Discrete Optimization Research Articles

An Efficient Algorithm for Exact Segmentation of Large Compositional and Categorical Time Series

ABSTRACTChange‐point detection, also known as signal segmentation, is an essential preprocessing step in many applications, ranging from industrial monitoring to bioinformatics. In short, it consists in finding the temporal boundaries of homogeneous regimes in long and non‐stationary time series. While this area of research is active, most existing methods are designed for Euclidean data. However, in many practical scenarios, the collected time series are compositional, meaning that each observation belongs to the probability simplex (the set of non‐negative vectors whose components sum to one). In this work, we propose an algorithm detecting change‐points in large compositional signals with an underlying piecewise stationary model. We cast the change‐point detection task as a discrete optimization problem, whose solution is shown to converge to the true change‐points. We introduce a new and time‐efficient dynamic programming algorithm that solves exactly this problem. To limit the number of operations, we describe a novel pruning rule that allows us to reduce the set of candidate change‐point indices. Our method is tested on a thorough simulation study, which confirms its efficiency. Additionally, we apply our method to a human activity segmentation task, highlighting the necessity for such novel techniques compared to standard algorithms.

Improved Monte Carlo tree search formulation with multiple root nodes for discrete sizing optimization of truss structures

This article proposes a novel reinforcement learning algorithm using an improved Monte Carlo tree search (IMCTS) formulation for the discrete optimum design of truss structures. IMCTS with multiple root nodes includes the update process, best reward, accelerating technique and terminal condition. The update process means that once a final solution is found, it is used as the initial solution for the next search tree. The best reward is used in the backpropagation step. The accelerating technique is introduced by decreasing the width of the search tree and reducing the maximum number of iterations. The agent is trained to minimize the total structural weight under various constraints until the terminal condition is satisfied. The optimal solution is the minimum value of all solutions found by the search trees. Numerical examples show that the agent can find the optimal solution with low computational cost, stably produces an optimal design, and is suitable for multi-objective structural optimization and large-scale structures.

Numerical computation of cut off wave number in polygonal wave guide by eight node finite element mesh generation approach

This study proposes a two-dimensional, eight-noded automated mesh generator for precise and efficient finite element analysis (FEA) in microwave applications. The suggested method for solving the Helmholtz problem employs an optimal domain discretization procedure. MAPLE-13 software's advanced automatic mesh generator was developed specifically for this work. To demonstrate the effectiveness of the approach, three distinct waveguide structures are analyzed, with the results compared to the best available analytical or numerical solutions. The findings indicate that the proposed method yields highly accurate and efficient finite element results, particularly for waveguide structures containing singularities. In microwave applications, this method can significantly enhance energy transmission efficiency.

Urban freight distribution with electric vehicles: comparing some solution procedures

The Vehicle Routing Problem (VRP) is a well-known discrete optimization problem that has an impact on theoretical and practical applications. In this paper, a freight distribution model that includes a charging system located at the depot, making it feasible for real world-implementation, is proposed. Two different solution methods are proposed and compared: a genetic algorithm (GA) and a population-based simulated annealing (PBSA) with the number of moves increasing during the iterations. Among the variety of algorithm used to solve the VRP, population-based search methods are the most useful, due to the ability to update the memory at each iteration. To demonstrate the practical aspects of the proposed solution a case study is solved using travel time on a real network to evaluate the potentiality for a real-world application.

Deep Learning Evidence for Global Optimality of Gerver’s Sofa

The moving sofa problem, introduced by Leo Moser in 1966, seeks to determine the maximal area of a 2D shape that can navigate an L-shaped corridor of unit width. Joseph Gerver’s 1992 solution, providing a lower bound of approximately 2.2195, is the best known, though its global optimality remains unproven. This paper leverages neural networks’ approximation power and recent advances in invexity optimization to explore global optimality. We propose two approaches supporting Gerver’s conjecture that his sofa is the unique global maximum. The first approach uses continuous function learning, discarding assumptions about the monotonicity, symmetry, and differentiability of sofa movements. The sofa area is computed as a differentiable function using our “waterfall” algorithm, with the loss function incorporating both differential terms and initial conditions based on physics-informed machine learning. Extensive training with diverse network initialization consistently converges to Gerver’s solution. The second approach applies discrete optimization to the Kallus–Romik upper bound, improving it from 2.37 to 2.3337 for five rotation angles. As the number of angles increases, our model asymptotically converges to Gerver’s area from above, indicating that no larger sofa exists.

