Solving Optimization Problems Research Articles

A Multigrid Solver for PDE-Constrained Optimization with Uncertain Inputs

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number N of samples used to discretized the probability space. We show that this reduced system can be solved with optimal O(N) complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.

A parallel social spider algorithm based on population mining

The social spider algorithm is a meta-heuristic algorithm inspired by the foraging behavior of biological social spiders, and is initially used to solve optimization problems with continuous functions. There exist many challenges when dealing with discrete combinatorial optimization problems. Therefore, a novel parallel social spider algorithm based on population mining is proposed. Its starting point is to mine dynamic heuristic information from the population of social spiders that cause vibrations to guide the spider’s next walk. It proposes the concept of vibration candidate sets and enhances the behavior of artificial spiders. Furthermore, it is implemented in parallel on the graphics processing unit. According to our investigation, it is the first parallel social spider algorithm. The classical traveling salesman problem is solved as an example. By solving 30 small-scale and medium-scale problems and comparing with other algorithms, it shows that its solution quality is significantly better. By solving 17 large-scale problems, compared with the parallel ant colony system and parallel iterative hill climbing algorithm on the GPU, it is found that both the solution quality and performance are also significantly better. Codes are available at https://github.com/BuptCIAGroup/P-SSA.

An integrated coupled oscillator network to solve optimization problems

Solving combinatorial optimization problems is essential in scientific, technological, and engineering applications, but can be very time and energy-consuming using classical algorithms executed on digital processors. Oscillator-based Ising machines offer a promising alternative by exploiting the analog coupling between electrical oscillators to solve such optimization problems more efficiently. Here we present the design and the capabilities of our scalable approach to solve Ising and quadratic unconstrained binary optimization problems. This approach includes routable oscillator connections to simplify the time-consuming embedding of the problem into the oscillator network. Our manufactured silicon chip, featuring 1440 oscillators implemented in a 28 nm technology, demonstrates the ability to solve optimization problems in 950 ns while consuming typically 319 μW per node. A frequency, phase, and delay calibration ensures robustness against manufacturing variations. The system is evaluated with multiple sets of benchmark problems to analyze the sensitivity for parameters such as the coupling strength or frequency.

Low-carbon operation strategy of virtual power plant considering progressive demand response

Adapting virtual power plants (VPPs) to the development of new power systems has gradually become a trend. Meanwhile, the economical and effective use of demand-side resources in VPPs has become a research hotspot in the field of new energy. This paper proposes a low-carbon operation strategy with progressive demand response (DR). A DR model id established by evaluating the scheduling potential of low-carbon and flexible loads and the DR transfer of different users motivated by incentive mechanisms. Because the changing carbon price in carbon trading also affects the response transfer amount, it is beneficial to reduce the operating cost of the energy system by using consumers’ surplus value according to consumer psychology to guide energy users to participate in DR. Finally, wind, light, user load data and VPP internal unit output are optimized in rolling mode over multiple time scales, and the optimal adjustment results of the previous day are taken as the basis. At the same time, the output deviation penalty function is added to further reduce the system cost. The nonlinear model of DR cost is transformed into a mixed integer linear model, which is modeled on MATLAB software and solved by a CPLEX optimization solver. The case analysis shows that the progressive DR strategy may increase the total operating cost of the system, but reduce the cost of the carbon emission. The proposed model and method can not only improve the feasibility and effectiveness of market supervision strategy, but also provides a useful reference for operators and demand-side users in energy systems to choose strategies.

Global Sensitivity Analysis via Optimal Transport

We examine the construction of variable importance measures for multivariate responses using the theory of optimal transport. We start with the classical optimal transport formulation. We show that the resulting sensitivity indices are well-defined under input dependence, are equal to zero under statistical independence, and are maximal under fully functional dependence. Also, they satisfy a continuity property for information refinements. We show that the new indices encompass Wagner’s variance-based sensitivity measures. Moreover, they provide deeper insights into the effect of an input’s uncertainty, quantifying its impact on the output mean, variance, and higher-order moments. We then consider the entropic formulation of the optimal transport problem and show that the resulting global sensitivity measures satisfy the same properties, with the exception that, under statistical independence, they are minimal, but not necessarily equal to zero. We prove the consistency of a given-data estimation strategy and test the feasibility of algorithmic implementations based on alternative optimal transport solvers. Application to the assemble-to-order simulator reveals a significant difference in the key drivers of uncertainty between the case in which the quantity of interest is profit (univariate) or inventory (multivariate). The new importance measures contribute to meeting the increasing demand for methods that make black-box models more transparent to analysts and decision makers. This paper was accepted by Baris Ata, stochastic models and simulation. Funding: A. Figalli acknowledges the support of the ERC [Grant 721675] “Regularity and Stability in Partial Differential Equations (RSPDE)” and of the Lagrange Mathematics and Computation Research Center. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01796 .

