Different Decomposition Strategies to Solve Stochastic Hydrothermal Unit Commitment Problems

Solving very-large-scale optimization problems frequently require to decompose them in smaller subproblems, that are iteratively solved to produce useful information. One such approach is the Lagrangian Relaxation (LR), a broad range technique that leads to many different decomposition schemes. The LR supplies a lower bound of the objective function and useful information for heuristics aimed at constructing feasible primal solutions. In this paper, we compare the main LR strategies used so far for Stochastic Hydrothermal Unit Commitment problems, where uncertainty mainly concerns water availability in reservoirs and demand (weather conditions). This problem is customarily modeled as a two-stage mixed-integer optimization problem. We compare different decomposition strategies (unit and scenario schemes) in terms of quality of produced lower bound and running time. The schemes are assessed with various hydrothermal systems, considering different configuration of power plants, in terms of capacity and number of units

A Lagrangian approach to Chance Constrained Routing with Local Broadcast

Mobile cellular networks play a pivotal role in emerging Internet of Things (IoT) applications, such as vehicular collision alerts, malfunctioning alerts in Industry-4.0 manufacturing plants, periodic distribution of coordination information for swarming robots or platooning vehicles, etc. All these applications are characterized by the need of routing messages within a given local area (geographic proximity) with constraints about both timeliness and reliability (i.e., probability of reception). This paper presents a Non-Convex Mixed-Integer Nonlinear Programming model for a routing problem with probabilistic constraints on a wireless network. We propose an exact approach consisting of a branch-and-bound framework based on a novel Lagrangian decomposition to derive lower bounds. Preliminary experimental results indicate that the proposed algorithm is competitive with state-of-the-art general-purpose solvers, and can provide better solutions than existing highly tailored ad-hoc heuristics to this problem

Solving Stochastic Hydrothermal Unit Commitment with a New Primal RecoverycTechnique Based on Lagrangian Solutions

The high penetration of intermittent renewable generation has prompted the development of Stochastic Hydrothermal Unit Commitmentc(SHUC) models, which are more difficult to be solved than their thermal-basedccounterparts due to hydro generation constraints and inflow uncertainties.cThis work presents a SHUC model applied in centralized cost-based dispatch, where the uncertainty is related to the water availability in reservoirs and demand. The SHUC is represented by a two-stage stochastic model, formulated as a large-scale mixed-binary linear programming problem. The solution strategy is divided into two steps, performed sequentially, with intercalated iterations to find the optimal generation schedule. The first step is the Lagrangian Relaxation (LR) approach. The second step is given by a Primal Recovery based on LR solutions and a heuristic based on Benders' Decomposition. Both steps benefit from each other, exchanging information over the iterative process. We assess our approach in terms of the quality of the solutions and running times on space and scenario LR decompositions. The results show the advantage of our primal recovery technique compared to solving the problem via MILP solver. This is true already for the deterministic case, and the advantage grows as the problem’s size (number of plants and/or scenarios) does

Improving problem reduction for 0-1 Multidimensional Knapsack Problems with valid inequalities

© 2016 Elsevier Ltd. All rights reserved. This paper investigates the problem reduction heuristic for the Multidimensional Knapsack Problem (MKP). The MKP formulation is first strengthened by the Global Lifted Cover Inequalities (GLCI) using the cutting plane approach. The dynamic core problem heuristic is then applied to find good solutions. The GLCI is described in the general lifting framework and several variants are introduced. A Two-level Core problem Heuristic is also proposed to tackle large instances. Computational experiments were carried out on classic benchmark problems to demonstrate the effectiveness of this new method

Solving Stochastic Hydrothermal Unit Commitment with a New Primal Recovery Technique Based on Lagrangian Solutions