Structure Design on Thermoplastic Composites Considering Forming Effects.

Carbon fiber reinforced polypropylene (CF/PP) thermoplastics integrate the superior formability of fabrics with the recoverable characteristics of polypropylene, making them a pivotal solution for achieving lightweight designs in new energy vehicles. However, the prevailing methodologies for designing the structural performance of CF/PP vehicular components often omit the constraints imposed by the manufacturing process, thereby compromising product quality and reliability. This research presents a novel approach for developing a stamping-bending coupled finite element model (FEM) utilizing ABAQUS/Explicit. Initially, the hot stamping simulation is implemented, followed by the transmission of stamping information, including fiber yarn orientation and fiber yarn angle, to the follow-up step for updating the material properties of the cured specimen. Then, the structural performance analysis is conducted, accounting for the stamping effects. Furthermore, the parametric study reveals that the shape and length of the blank holding ring exerted minimal influence on the maximum fiber angle characteristic. However, it is noted that the energy absorption and crushing force efficiency metrics of the CF/PP specimens can be enhanced by increasing the length of the blank holding ring. Finally, a discrete optimization design is implemented to enhance the bending performance of the CF/PP specimen, accounting for the constraint of the maximum shear angle resulting from the stamping process. The optimized design resulted in a mass reduction of 14.3% and an improvement in specific energy absorption (SEA) by 17.5% compared to the baseline sample.

Optimal Test Design for Estimation of Mean Ability Growth

The design of an achievement test is crucial for many reasons. This article focuses on a population’s ability growth between school grades. We define design as the allocating of test items concerning the difficulties. The objective is to present an optimal test design method for estimating the mean and percentile ability growth with good precision. We use the asymptotic expression of the variance in terms of the test information. With that criterion for optimization, we propose to use particle swarm optimization to find the optimal design. The results show that the allocation of the item difficulties depends on item discrimination and the magnitude of the ability growth. The optimization function depends on the examinees’ abilities, hence, the value of the unknown mean ability growth. Therefore, we will also use an optimum in-average design and conclude that it is robust to uncertainty in the mean ability growth. A test is, in practice, assembled from items stored in an item pool with calibrated item parameters. Hence, we also perform a discrete optimization using simulated annealing and compare the results to the particle swarm optimization.

An Enhanced Algorithm for Hybrid Flow Shop Scheduling Using Discrete Sparrow Optimization and Rough Set Theory

Aiming at the shortcomings of the sparrow search algorithm, such as it is easy to fall into local optimum and unable to solve discrete optimization problems, an improved discrete sparrow search algorithm is proposed. Firstly, the position update formula of the original sparrow search algorithm is abstracted, a new discrete heuristic position update strategy is designed according to the different identities of individuals, and the encoding and decoding methods are designed for the hybrid flow shop scheduling problem; Secondly, the rough data-deduction theory is introduced, and the feasibility and rationality of the above theory are explained by mathematical proofs, which provides theoretical support for the algorithm and improves the interpretability; Then, the nature of upper approximation is adopted to expand the search space, improve the population diversity, avoid prematurity of the algorithm, combine division and rough data-deduction to propose three strategies to promote information sharing among populations, regulate the exploitation ability and exploration ability of populations, and reduce the probability of the algorithm falling into local optimum; Finally, the improved discrete sparrow search algorithm is used to solve the hybrid flow shop scheduling problem. Simulation experiments are carried out on three small-scale practical examples and Liao's classic test set to verify the feasibility of the improved discrete sparrow search algorithm to solve the hybrid flow shop scheduling problem, and to prove the superiority of the proposed algorithm and the effectiveness of the improved strategy through comparison experiments with other algorithms.

