Dual Programming Research Articles

A Stochastic Dual Dynamic Integer Programming for the Uncapacitated Lot-Sizing Problem with Uncertain Demand and Costs

We study the uncapacitated lot-sizing problem with uncertain demand and costs. We consider a multi-stage decision process and rely on a scenario tree to represent the uncertainty. We propose to solve this stochastic combinatorial optimization problem thanks to a new extension of the stochastic dual dynamic integer programming algorithm. Our results show that this approach can provide good quality solutions in a reasonable time for large-size instances.

Research on Multiobjective Location of Urban Emergency Logistics under Major Emergencies

After the outbreak of major emergencies, the scheduling of emergency supplies is the key to the emergency rescue work, and the establishment of appropriate emergency logistics centers plays a crucial supporting role. In order to deal with the problem of emergency facility location and material distribution in urban emergency logistics system, this paper establishes a dual objective mixed integer nonlinear programming (MINLP) model with the objective of minimizing the emergency rescue time and maximizing the satisfaction rate of emergency material demand, designs a genetic algorithm to solve the emergency logistics location and allocation model, and obtains the Pareto optimal solution set of the model. Finally, a case study of COVID-19 epidemic in Beijing-Tianjin-Hebei region was carried out to verify the feasibility and effectiveness of the model and algorithm in the actual application, which can provide reference and suggestions for the location and material distribution of urban emergency logistics centers.

Enhancing the Flexibility of Storage Integrated Power System by Multi-Stage Robust Dispatch

The ever-increasing integration of variable wind energy requests for a power system with high flexibility. In this paper, we formulate the real-time economic dispatch problem as a multi-stage robust program to leverage flexible resources, especially energy storage systems, in a broader timescale, and also to avoid the risk of high costs in critical scenarios. The multi-stage robust program is decomposed to the dynamic programming form and the fast robust dual dynamic programming (FRDDP) algorithm is developed to solve it efficiently. To tackle the nonconvexities caused by the inter-temporal constraints of ESSs, an estimation-correction method is developed, and measures of the approximation accuracy are presented. Comprehensive case studies comprising units up to 1371 are carried out to verify the effectiveness of the proposed model. Numerical results show that compared with the existing ones, the proposed multi-stage robust model achieves lower cost under the worst-case scenario and shows better scalability in large-scale systems.

A Mathematical Model for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers via Interval Linear Programming Problems

We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

Stochastic Dynamic Cutting Plane for Multistage Stochastic Convex Programs

We introduce Stochastic Dynamic Cutting Plane (StoDCuP), an extension of the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve multistage stochastic convex optimization problems. At each iteration, the algorithm builds lower bounding affine functions not only for the cost-to-go functions, as SDDP does, but also for some or all nonlinear cost and constraint functions. We show the almost sure convergence of StoDCuP. We also introduce an inexact variant of StoDCuP where all subproblems are solved approximately (with bounded errors) and show the almost sure convergence of this variant for vanishing errors. Finally, numerical experiments are presented on nondifferentiable multistage stochastic programs where Inexact StoDCuP computes a good approximate policy quicker than StoDCuP while SDDP and the previous inexact variant of SDDP combined with Mosek library to solve subproblems were not able to solve the differentiable reformulation of the problem.

Optimal Corrective Rescheduling of Real and Reactive Powers for Power System Security Enhancement.(Dept.E)

This paper presents an optimal technique for solving the corrective rescheduling problem in p ower system. A successive dual linear programming approach is used with linearized p ower flow model. The system real and reactive power generations and transformer tap ratios are optimized within their operating limits so that no limit voilations of the line flow and bus voltage magnitudes in either the base case and contengency cases. The rest results on IEEE-30 bus test system show that the proposed technique can be successfully employed in conjunction with any fast security assessment scheme for a fast on-line security assessment and control.

