Shortcomings of linear programming in optimizing river basin allocation

River Basin Restoration and Management is the result of two workshops that took place at the 4th IWA World Water Congress: The Restoration of Degraded River Basins and River Basin Management Using Machine Learning. The Restoration of Degraded River Basins set out to share experience in the institutional, policy, and public participation elements of restoration programmes, the ‘soft’ issues surrounding restoration of a degraded river basin and the development of the river basin plan. The resulting papers include a number of case studies from a variety of river basins in Israel, South Africa, United Kingdom, Australia and Central Europe. The River Basin Management Using Machine Learning workshop highlighted and compared the two different approaches to watershed management: the physically based modelling approach relying on the system physics versus the data driven modelling approach based on exploring the system ‘data behaviour’. The workshop was motivated by the recent rapid advance in information processing systems. These have pushed the hydrological research community to explore the possibilities of using intelligent systems aimed at automatically-evolving models of natural phenomena. This is the discipline of machine learning (ML), the study of computer algorithms that improve automatically through experience. This title belongs to Water and Environmental Management Series (WEMS) ISBN: 9781843395102 (Print) ISBN: 9781780402581 (eBook)

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

SMT solvers combine SAT reasoning with specialized theory solvers either to find a feasible solution to a set of constraints or to prove that no such solution exists. Linear programming (LP) solvers come from the tradition of optimization, and are designed to find feasible solutions that are optimal with respect to some optimization function. Typical LP solvers are designed to solve large systems quickly using floating point arithmetic. Because floating point arithmetic is inexact, rounding errors can lead to incorrect results, making inexact solvers inappropriate for direct use in theorem proving. Previous efforts to leverage such solvers in the context of SMT have concluded that in addition to being potentially unsound, such solvers are too heavyweight to compete in the context of SMT. In this paper, we describe a technique for integrating LP solvers that improves the performance of SMT solvers without compromising correctness. These techniques have been implemented using the SMT solver CVC4 and the LP solver GLPK. Experiments show that this implementation outperforms other state-of-the-art SMT solvers on the QF_LRA SMT-LIB benchmarks and is competitive on the QF_LIA benchmarks.

The row pivoting method for linear programming

In metro systems, a tactical train schedule with the rolling stock circulation plan aims to determine the movements of all physical trains. To utilize the regenerative energy as much as possible, this paper proposes an integrated model to simultaneously generate the optimal train schedule and rolling stock circulation plan, in which the brake-traction overlapping time at stations is maximized. In particular, our model rigorously considers the train turn-around constraints, train circulation constraints, and dynamic passenger demands to tackle the train loading capacity constraints. To eliminate the effect of non-linear constraints, we reformulate the original model into its equivalent linear model that can be efficiently solved by linear programming solvers. Finally, the numerical experiments based on Beijing Yizhuang Metro Line are implemented to demonstrate the effectiveness of our proposed model.

When developing linear programming models, issues such as budget limitations, customer requirements, or licensing may preclude the use of commercial linear programming solvers. In such cases, one option is to use an open-source linear programming solver. A survey of linear programming tools was conducted to identify potential open-source solvers. From this survey, four open-source solvers were tested using a collection of linear programming test problems and the results were compared to IBM ILOG CPLEX Optimizer (CPLEX) [1], an industry standard. The solvers considered were: COIN-OR Linear Programming (CLP) [2], [3], GNU Linear Programming Kit (GLPK) [4], lp_solve [5] and Modular In-core Nonlinear Optimization System (MINOS) [6]. As no open-source solver outperforms CPLEX, this study demonstrates the power of commercial linear programming software. CLP was found to be the top performing open-source solver considered in terms of capability and speed. GLPK also performed well but cannot match the speed of CLP or CPLEX. lp_solve and MINOS were considerably slower and encountered issues when solving several test problems.