Hub production scheduling problem with mold availability constraints

This research delves into a scheduling challenge inherent in the wheel hub casting process, a critical stage in automotive wheel manufacturing. The process involves shaping molten metal into specific wheel hub designs using dedicated molds. Diverse customer demands necessitate a unique mold for each wheel hub design, posing a scheduling challenge due to the limited availability of these expensive and long-lasting molds. There are 2 machines available for casting. Each wheel hub design requires a specific mold. These molds are expensive, long-lasting, and need periodic maintenance. The objective function is to minimize the total time (makespan) to complete all casting jobs. Only one job requiring a specific mold can be processed at a time (due to mold limitations). Molds are unavailable during maintenance periods. The challenge lies in scheduling jobs considering both machine availability and mold constraints to minimize the overall production time. We formulate a mathematical model using mixed integer programming to precisely represent the problem and its constraints. Inspired by the Longest Processing Time (LPT) rule, we propose a heuristic named LRTPT to efficiently schedule jobs on the two machines. For large-scale problems, we employ a novel SIAIS algorithm to find near-optimal solutions effectively. We also present a dedicated branch and bound algorithm specifically tailored to solve smaller-sized instances of the problem. This algorithm leverages two lower bounds and several dominance properties to expedite the search for the optimal solution. To evaluate the effectiveness of our proposed approaches, we conduct a series of comprehensive experiments. The results demonstrate that the branch and bound algorithm significantly outperforms the widely used optimization solver CPLEX for smaller problems. Furthermore, the SIAIS algorithm proves to be more efficient than existing scheduling heuristics.

Estimation of the error of digital modeling of vibration ef ects on onboard radio-electronic equipment

An analysis of the destructive effect of external vibrations on radio-electronic equipment located on moving objects was carried out. The relevance of digital modeling of vibration impacts, aimed at long-term forecasting of the processes of origin and development of defects in on-board radio-electronic equipment, is substantiated. Vibration effects are considered from two points of view: on the one hand, as a factor in the destruction of equipment, on the other, as an informative signal indicating the nature and extent of this destruction. As equipment becomes more complex, the requirements for the adequacy of models and methods for assessing modeling errors increase. Digital modeling uses matrices constructed based on the equilibrium equations of elasticity theory. Based on these matrices, Lame coeffi cients are calculated and boundary conditions are formed. In the article, an explicit difference scheme is obtained, intended for the numerical solution of the system of equilibrium equations of the theory of elasticity. Factors infl uencing the modeling error have been identifi ed. The principles for calculating the vibration modeling error are formulated. Functionals are presented and justifi ed that connect the amplitude modeling error with the speed of vibration propagation in the medium under study and with the number of nodes of the discrete model for each projection of the vibration wave propagation vector. A specifi c numerical example shows how these functionals allow solving optimization problems.

The method of decomposition of the values of two-dimensional spectral features of remote sensing based on the analysis of the components of the correlation coef cient

A technique has been developed for decomposing the values of two-dimensional spectral features according to their components of correlation coeffi cients. A close analogue of the proposed methodology are automatic classification algorithms. The basis of the methodology is the analysis of the proposed indicator – the products of normalized values of spectral features and their probability density. A nonparametric Rosenblatt-Parzen estimate is used to reconstruct the probability density from the initial statistical data. The peculiarity of the proposed indicator and the user-selected threshold values of the indicator make it possible to form variants of the decomposition of the initial statistical data and mapping of the results obtained during the computational experiment. Using a human-machine decomposition procedure for the values of two-dimensional spectral features, it is possible to circumvent the problem of solving optimization problems when using automatic classifi cation algorithms and use information about the relationship between spectral features in the elements of the earth’s surface. The results of the application of the technique in the processing of remote sensing data of the forest area and their comparison with the initial information are considered. Spectral features have been established, which mainly determine the decomposition between dead wood and other forest conditions. The obtained results reveal their development in the formation of sets of spectral features in the assessment of the states of natural objects.