Water-based resistive switches for neuromorphic long-range connections

The brain’s small-world network utilizes its short-range and long-range synaptic connections to process information in a complex and energy-efficient manner. To emulate the former, neuromorphic hardware typically leverages the conductance switching properties of thin-film dielectrics and semiconductors. Because these materials offer low ion mobilities, long-range connections built from thicker dielectrics require impractically-large forming voltages. To overcome this intrinsic shortcoming of solid-state active media, we present in this paper a simple Ag–H2O–Au cell that takes advantage of the relatively high ion mobility offered by deionized water to enable programmable connectivity switches between neurons separated by large gaps (∼40 µm). We introduce dual voltage programming schemes that allow the switch conductance to be modulated in analog and digital steps. When operating in the analog mode, the switch conductance could be potentiated and depressed over a relatively large (3.5×) range. In the digital mode, the Ag–H2O–Au switch delivered a high ON/OFF current ratio of ∼600 and sustained this margin over 200 switching cycles. Additionally, both switch states could be maintained for at least 3 h without external power. We show that unlike their solid-state counterparts, the water-gap in the Ag–H2O–Au cell can be easily refreshed without compromising the switching functionality. These attributes of Ag–H2O–Au switches in addition to their biocompatibility and simple design make them attractive for neuromorphic wetware implementations.

Synthesis of a Path-Planning Algorithm for Autonomous Robots Moving in a Game Environment during Collision Avoidance

This paper describes and illustrates the optimization of a safe mobile robot control process in collision situations using the model of a multistep matrix game of many participants in the form of a dual linear programming problem. The synthesis of non-cooperative and cooperative game control software was performed in Matlab/Simulink software to determine the safe path of the robot when passing a greater number of other robots and obstacles. The operation of the game motion control algorithm of a mobile robot is illustrated by computer simulations made in the Matlab/Simulink program of two real previously recorded navigation situations while passing dozens of other autonomous mobile robots.

Dual dynamic programming for multi-scale mixed-integer MPC

We propose a dual dynamic integer programming (DDIP) framework for solving multi-scale mixed-integer model predictive control (MPC) problems. Such problems arise in applications that involve long horizons and/or fine temporal discretizations as well as mixed-integer states and controls (e.g., scheduling logic and discrete actuators). The approach uses a nested cutting-plane scheme that performs forward and backward sweeps along the time horizon to adaptively approximate cost-to-go functions. The DDIP scheme proposed can handle general MPC formulations with mixed-integer controls and states and can perform forward-backward sweeps over block time partitions. We demonstrate the performance of the proposed scheme by solving mixed-integer MPC problems that arise in the scheduling of central heating, ventilation, and air-conditioning (HVAC) plants. We show that the proposed scheme is scalable and dramatically outperforms state-of-the-art mixed-integer solvers.

CCR model-based evaluation on the effectiveness and maturity of technological innovation

<p style='text-indent:20px;'>As there are many indexes for evaluating technological innovation in enterprises, it is hard to quantify all those indexes. Therefore, common evaluation methods cannot be applied to solve the absolute value of the evaluation indexes. Therefore, this study used the nonparametric CCR model based on input-output to estimate the relative value of evaluation index, and took dual programming tool to obtain the judgment basis for the most effective and optimal solution. Based on the software evaluation criteria, this paper proposed the concept of "maturity in technological innovation, " its four levels, and an evaluation standard for maturity. Based on the homogeneity, the paper selected four Beijing enterprises as evaluation samples. After comparing and analyzing the efficiency, scale return, production surface projection and maturity, we found that the evaluation results conform to the reality of sampling enterprises. CCR model was used to evaluate decision-making units with multiple inputs and outputs. The results show that this method can help accurately obtain the relative order and the enterprises' ability to make technological innovation. Thus, CCR model is able to help enterprises formulate policies on technological innovation.</p>