A number of computer models for river basin planning and management have been developed by various agencies and used extensively since the mid-1970s. Most of the early developments have been based on the use of heuristic weight factors to represent priorities of allocation, and specialized optimization algorithms that were based on the use of network flow algorithms (NFAs). While these algorithms were at first considerably faster than the standard Simplex solvers, their handling of flow constraints was simplistic, which eventually led to the use of iterative schemes for handling nonnetwork constraints. This paper critically examines the notion that iterations applied in combination with NFA are a good vehicle for handling nonnetwork constraints. The failures are demonstrated on several variants of a simple problem with two reservoirs in series.

COHO is a verification tool for systems modeled by nonlinear ordinary differential equations (ODEs). Verification is performed using reachability analysis. The reachable space is represented by projectagons which are the polyhedron described by their projection onto two dimensional subspace. COHO approximates nonlinear ODEs with linear differential inclusions to compute the reachable state. We re-engineer the COHO system to provide a robust implementation with a clear interface and well-organized structure. We demonstrate the soundness and robustness of our approach by applying it to several examples, including a three dimensional, non-linear systems for which previous version of COHO failed due to numerical stability problems. The correctness of COHO strongly depends on the accuracy, efficiency and robustness of the linear program solver that is used throughout the analysis. Our COHO linear program solver is implemented by integrating the Simplex algorithm with interval arithmetic based on the framework set up by Laza [Laz01]. For highly ill-conditioned problems, an approximation that is very close to the optimal result is computed by our algorithm. This results in a very small error bound that allows COHO to successfully verify interesting examples. This thesis also presents an algorithm to solve the problem of projecting the feasible region of a linear program onto two dimensional subspaces. This algorithm uses the COHO linear program solver for efficiency and accuracy. We derive an analytical upper bound for the error and present experimental results to show that the errors are negligible in practice.

This paper outlines the importance of multiple time step optimization (MTO) in river basin allocation. The principal novelty of the work presented here is to provide a methodology for how to use MTO solutions in river basin planning and real-time operation. Two approaches for using the MTO results were tested on Damodar River basin in India and are presented in the paper. Using the proposed approach, the model managed flood flows without exceeding the downstream full bank channel capacity in 35 years of available historical data, while at the same time increasing generated hydropower on average by 63% annually, and supplying an additional 350 million m3 to irrigation and industry compared to the historical levels. The results presented in this study were obtained using the new Web-based Basin Management (WEB.BM) water management model, the only water allocation model with full Linear Programming (LP) optimization capabilities available online free of charge.

ABSTRACTRecently, the government of India has proposed the River Basin Management Bill, 2018. This Bill seeks to establish a River Basin Authority for the management and development of inter-state rivers and river basins in India. The Bill also seeks to entrench the tenets of Integrated River Basin Management in the Indian social and legal setup. This note attempts to normatively analyse the proposed Bill in light of the environmental justice principles, namely, distribution, participation and recognition. Through such analysis, this note argues that integrated river basin planning must not only focus on developing water resources but also human resources living within a river basin. This note also puts forth the proposition that water resource development within a river basin must be carried out taking into consideration the wider social, economic and political aspects of the people living within a basin area.

The equitable dissemination of COVID-19 vaccinations in lower-income nations presents a crucial worldwide dilemma, necessitating smart allocation techniques to guarantee impartial and effective coverage. This paper introduces a mathematical approach that utilizes Linear Programming (LP) to efficiently distribute vaccination doses among lower-income countries. The model incorporates crucial variables such population magnitude, healthcare infrastructure capability, susceptibility to the virus, financial limitations, and worldwide vaccine availability. The linear programming (LP) model is constructed using decision variables that indicate the quantity of vaccination doses assigned to each country. The goal function seeks to optimize the fair allocation of resources by giving weights to countries according to certain criteria. Constraints are imposed to ensure compliance with global vaccine supply, population coverage targets, healthcare capacity, vulnerability concerns, and budget limits. The non-negativity constraints guarantee that the vaccination allocations will always be greater than or equal to zero. The model is designed to be flexible, taking into account dynamic factors such as emerging viral strains and fluctuating vaccination availability. It highlights the significance of timely data updates to facilitate decision-making. The LP solver, such as the simplex method or interior-point approaches, is used to determine the best solution that achieves a fair distribution while considering logistical limitations. The results obtained from the linear programming (LP) model offer practical and effective guidance for policymakers and health organizations. These insights help in distributing COVID-19 vaccinations in a way that maximizes their impact and reduces inequalities among lower-income countries. The approach recognizes the interdependence of global health and emphasizes the need for coordinated endeavors in addressing the pandemic. This mathematical model is a useful tool in the ongoing efforts to distribute COVID-19 vaccinations fairly and efficiently. It contributes to the worldwide aim of obtaining universal immunity and overcoming the problems posed by the epidemic in lower-icnome countries.

