Comparison of open-source linear programming solvers.

Numerous computer models for river basin planning and management have been developed and used extensively since the mid‐1970s. Early developments have relied on the use of network flow algorithms (NFA), due mainly to higher execution speed than the standard Simplex solvers. However, subsequent efforts to include proper modeling of hydraulic and hydrologic constraints introduced iterative schemes into the NFA‐based models, which diminished the initial advantages in execution speed and which also caused concerns over the accuracy of the convergence schemes. Hence full‐blown commercial linear programming (LP) solvers were introduced as a replacement to the iterative solution strategy of the NFA approach. This paper demonstrates one possible failure to solve a simple allocation problem using the NFA‐based model and shows how this problem can be solved using the standard LP approach. It then identifies cases when even a full‐blown LP approach cannot properly model two critical aspects of river basin management, one related to reservoirs with multiple outflows and the other one related to modeling of hydrologic channel routing. For NFA‐based models the failures are the result of the inability to include relationships between flows on different model components directly into the search process. For the models based on LP solvers, the failures are caused by the fact that integrated reservoir outflow capacity between the starting and the ending storage levels is assumed over the entire length of the assumed time step, while the actual outflow can only take place during the portion of the time step when the storage level is above the invert of the outlet structure.

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.

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.

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.

Voltage dips represent a significant power quality problem. The main cause of voltage dips and short-term interruptions is an electrical short circuit that occurs in transmission or distribution networks. Faults in the power system are stochastic by nature and the main cause of voltage dips. As faults in the transmission system can affect more customers than faults in the distribution system, to reduce the number of dips, it is not enough to invest in a small part of the transmission or distribution system. Only targeted investment in the whole (or a large part of the) power system will reduce voltage dips. Therefore, monitoring parts of the power system is very important. The ideal solution would be to cover the entire system so that a power quality (PQ) monitor is installed on each bus, but this method is not economically justified. This paper presents an advanced method for determining the optimal location and the optimal number of voltage dip measuring devices. The proposed algorithm uses a monitor reach area matrix created by short-circuit simulations, and the coefficient of the exposed area. Single-phase and three-phase short circuits are simulated in DIgSILENT software on the IEEE 39 bus test system, using international standard IEC 60909. After determining the monitor reach area matrix of all potential monitor positions, the binary bat algorithm with a coefficient of the exposed area of the system bus is used to minimize the proposed objective function, i.e., to determine the optimal location and number of measuring devices. Performance of the binary bat algorithm is compared to the mixed-integer linear programming algorithm solved by using the GNU Linear Programming Kit (GLPK).

A linear programming solver which can find the optimal solution for primal-dual linear programming (PDLP) problems is presented. The neural net solver is based on an extension model of the Kosko's bidirectional associative memory (BAM). The proposed neural net solver uses the same number of neurons and connections as does the Hopfield model or Chua's canonical nonlinear programming model while it resolves the problems of (1) getting infeasible solutions and (2) employing a large penalty term in the objective function. Simulation results based on RC-active circuit implementation of the neural net solver are presented. The results indicate that when the solver is implemented in hardware with full parallelism, it is capable of solving linear programming problems very quickly and this computation speed seems to be insensitive to the dimension of the underlying problem. >

The row pivoting method for linear programming

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].

This paper presents primal heuristics for the computation of Wasserstein Barycenters of a given set of discrete probability measures. The computation of a Wasserstein Barycenter is formulated as an optimization problem over the space of discrete probability measures. In practice, the barycenter is a discrete probability measure which minimizes the sum of the pairwise Wasserstein distances between the barycenter itself and each input measure. While this problem can be formulated using Linear Programming techniques, it remains a challenging problem due to the size of real-life instances. In this paper, we propose simple but efficient primal heuristics, which exploit the properties of the optimal plan obtained while computing the Wasserstein Distance between a pair of probability measures. In order to evaluate the proposed primal heuristics, we have performed extensive computational tests using random Gaussian distributions, the MNIST handwritten digit dataset, and the Fashion MNIST dataset introduced by Zalando. We also used Translated MNIST, a modification of MNIST which contains original images, rescaled randomly and translated into a larger image. We compare the barycenters computed by our heuristics with the exact solutions obtained with a commercial Linear Programming solver, and with a state-of-the-art algorithm based on Gaussian convolutions. Our results show that the proposed heuristics yield in very short run time and an average optimality gap significantly smaller than 1%.

CFD and potential flow assessment of the hydrodynamics of a kitefoil

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.

Advances in technology for the manufacturing of integrated circuits have resulted in extremely large, and time consuming, problems on how to lay out components for optimal circuit performance. These problems can be written as mixed integer programs which are easily relaxed to linear programs with a very high number of variables and constraints. The relaxed programs can often be solved by applying state-of-the-art linear programming software, however these solutions come at the expense of long solution time. In this paper we show that, by considering the structure inherent in VLSI problems, one can specialize classical preprocessing algorithms to take into account the standard form of the constraint matrix for VLSI problems, thereby achieving improved preprocessing results with relatively little effort. We provide analysis showing our preprocessing techniques are accurate and provide some numerical testing demonstrating the increased efficiency. The numerical tests also demonstrate that using our preprocessing in conjunction with internal preprocessing methods that come with many linear program solvers, can improve the overall performance of the linear program solver and its preprocessor.

A Dantzig–Wolfe decomposition algorithm for linear economic model predictive control of dynamically decoupled subsystems

Online advertising has been the mainstream monetization approach for internet-based companies, in which bid optimization plays a crucial role in enhancing advertising performance. Currently, the bid optimization problem has narrowed down to two specific forms: Budget-constrained bidding (BCB) and Multi-constraint bidding (MCB). Existing solutions try to solve BCB/MCB via linear programming solvers, learning methods, or feedback control. However, in large-scale complex e-commerce, they still suffer from inefficiency, poor convergence, or slow adaptation to the changing market. This research presents an online receding optimization method as a solution for practical bid optimization problems. We conduct a theoretical analysis of the optimal bidding strategy's structure. Further, an online receding optimization process is designed based on open-loop feedback control, which periodically updates a constructed optimal bid formulation that can be solved by linear programming. Then, considering large-scale linear programming problems, we propose an efficient down sampling scheme. Besides, a neural-network-based auction scale prediction is used to adapt to the changing market. Finally, a series of online A/B experiments on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Taobao Sponsored Search</i> compare our work to industrial methods and state-of-the-art from several aspects. The proposed method has been implemented on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Taobao</i>, a billion-scaled online advertising business, for over a year.

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.