Constant Depth Decision Rules for multistage optimization under uncertainty

In this paper, we introduce a new class of decision rules, referred to as Constant Depth Decision Rules (CDDRs), for multistage optimization under linear constraints with uncertainty-affected right-hand sides. We consider two uncertainty classes: discrete uncertainties which can take at each stage at most a fixed number d of different values, and polytopic uncertainties which, at each stage, are elements of a convex hull of at most d points. Given the depthμ of the decision rule, the decision at stage t is expressed as the sum of t functions of μ consecutive values of the underlying uncertain parameters. These functions are arbitrary in the case of discrete uncertainties and are poly-affine in the case of polytopic uncertainties. For these uncertainty classes, we show that when the uncertain right-hand sides of the constraints of the multistage problem are of the same additive structure as the decision rules, these constraints can be reformulated as a system of linear inequality constraints where the numbers of variables and constraints is O(1)(n+m)dμN2 with n the maximal dimension of control variables, m the maximal number of inequality constraints at each stage, and N the number of stages.As an illustration, we discuss an application of the proposed approach to a Multistage Stochastic Program arising in the problem of hydro-thermal production planning with interstage dependent inflows. For problems with a small number of stages, we present the results of a numerical study in which optimal CDDRs show similar performance, in terms of optimization objective, to that of Stochastic Dual Dynamic Programming (SDDP) policies, often at much smaller computational cost.

Single cut and multicut stochastic dual dynamic programming with cut selection for multistage stochastic linear programs: convergence proof and numerical experiments

We introduce a variant of Multicut Decomposition Algorithms, called CuSMuDA (Cut Selection for Multicut Decomposition Algorithms), for solving multistage stochastic linear programs that incorporates a class of cut selection strategies to choose the most relevant cuts of the approximate recourse functions. This class contains Level 1 (Philpott et al. in J Comput Appl Math 290:196–208, 2015) and Limited Memory Level 1 (Guigues in Eur J Oper Res 258:47–57, 2017) cut selection strategies, initially introduced for respectively Stochastic Dual Dynamic Programming and Dual Dynamic Programming. We prove the almost sure convergence of the method in a finite number of iterations and obtain as a by-product the almost sure convergence in a finite number of iterations of Stochastic Dual Dynamic Programming combined with our class of cut selection strategies. We compare the performance of Multicut Decomposition Algorithms, Stochastic Dual Dynamic Programming, and their variants with cut selection (using Level 1 and Limited Memory Level 1) on several instances of a portfolio problem. On these experiments, in general, Stochastic Dual Dynamic Programming is quicker (i.e., satisfies the stopping criterion quicker) than Multicut Decomposition Algorithms and cut selection allows us to decrease the computational bulk with Limited Memory Level 1 being more efficient (sometimes much more) than Level 1.

Pointwise residual method for solving primal and dual ill-posed linear programming problems with approximate data

We propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved. doi:10.1017/S1446181120000243

Intelligent adaptive optimal control using incremental model-based global dual heuristic programming subject to partial observability

The scarcity of information regarding dynamics and full-state feedback increases the demand for a model-free control technique that can cope with partial observability. To deal with the absence of prior knowledge of system dynamics and perfect measurements, this paper develops a novel intelligent control scheme by combining global dual heuristic programming with an incremental model-based identifier. An augmented system consisting of the unknown nonlinear plant and unknown varying references is identified online using a locally linear regression technique. The actor–critic is implemented using artificial neural networks, and the actuator saturation constraint is addressed by exploiting a symmetrical sigmoid activation function in the output layer of the actor network. Numerical experiments are conducted by applying the proposed method to online adaptive optimal control tasks of an aerospace system. The results reveal that the developed method can deal with partial observability with performance comparable to the full-state feedback control, while outperforming the global model-based method in stability and adaptability.

Event-triggered design for discrete-time nonlinear systems with control constraints

In order to solve the constrained-input problem and reduce the computing resources, a novel event-triggered optimal control method is proposed for a class of discrete-time nonlinear systems. In the proposed method, the event-triggered control policy is applied to the globalized dual heuristic dynamic programming (GDHP) algorithm. Compared with the traditional adaptive dynamic programming (ADP) control, the event-triggered GDHP control can reduce the computation while ensuring the system performance. In this paper, a non-quadratic function is given to code the control constraints and the trigger condition with the stability analysis is provided. Neural networks (NN) are constructed in the GDHP structure, where the model network is designed to identify the unknown nonlinear system, the critic network is used to learn the cost function and its partial derivative, and the action network is designed to obtain the approximate optimal control law. Three simulation examples are presented to demonstrate the performance of the proposed event-triggered design for constrained discrete-time nonlinear systems.