Quantum Algorithms for optimizing problems

Quantum computing is quickly becoming a field that can change the game. It can completely change how businesses solve optimisation problems. We will be looking at three different quantum algorithms in great detail: the Quantum Approximate Optimisation Algorithm (QAOA), the Variational Quantum Eigensolver (VQE), and Grover's Algorithm. We look into how these algorithms work on the inside, how they compare to more traditional methods, and how they might be used in areas like energy, banking, and logistics. The piece then talks about current research projects that are trying to fix the technical issues and hardware limits of quantum technology. In the end, we look ahead to possible future developments that might help solve optimisation problems, such as better quantum gear and more complex quantum algorithms. By combining what has already been written with what is new, this study aims to shed light on how quantum computing could help solve tough optimisation problems and spark new ideas.

Segmented distribution of gas diffusion layer porosity and catalyst layer ionomer content in a polymer electrolyte membrane fuel cell

Gas diffusion layer porosity and ionomer content inside catalyst layer both can affect cell performance. Electrochemical reaction rate is non-uniform inside the cell. Non-uniform design of components or structures can improve the electrochemical reaction rate distribution and thus promote cell performance. Thus, the investigations on the optimal distribution are done using the optimization solver within COMSOL Multiphysics 5.3a to seek the optimal segmented distribution of porosity or ionomer content to obtain higher current density. The distributions in cathode side along three directions are all investigated. The optimized results show that the porosity increases along main gas flow direction and the ionomer content reduces. With voltage decreasing, the value of porosity around outlet region reduces and the value around outlet region increases, ionomer content shows opposite trend. The porosity under gas channel is higher than that under land. Ionomer content increases under land and reduces under gas channel with voltage decreasing. The porosity is high near catalyst layer and ionomer content near proton exchange membrane is higher at low voltages. The ionomer content distribution along thickness direction can improve current density by 5.596 % at 0.2 V. For better overall cell performance, the porosity changing along x-axis can be selected, the values from under land to under gas channel are 0.384, 0.499, 0.618. These results can provide theoretical guidance for cell components design to obtain higher cell output performance.

Neural quantile optimization for edge–cloud networking

We seek the best traffic allocation scheme for the edge–cloud networking subject to SD-WAN architecture and burstable billing. First, we formulate a family of quantile-based integer programming problems for a fixed network topology with random parameters describing the traffic demands. Then, to overcome the difficulty caused by the discrete feature, we generalize the Gumbel-softmax reparameterization method to induce an unconstrained continuous optimization problem as a regularized continuation of the discrete problem. Finally, we introduce the Gumbel-softmax sampling neural network to solve optimization problems via unsupervised learning. The neural network structure reflects the edge–cloud networking topology and is trained to minimize the expectation of the cost function for unconstrained continuous optimization problems. The trained network works as an efficient traffic allocation scheme sampler, outperforming the random strategy in feasibility and cost value. Besides testing the quality of the output allocation scheme, we examine the generalization property of the network by increasing the time steps and the number of users. We also feed the solution to existing integer optimization solvers as initial conditions and verify the warm-starts can accelerate the short-time iteration process. The framework is general, and the decoupled feature of the random neural networks is adequate for practical implementations.

Bayesian optimization algorithms for accelerator physics

Accelerator physics relies on numerical algorithms to solve optimization problems in online accelerator control and tasks such as experimental design and model calibration in simulations. The effectiveness of optimization algorithms in discovering ideal solutions for complex challenges with limited resources often determines the problem complexity these methods can address. The accelerator physics community has recognized the advantages of Bayesian optimization algorithms, which leverage statistical surrogate models of objective functions to effectively address complex optimization challenges, especially in the presence of noise during accelerator operation and in resource-intensive physics simulations. In this review article, we offer a conceptual overview of applying Bayesian optimization techniques toward solving optimization problems in accelerator physics. We begin by providing a straightforward explanation of the essential components that make up Bayesian optimization techniques. We then give an overview of current and previous work applying and modifying these techniques to solve accelerator physics challenges. Finally, we explore practical implementation strategies for Bayesian optimization algorithms to maximize their performance, enabling users to effectively address complex optimization challenges in real-time beam control and accelerator design. Published by the American Physical Society 2024

