div>

TSO-DSO Operational Planning Coordination through "l1-Proximal" Surrogate Lagrangian Relaxation

The proliferation of distributed energy resources (DERs), located at the Distribution System Operator (DSO) level, bring new opportunities as well as new challenges to the operations within the grid, specifically, when it comes to the interaction with the Transmission System Operator (TSO). To enable interoperability, while ensuring higher flexibility and cost-efficiency, DSOs and the TSO need to be efficiently coordinated. Difficulties behind creating such TSO-DSO coordination include the combinatorial nature of the operational planning problem involved at the transmission level as well as the nonlinearity of AC power flow within both systems. These considerations significantly increase the complexity even under the deterministic setting. In this paper, a deterministic TSO-DSO operational planning coordination problem is considered and a novel decomposition and coordination approach is developed. Within the new method, the problem is decomposed into TSO and DSO subproblems, which are efficiently coordinated by updating Lagrangian multipliers. The nonlinearities at the TSO level caused by AC power flow constraints are resolved through a dynamic linearization while guaranteeing feasibility through ``l1-proximal'' terms. Numerical results based on the coordination of the 118-bus TSO system with up to 32 DSO 34-bus systems indicate that the method efficiently overcomes the computational difficulties of the problem.<br>

Calculations of Complex Chemical Reaction Equilibria using Free Energy Minimization and Arc-Length Continuation

In the present paper, we determine the chemical equilibrium compositions for two combustion reactions involving either hydrazine or propane at fixed high pressure and temperature values using several computational approaches. Then, we compute the chemical equilibria for reacting systems under a multitude of temperature and pressure conditions and various initial system compositions. These sensitivity analyses are based on a combination of the method of Lagrangian multipliers and the arc-length continuation technique. Indeed, three industrially relevant case studies are elucidated: (1) the synthesis of ammonia using the Haber process, (2) the gasification of a typical biomass surrogate: glucose using steam and (3) the gasification of cellulose using steam. For all the above reacting systems, our results are benchmarked against their counterparts obtained either from the ubiquitous process simulator: ASPEN-Plus® or from data available in the open literature.

Distributed and Asynchronous Coordination of a Mixed-Integer Linear System via Surrogate Lagrangian Relaxation

With the emergence of Internet of Things that allows communications and local computations, and with the vision of Industry 4.0, a foreseeable transition is from centralized system planning and operation toward decentralization with interacting components and subsystems, e.g., self-optimizing factories. In this paper, a new “price-based” decomposition and coordination methodology is developed to efficiently coordinate subsystems such as machines and parts, which are described by Mixed-Integer Linear Programming (MILP) formulations, in a distributed and asynchronous way. To ensure low communication requirements, exchanges between the “coordinator” and subsystems are limited to “prices” (Lagrangian multipliers) broadcast by the coordinator, and to subsystem solutions sent to the coordinator. Asynchronous coordination, however, may lead to convergence difficulties since the order in which subsystem solutions arrive at the coordinator is not predefined as a result of uncertainties in communication and solving times. Under realistic assumptions of finite communication and solve times, convergence of our method is proved by innovatively extending Lyapunov Stability Theory. Numerical testing of generalized assignment problems through simulation demonstrates that the method converges fast and provides near-optimal results, paving the way for self-optimizing factories in the future.

TSO-DSO Operational Planning Coordination through Surrogate Lagrangian Relaxation

With the proliferation of distributed energy resources (DERs), located at the Distribution System Operator (DSO) level, uncertainties propagate from Transmission System Operator (TSO) to DSOs and vice versa. Therefore, to enable interoperability, while ensuring higher flexibility and cost-efficiency, both systems need to be efficiently coordinated to operate in sync. Moreover, because of the intermittency of renewable generation, voltages exhibit fluctuations thus necessitating the inclusion of AC power flow. Difficulties behind creating such TSO-DSO coordination include the combinatorial nature of the operational planning problem involved at the transmission level as well as the nonlinearity of AC power flow within both systems. These considerations significantly increase the complexity even under the deterministic setting. In this paper, a deterministic TSO-DSO operational planning coordination problem is considered and a novel decomposition and coordination approach is developed. Within the new method, the problem is decomposed into TSO and DSO subproblems, which are efficiently coordinated by updating Lagrangian multipliers. The nonlinearities at the TSO level caused by AC power flow constraints are resolved through a novel dynamic linearization. Numerical results based on the coordination of 118-bus TSO system with 32 DSO 34-bus systems indicate that both systems benefit from the coordination.<br>