Fenchel–Lagrange duality for DC infinite programs with inequality constraints

In this paper, we consider a composite DC optimization problem with constraints described by an arbitrary (possibly infinite) number of DC inequalities. By using the properties of the epigraph of the conjugate functions, we introduce some new constraint qualifications, which completely characterize the weak (zero/strong/stable) Fenchel–Lagrange duality between the primal problem and its dual problem. As applications, we obtain the corresponding results for a conic programming problem with DC composite function, a DC infinite programming problem with inequality constraints, and a DC optimization problem with composite functions, respectively.

Optimality conditions of fenchel-lagrange duality and farkas-type results for composite dc infinite programs

<p style='text-indent:20px;'>This paper is concerned with a DC composite programs with infinite DC inequalities constraints. Without any topological assumptions and generalized increasing property, we first construct some new regularity conditions by virtue of the epigraph technique. Then we give some complete characterizations of the (stable) Fenchel-Lagrange duality and the (stable) Farkas-type assertions. As applications, corresponding assertions for the DC programs with infinite inequality constraints and the conic programs with DC composite function are also given.</p>

Benefit of PARMA Modeling for Long-Term Hydroelectric Scheduling Using Stochastic Dual Dynamic Programming

AbstractIn the long-term management of large hydropower systems, operators generally have to maximize the benefits from energy production while ensuring they satisfy a minimal energy profile throug...

Energy-Efficient Power Allocation for D2D Communication underlaying Cellular Networks

This paper considers the power allocation problem in device-to-device (D2D) communication underlaying a cellular network and investigates the impact of different transmitting and interference power constraints on the energy efficiency and spectral efficiency of the network. We formulate the power allocation problem in D2D communication as a nonlinear fractional programming problem with an objective to maximize the energy efficiency of a D2D communication link subject to four different combinations of transmitting and interference power constraints. To solve the original formulated nonlinear fractional programming problem, we first convert it into a dual nonlinear parametric programming problem, and then decouple the dual problem into several solvable concave problems. Further, a closed-form solution is derived to each dual problem and an efficient power allocation algorithm is proposed to find a numerical solution to the formulated problem. Simulation results show that the proposed power allocation algorithm outperforms an ergodic capacity maximization algorithm and a uniform power distribution algorithm in terms of the energy efficiency of a D2D link, and can efficiently improve the overall ergodic capacity of a cellular network.

Global warming impact to River Basin of Blue Nile and the optimum operation of its multi-reservoir system for hydropower production and irrigation

The water resource of the Blue Nile River basin (BNRB) has been under pressure due to growing demands from many users, and the climate change impact. Potential impact of climate change for the maximum, median and minimum projected changes in the simulated streamflow of BNRB by a hydrologic model, VIC, driven by Representative Concentration Pathways climate scenarios, RCP4.5 and RCP8.5, of 4 GCMs (global climate models) downscaled dynamically by a regional climate model, WRF (Weather Research Forecasting) using a one-domain framework that covers the entire NRB for 2041–2070 and 2071–2100. These projected changes in streamflow were used to assess its future water allocations using a stochastic Dual Dynamic Programming (SDDP) algorithm and a hydro-economic model to optimize hydropower production and irrigated agriculture. Overall, it seems the Grand Ethiopian Renaissance Dam (GERD) reservoir will likely not operate at full storage level because the streamflow of BNRB is assumed to be regulated by three upstream reservoirs. The outflow from the reservoir of GERD or BNRB's annual flow at Khartoum is projected to increase under maximum, but is expected to decrease under minimum and median projected changes in streamflow for 2041–2070 and 2071–2100, respectively. Given the annual net benefit obtained from hydropower production and irrigated agriculture of the reservoir is projected to increase (decrease) under the maximum (median and minimum) projected changes in streamflow, the potential climate change impact should be considered in designing and developing the future water resources of BNRB.