Solving Boltzmann optimization problems with deep learning

Decades of exponential scaling in high-performance computing (HPC) efficiency is coming to an end. Transistor-based logic in complementary metal-oxide semiconductor (CMOS) technology is approaching physical limits beyond which further miniaturization will be impossible. Future HPC efficiency gains will necessarily rely on new technologies and paradigms of computing. The Ising model shows particular promise as a future framework for highly energy-efficient computation. Ising systems are able to operate at energies approaching thermodynamic limits for energy consumption of computation. Ising systems can function as both logic and memory. Thus, they have the potential to significantly reduce energy costs inherent to CMOS computing by eliminating costly data movement. The challenge in creating Ising-based hardware is in optimizing useful circuits that produce correct results on fundamentally nondeterministic hardware. The contribution of this paper is a novel machine learning approach, a combination of deep neural networks and random forests, for efficiently solving optimization problems that minimize sources of error in the Ising model. In addition, we provide a process to express a Boltzmann probability optimization problem as a supervised machine learning problem.

SAMP Rao algorithm based minimization of the roughness of milled surface of Ti-6Al-4V

In order to solve optimization problems including machining responses as objectives, this study suggests a parameter-less method called the self-adaptive multi population (SAMP) Rao algorithm that does not rely on metaphors. When machining titanium alloys, achieving a good surface quality is a difficult process. In the current study, an effort has been made to reduce surface roughness during milling Ti-6Al-4V. Response surface methodology (RSM) was used in the experiment design to create a model for surface roughness using cutting parameters as variables. The developed model was tested in additional tests in addition to the primary experiments. It was shown that cutter speed and feed rate had the biggest effects on milled roughness, whereas depth of cut had very little of an impact. The model’s quality is demonstrated by the correlation coefficient (R2) 98%, which indicates that the model can explain 98% of the data. Later, a response surface-based desirability technique was used to minimize roughness of milled surface. The outcome of the proposed algorithm is compared with RSM optimizer. It has been noted that the outcomes achieved with the SAMP approach are more advantageous than RSM approach and found 18% minimum value of roughness of milled surface. Main reason to get better result in case of SAMP Rao algorithm is that it is based on the iterative search process with random values of cutting parameters with in defined range. SAMP Rao Algorithm provides cutting settings of 133.5 m min−1, 0.13 mm/tooth feed rate, and 2.06 mm of milling depth along with a minimal roughness of milled surface of 0.37 μm.

Diagnosing infeasible optimization problems using large language models

Decision-making problems can be represented as mathematical optimization models, finding wide applications in fields, such as economics, engineering, transportation, and health care. One of the primary barriers to deploying these models in practice is the challenge of helping practitioners understand and interpret such models, particularly when they are infeasible, meaning no decision satisfies all the constraints. Existing methods for diagnosing infeasible optimization models often rely on expert systems, necessitating significant background knowledge in optimization. In this paper, we introduce OptiChat, a first-of-its-kind natural language-based system equipped with a chatbot GUI for engaging in interactive conversations about infeasible optimization models. OptiChat can provide natural language descriptions of the optimization model itself, identify potential sources of infeasibility, and offer suggestions to make the model feasible. The implementation of OptiChat is built on GPT-4, which interfaces with an optimization solver to identify the minimal subset of constraints that render the entire optimization problem infeasible, known as the Irreducible Infeasible Subset (IIS). We utilize few-shot learning, expert chain-of-thought, key-retrieve, and sentiment prompts to enhance OptiChat’s reliability. Our experiments demonstrate that OptiChat assists both expert and non-expert users in improving their understanding of the optimization models, enabling them to quickly identify the sources of infeasibility.

Zonewise surrogate-based optimization of box-constrained systems

Complex physical or numerical systems may exhibit distinct behaviors in various zones of their design spaces. We present an algorithm that uses multiple cluster-based surrogates for optimizing such box-constrained systems. It partitions the design space into multiple clusters using K-means clustering and develops a separate surrogate for each cluster. It then uses these surrogates to sample additional points in the design space whose function evaluations guide the search for a global optimum. Clustering, surrogate construction, and smart sampling are employed iteratively to add sample points until a pre-defined threshold. The best solution from these points estimates a global optimum. An extensive test bed of 52 box-constrained functions was used to evaluate and compare the algorithm's performance and computational requirements with sixteen derivative-free optimization solvers. The best version of our algorithm surpassed all sixteen solvers in optimization accuracy for a fixed number of evaluations and demanded lower computational effort than fifteen.