A meta-heuristic extension of the Lagrangian heuristic framework

Lagrangian heuristics for discrete optimization work by modifying Lagrangian relaxed solutions into feasible solutions to an original problem. They are designed to identify feasible, and hopefully also near-optimal, solutions and have proven to be highly successful in many applications. Based on a primal-dual global optimality condition for non-convex optimization problems, we develop a meta-heuristic extension of the Lagrangian heuristic framework. The optimality condition characterizes (near-)optimal solutions in terms of near-optimality and near-complementarity measures for Lagrangian relaxed solutions. The meta-heuristic extension amounts to constructing a weighted combination of these measures, thus creating a parametric auxiliary objective function, which is a close relative to a Lagrangian function, and embedding a Lagrangian heuristic in a search procedure in the space of the weight parameters. We illustrate and make a first assessment of this meta-heuristic extension by applying it to the generalized assignment and set covering problems. Our computational experience show that the meta-heuristic extension of a standard Lagrangian heuristic can significantly improve upon solution quality.

Design and optimization of reaction-separation-recycle systems using a pseudo-transient continuation model

The reaction-separation-recycle (RSR) systems are an important part of chemical processes. Due to the interaction between reaction and separation sections, the behavior of RSR processes becomes highly complex and non-linear, posing different challenges for their design and optimization. The purpose of this study is to design and optimize RSR processes for irreversible liquid phase reaction systems using the pseudo-transient continuation (PTC) approach within an equation-oriented programing environment. This approach was applied to investigate one binary system and four ternary systems. The differential-algebraic models of the proposed process flowsheets were solved until the steady-state conditions were reached. The tray bypass efficiency method was incorporated into the PTC to circumvent the need for discrete optimization. The results demonstrated that in those cases where the product was heavier than the reactants, employing a stripping column was more economical than using a conventional distillation column for both the binary and ternary systems. In the ternary systems with two recycle streams, when the product was the intermediate component in terms of boiling point, the utilization of a divided-wall distillation column (DWC) resulted in a total annual cost saving of 35 % and 41 % compared to direct and indirect separation methods, respectively.

Influence maximization based on discrete particle swarm optimization on multilayer network

Influence maximization (IM) aims to strategically select influential users to maximize information propagation in social networks. Most of the existing studies focus on IM in single-layer networks. However, we have observed that individuals often engage in multiple social platforms to fulfill various social needs. To make better use of this observation, we consider an extended problem of how to maximize influence spread in multilayer networks. The Multilayer Influence Maximization (MLIM) problem is different from the IM problem because information propagation behaves differently in multilayer networks compared to single-layer networks: users influenced on one layer may trigger the propagation of information on another layer. Our work successfully models the information propagation process as a Multilayer Independent Cascade model in multilayer networks. Based on the characteristics of this model, we introduce an approximation function called Multilayer Expected Diffusion Value (MLEDV) for it. However, the NP-hardness of the MLIM problem has posed significant challenges to our work. To tackle the issue, we devise a novel algorithm based on Discrete Particle Swarm Optimization. Our algorithm consists of two stages: 1) the candidate node selection, where we devise a novel centrality metric called Random connectivity Centrality to select candidate nodes, which assesses the importance of nodes from a connectivity perspective. 2)the seed selection, where we employ a discrete particle swarm algorithm to find seed nodes from the candidate nodes. We use MLEDV as a fitness function to measure the spreading power of candidate solutions in our algorithm. Additionally, we introduce a Neighborhood Optimization strategy to increase the convergence of the algorithm. We conduct experiments on four real-world networks and two self-built networks and demonstrate that our algorithms are effective for the MLIM problem.