In our previous work we have presented an approach to learn interpretable classification rules using a Boolean compressed sensing formulation. Our approach uses a linear programming (LP) relaxation and allows us to find interpretable (sparse) classification rules that achieve good generalization accuracy. However, the resulting LP representation for problems with either a large number of samples or large number of continuous features tends to become challenging for off-the-shelf LP solvers. We have explored a screening approach which allows us to dramatically reduce the number of active features without sacrificing optimality. In this work we explore reducing the number of samples in a sequential setting where we can certify reaching a near-optimal solution while only solving the LP on a small fraction of the available data points. In a batch setting this approach can dramatically reduce the computational complexity of the rule-learning LP formulation. In an online setting we derive stochastic upper and lower bounds on the the LP objective for unseen samples. This allows early stopping when we detect that the classifier will not change significantly with additional samples. The upper bounds are related to the learning curve literature in machine learning, and our lower bounds appear not to have been explored. Finally, we discuss a quick approach to compute the complete regularization path balancing rule interpretability versus accuracy.

This paper proposes a linear programming approach for onboard optimization of rocket trajectories, motivated by the need for reliable and computationally efficient trajectory optimization methods in real-time applications. To reformulate rocket trajectory optimization problems within a linear programming framework, a polytopal approximation method is developed. The polytopal approximation method approximates the thrust magnitude constraint using a set of artificial variables and linear constraints. Compared with a state-of-the-art approximation method, the polytopal approximation method demonstrates a higher accuracy-to-size ratio, enabling rocket trajectory optimization problems to be cast into linear subproblems without significantly increasing problem sizes. As linear subproblems can be readily solved by linear programming solvers, the proposed trajectory optimization approach inherits the solver’s reliability and computational efficiency, making it potentially useful for critical onboard applications. An ascent trajectory optimization problem is provided as an illustrative example to demonstrate the effectiveness of the proposed linear programming approach.

There is a growing research interest on the transdisciplinary measurement of vulnerability to climatic hazards from the perspective of integrated river basin management. However, the incorporation of stakeholders’ participation, local knowledge and locally spatial characteristics into the process of such vulnerability assessment is one of the challenges faced by decision-makers, especially in developing countries. This article proposes a novel methodology for assessing and communicating vulnerability to policymaking at the river basin level through a case study of Tachia River basin in Taiwan. The authors used a multicriteria decision analysis to develop an integrated vulnerability index applied to a participatory geographic information system (GIS) to map vulnerability to climatic hazards. Using a GIS-based spatial statistics technique and multivariate analysis, we test the degree to which vulnerabilities are spatially autocorrelated throughout the river basin, explain why clustering of vulnerable areas occurs in specific locations, and why some regions are particularly vulnerable. Results demonstrate that vulnerable areas are spatially correlated across the river basin. Moreover, exposure, biophysical sensitivity, land uses and adaptive capacity are key factors contributing to the formation of localized ‘hot spots’ of similarly and particularly vulnerable areas. Finally, we discuss how the findings provide direction for more effective approaches to river basin planning and management.