The high penetration of intermittent renewable generation has prompted the development of Stochastic Hydrothermal Unit Commitment (SHUC) models, which are more difficult to be solved than their thermal-based counterparts due to hydro generation constraints and inflow uncertainties. This work presents a SHUC model applied in centralized cost-based dispatch. The SHUC is represented by a two-stage stochastic model, formulated as a large-scale mixed-binary linear programming problem. The solution strategy is divided into two steps. The first step is the Lagrangian Relaxation (LR) approach, which is applied to solve the dual problem and generate a lower bound for SHUC. The second step is given by a Primal Recovery where we use the solution of the LR dual problem with heuristics based on Benders’ Decomposition to obtain the primal-feasible solution. Both steps benefit from each other, exchanging information over the iterative process. We assess our approach in terms of the quality of the solutions and running times on space and scenario LR decompositions. The computational instances use various power systems, considering the different configuration of plants (capacity and number of units). The results show the advantage of our primal recovery technique compared to solving the problem via MILP solver. This is true already for the deterministic case, and the advantage grows as the problem’s size (number of plants and/or scenarios) does. The space decomposition provides better solutions, although scenario one provides better lower bounds, but the main idea is to encourage researchers to explore LR decompositions and heuristics in other relevant problems

A Matheuristic for Integrated Timetabling and Vehicle Scheduling

Planning a public transportation system is a complex process, which is usually broken down in several phases, performed in sequence. Most often, the trips required to cover a service with the desired frequency (headway) are decided early on, while the vehicles needed to cover these trips are determined at a later stage. This potentially leads to requiring a larger number of vehicles (and, therefore, drivers) that would be possible if the two decisions were performed simultaneously. We propose a multicommodity-flow type model for integrated timetabling and vehicle scheduling. Since the model is large-scale and cannot be solved by off-the-shelf tools with the efficiency required by planners, we propose a diving-type matheuristic approach for the problem. We report on the efficiency and effectiveness of two variants of the proposed approach, differing on how the continuous relaxation of the problem is solved, to tackle real-world instances of bus transport planning problem originating from customers of M.A.I.O.R., a leading company providing services and advanced decision-support systems to public transport authorities and operators. The results show that the approach can be used to aid even experienced planners in either obtaining better solutions, or obtaining them faster and with less effort, or both

Incremental bundle methods using upper models

We propose a family of proximal bundle methods for minimizing sum-structured convex nondifferentiable functions which require two slightly uncommon assumptions, that are satisfied in many relevant applications: Lipschitz continuity of the functions and oracles which also produce upper estimates on the function values. In exchange, the methods: i) use upper models of the functions that allow to estimate function values at points where the oracle has not been called; ii) provide the oracles with more information about when the function computation can be interrupted, possibly diminishing their cost; iii) allow to skip oracle calls entirely for some of the component functions, not only at "null steps" but also at "serious steps"; iv) provide explicit and reliable a-posteriori estimates of the quality of the obtained solutions; v) work with all possible combinations of different assumptions on how the oracles deal with not being able to compute the function with arbitrary accuracy. We also discuss the introduction of constraints (or, more generally, of easy components) and use of (partly) aggregated models

A Stabilized Structured Dantzig-Wolfe Decomposition Method

We discuss an algorithmic scheme, which we call the stabilized structured Dantzig-Wolfe decomposition method, for solving large-scale structured linear programs. It can be applied when the subproblem of the standard Dantzig-Wolfe approach admits an alternative master model amenable to column generation, other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron. Stabilization is achieved by the same techniques developed for the standard Dantzig-Wolfe approach and it is equally useful to improve the performance, as shown by computational results obtained on an application to the multicommodity capacitated network design problem

Minimizing value-at-risk in single-machine scheduling

The vast majority of the machine scheduling literature focuses on deterministic problems in which all data is known with certainty a priori. In practice, this assumption implies that the random parameters in the problem are represented by their point estimates in the scheduling model. The resulting schedules may perform well if the variability in the problem parameters is low. However, as variability increases accounting for this randomness explicitly in the model becomes crucial in order to counteract the ill effects of the variability on the system performance. In this paper, we consider single-machine scheduling problems in the presence of uncertain parameters. We impose a probabilistic constraint on the random performance measure of interest, such as the total weighted completion time or the total weighted tardiness, and introduce a generic risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VaR) of the random performance measure at a specified confidence level. We propose a Lagrangian relaxation-based scenario decomposition method to obtain lower bounds on the optimal VaR and provide a stabilized cut generation algorithm to solve the Lagrangian dual problem. Furthermore, we identify promising schedules for the original problem by a simple primal heuristic. An extensive computational study on two selected performance measures is presented to demonstrate the value of the proposed model and the effectiveness of our solution method