Synthesize Personalized Training for Robot-Assisted Upper Limb Rehabilitation With Diversity Enhancement.

For upper limb rehabilitation, the robot-assisted technique in combination with serious games requires well-specified training plans. For the best quality of the rehabilitation process, customized game levels for each user are desired, while it is labor-intensive to design and adjust game levels for different individuals. We work on generating training content for a desktop end-effector rehabilitation robot and propose a method to automatically generate individualized training plans. By modeling the search of the training motions as finding optimal hand paths and trajectories, we introduce solving the design problem with a multi-objective optimization (MO) solver. We further improve the MO solver to enhance the diversity of the solutions. With the proposed approach, our system is capable of automatically generating various training plans considering the training intensity and dexterity of each joint in the upper limb. In addition, the enhanced diversity avoids repeated training plans, which helps motivate the user in the rehabilitation. We test our method with different requirements on the training plans and validate the solutions.

Artificial bee colony-based nonrigid demons registration

The artificial bee colony (ABC) algorithm has gained popularity in recent years for its ability to solve optimization problems. The accuracy and resilience of ABC-based image processing techniques have demonstrated encouraging outcomes. The ABC method is an excellent solution for image processing issues since it has the ability to swiftly and effectively explore the search space. The current research intends to address image registration issues by refining the existing image registration strategy using ABC algorithm. The process of nonrigid demons registration is frequently employed in the processing of medical images. The combination of these two techniques is referred to as the ABC-based nonrigid demons registration method. The proposed method has shown superior performance in registration accuracy and efficiency compared to other existing methods. Applications in medical image analysis and computer-assisted diagnosis are highly promising for the ABC-based nonrigid demons registration. Particle swarm optimization (PSO) and frameworks based on genetic algorithms (GA) have been compared with the suggested framework. The observed results showed improved accuracy and faster convergence in ABC-based demons registration.

Optimisasi Jumlah Produksi Roti Kacang Dengan Metode Cutting Plane

Abstract. This research aims to optimize the production of peanut bread at UKM Peanut Roti Hj. Eliya Lubis in Tebing Tinggi City. The main challenges faced are fluctuating raw material prices and unstable consumer demand, which affect the production process. The research uses the cutting plane method to complete an integer linear program to determine optimal production quantities. This method was chosen because it is effective in solving optimization problems with integer results, which are relevant for in-house production. The research results show that the cutting plane method is able to produce better optimal solutions than other methods, such as branch and bound, thereby increasing production efficiency and company profits. This research makes a significant contribution in helping SME entrepreneurs face intense business competition by managing resources more effectively and efficiently.

Dynamic security constrained AC optimal power flow for microgrids

To ensure the continuous operation of a microgrid, proactive planning is essential, especially when contemplating possible dynamic events that alter the scheduled scenarios for the islanded operation or during its transition from connected to islanded mode. This paper introduces an innovative mathematical programming model for the AC optimal power flow (AC-OPF) with dynamic security constraints (DSCs), considering two scenarios: the islanded operation and the transition. This model uses positive-sequence equations to represent the inverter-based resources (IBRs) for grid-forming and grid-following roles, along with a fourth-order synchronous generator model equipped with excitation and frequency control systems. They assess the dynamic response to specific events such as short-circuits, decreases in PV generation, increases in load demand during the islanded operation, and the transition itself. The DSCs are applied to dispatchable distributed energy resources (DERs), which react to variations in the microgrid, considering generation and opportunity costs to minimize the discrepancy between planned and secure operating points. The mathematical programming model is implemented using AMPL, and solutions are obtained through the nonlinear optimization solver IPOPT. The tests are conducted in an adapted version of the microgrid being developed at the University of Campinas that includes a synchronous generator, photovoltaic (PV) generation, and a battery energy storage system (BESS). Results demonstrate the model’s effectiveness in adjusting generation dispatch to withstand defined events and optimizing generation resources, even when limits are not reached.