The concept of “river basin management” calls for managing water resources at the river basin level in order to promote the sustainable use of water resources. Often the concept of river basin management is associated with the introduction of “river basin organisations” (RBOs) as special purpose organisations. However, in many developing countries, RBOs remain underfunded. Fiscal decentralisation involves shifting certain responsibilities for expenditures or revenues to lower levels of government. It usually involves a reallocation of functions and revenues so that functions are adequately funded. Fiscal decentralisation is supported by the principles of subsidiarity, disentanglement, cooperation, accountability and fiscal equivalence, with the latter asking for a congruence of those who pay for, who decide upon and who benefit from public goods. On the one hand, fiscal decentralisation can support RBOs if sufficient financial resources are allocated to them to fulfil their tasks. However, if RBOs are excluded from resources at the sub-national level, it can also be counterproductive and lead to increased competition with general-purpose jurisdictions at the sub-national level. <br> This discussion paper, therefore, asks the question of whether fiscal decentralisation supports or counteracts the funding of river basin management in the case of Mongolia, which has undergone parallel processes of institutionalising RBOs and of fiscal decentralisation in the last decade. In order to answer the overriding question, the paper analyses i) how competencies for various water governance functions between RBOs and other bodies at the sub-national level are formally allocated; ii) the funding arrangements in place for water-related functions of RBOs and other sub-national bodies; and iii) how RBOs and other sub-national authorities coordinate or compete for water governance funding, and what this implies for sustainable water resource use. The paper compares the responsibilities of national and various sub-national entities and assesses the extent to which the principles of disentanglement, cooperation, subsidiarity and accountability are applied in Mongolian water governance. It also analyses the formal and de facto funding sources and budgeting systems of the sub-national entities responsible for water management, in order to assess to what extent the principle of fiscal equivalence is realised. <br> The discussion paper finds that there is still considerable room for improving the realisation of the principles of disentanglement, cooperation, subsidiarity and accountability in the allocation of competencies in Mongolian water governance. At the sub-national level, a high number of actors are involved in water governance, and the allocation of various water governance functions between river basin authorities (RBAs), river basin councils (RBCs), Aimag environmental agencies (AEAs), and governors and assemblies at the levels of provinces (Aimags), districts (Soums) and communities (Bahgs) remains complex and convoluted. Despite a broad division of labour, a high level of overlap exists in terms of data management; monitoring water resources, water uses and law implementation; law enforcement; and to a certain degree the implementation of measures among these actors. <br> The principle of fiscal equivalence is also not fully realised in Mongolian water resources management. In terms of financing water governance, RBAs are primarily financed through the national budget and AEAs through sub-national province budgets. However, uncertainties exist regarding the allocation of water-use fees due to inconsistent legal provisions. In practice, available funds to RBAs only cover fixed costs. Thus, RBAs are largely assigned unfunded mandates for planning, monitoring and implementing water protection. This considerably limits the effectiveness of the river basin management approach. Overall, AEAs have somewhat higher budgets for environmental protection than RBAs. However, the case study shows, for example, that AEAs sometimes do not allocate any funds for water-related projects at all and do not necessarily earmark 35 per cent of water-use fees for environmental protection. <br> Furthermore, problems of underfinancing exacerbate problems of overlapping responsibilities. The inconsistent legal provisions on water-use fees have led to competition between AEAs and RBAs, which undermines trust between the different agencies and makes it more difficult to perform the tasks that they share as well as to coordinate on those tasks where their responsibilities overlap. Still, in the case analysed, recent first attempts have been undertaken to clarify responsibilities in Memorandums of Understanding. While the effectiveness of these Memorandums of Understanding should be subject to future research, they alone will hardly solve the problem of underfinancing. Therefore, the paper concludes that in Mongolia, fiscal decentralisation and the institutionalisation of river basin management are, so far, hardly mutually supportive, but rather competitive processes. It recommends a number of legal adjustments as well as the financial empowerment of RBAs and RBCs in order to support the sustainable use of water resources.