Discrete Grey Wolf Optimizer for Solving Urban Traffic Light Scheduling Problem

Discrete Grey Wolf Optimizer for Solving Urban Traffic Light Scheduling Problem

Power flow analysis using quantum and digital annealers: a discrete combinatorial optimization approach

Power flow (PF) analysis is a foundational computational method to study the flow of power in an electrical network. This analysis involves solving a set of non-linear and non-convex differential-algebraic equations. State-of-the-art solvers for PF analysis, therefore, face challenges with scalability and convergence, specifically for large-scale and/or ill-conditioned cases characterized by high penetration of renewable energy sources, among others. The adiabatic quantum computing paradigm has been proven to efficiently find solutions for combinatorial problems in the noisy intermediate-scale quantum (NISQ) era, and it can potentially address the limitations posed by state-of-the-art PF solvers. For the first time, we propose a novel adiabatic quantum computing approach for efficient PF analysis. Our key contributions are (i) a combinatorial PF algorithm and a modified version that aligns with the principles of PF analysis, termed the adiabatic quantum PF algorithm (AQPF), both of which use Quadratic Unconstrained Binary Optimization (QUBO) and Ising model formulations; (ii) a scalability study of the AQPF algorithm; and (iii) an extension of the AQPF algorithm to handle larger problem sizes using a partitioned approach. Numerical experiments are conducted using different test system sizes on D-Wave’s Advantage™ quantum annealer, Fujitsu’s digital annealer V3, D-Wave’s quantum-classical hybrid annealer, and two simulated annealers running on classical computer hardware. The reported results demonstrate the effectiveness and high accuracy of the proposed AQPF algorithm and its potential to speed up the PF analysis process while handling ill-conditioned cases using quantum and quantum-inspired algorithms.

PTE: Prompt tuning with ensemble verbalizers

Prompt tuning has achieved remarkable success in facilitating the performance of Pre-trained Language Models (PLMs) across various downstream NLP tasks, particularly in scenarios with limited downstream data. Reframing tasks as fill-in-the-blank questions represents an effective approach within prompt tuning. However, this approach necessitates the mapping of labels through a verbalizer consisting of one or more label tokens, constrained by manually crafted prompts. Furthermore, most existing automatic crafting methods either introduce external resources or rely solely on discrete or continuous optimization strategies. To address this issue, we have proposed a methodology for optimizing discrete verbalizers based on gradient descent, which we refer to this approach as PTE. This method integrates discrete tokens into verbalizers that can be continuously optimized, combining the distinct advantages of both discrete and continuous optimization strategies. In contrast to prior approaches, ours eschews reliance on prompts generated by other models or prior knowledge, merely augmenting a matrix. This approach boasts remarkable simplicity and flexibility, enabling prompt optimization while preserving the interpretability of output label tokens without constraints imposed by discrete vocabularies. Finally, employing this method in text classification tasks, we observe that PTE achieves results comparable to, if not surpassing, previous methods even under extreme conciseness. This furnishes a simple, intuitive, and efficient solution for automatically constructing verbalizers. Moreover, through quantitative analysis of optimized verbalizers, we uncover that language models likely rely not only on semantic information but also on other features for text classification. This revelation unveils new avenues for future research and model enhancements.

Validation of numerical method of unsteady Reynolds-averaged Navier–Stokes coupled with discrete phase model and optimization for the snow-resistance performance of bogies of a high-speed train

This paper presents an investigation aimed at enhancing the snow resistance of bogies in a three-car grouped alpine high-speed train (HST) that operates in a snowy region. The study employs a combination of the unsteady Reynolds-averaged Navier–Stokes model, coupled with the discrete phase model. This modeling approach has been validated through wind tunnel tests, including flow field tests, two-phase flow tests, and snow-ice tests. The study comprehensively compares the characteristics of the wind-snow flow, spatial snow concentration, and the distribution of snow on the train's underbody between two scenarios: the original train's underbody structure and the multifactor collaborative optimization scheme (MCOS). The results indicate that the MCOS case facilitates the escape of snow particles from the bogie cavities by weakening the upward and reverse flows. Furthermore, the MCOS case significantly reduces the concentration of snow particles around the heat-producing elements and decreases the accumulation of snow on both the lower and upper surfaces of the bogies. As a result, it reduces snow accumulation on the bogies by 50.6%, 56.4%, 60.8%, and 62.4% at train speeds of 200, 250, 300, and 350 km/h, respectively. In summary, this research provides valuable insights for improving the snow resistance of HSTs operating in snowy regions, with potential applications in enhancing railway transportation safety and efficiency.

Improving Transonic Performance with Adjoint-Based NACA 0012 Airfoil Design Optimization

This study delves into the discrete adjoint method, a powerful tool in high-fidelity aerodynamic shape optimization that efficiently computes derivatives of a target function for different design variables. It offers a theoretical exploration of its implementation as an innovative tool for calculating partial derivatives (sensitivities) related to objective functions and design variables. It is implemented to a NACA0012 airfoil at a transonic speed. This designated test case is qualitatively evaluated, considering specified Mach number and Reynolds number values of 0.7 and 15.96 million, respectively. The standard Spalart-Allmaras turbulence model is adapted to improve computational cost efficiency. The results validate the efficiency of Discrete Adjoint with OpenFOAM, showcasing its capability to generate optimal configurations. The achieved optimized performance is evidenced by minimizing the drag coefficient value (CD) to an impressive value of 0.00871, 62.2% lower than the previous studies, thus securing a novelty. While this research does not consider the post-processing of sensitivity calculations, it enables the potential for future research. The primary target of this paper is to assess the educational and training needs of researchers, engineers, and graduate students, offering a comprehensive introduction to discrete adjoint aerodynamic design optimization, presenting a novel methodology, and contributing to future advancements in the design optimization field.

A temperature-sensitive points selection method for machine tool based on rough set and multi-objective adaptive hybrid evolutionary algorithm

Reasonable deployment of temperature sensors is the key to accurately monitoring the temperature field of machine tools and improving the accuracy of thermal error prediction and compensation models. To determine the optimal deployment location of sensors, this paper proposes a temperature-sensitive points selection method tightly coupled with rough set and multi-objective optimization. Firstly, the importance of each temperature measurement point to the thermal error is calculated based on the rough set, and information entropy is introduced to amplify the importance difference among adjacent measurement points at the same heat source. Then, with the temperature measurement points groups as the variables, the number of temperature measurement points in the group, and the information importance of the group as the objectives, a multi-objective attribute reduction model is established, which transforms the temperature-sensitive points selection problem into a discrete multi-objective optimization problem. Finally, a multi-objective adaptive hybrid evolutionary algorithm is proposed, which designs a population initialization method based on mutual information and interval probability, and dynamic adaptive evolutionary parameters to achieve optimal temperature-sensitive points selection. Experiments on the high-speed dry hobbing machine verify the superiority and effectiveness of the proposed method.

Global and local semantic enhancement of samples for cross-modal hashing

Hashing becomes popular in cross-modal retrieval due to its exceptional performance in both search and storage. However, existing cross-modal hashing (CMH) methods may (a) neglect to learn sufficient modal-specific information, and (b) fail to fully exploit sample semantics. To address these issues, we propose a method called Semantic Enhancement of Sample Hashing (SESH). First, SESH employs a global modal-specific learning strategy to draw overall shared information and global modal-specific information by factoring the mapping matrix. Second, SESH introduces manifold learning and a local modal-specific learning strategy to extract additional local modal-specific and modal-shared data under label guidance. Meanwhile, local modal-specific information is integrated with global modal-specific details to add rich modal-specific information. Third, SESH uses discrete maximum similarity and orthogonal constraint transformation to enhance both global and local semantic information, embedding more discriminative information into the Hamming space. Finally, an efficient discrete optimization algorithm is proposed to generate the hash codes directly. Experiments on three datasets demonstrate the superior performance of SESH. The source code will be available at https://github.com/kokorording/SESH.

Stress-constrained concurrent multiscale topological design of porous composites based on discrete material optimisation

Porous composites have attracted increasing attention in recent decades. This study develops a concurrent multiscale topology optimisation (CMTO) method under a prescribed stress constraint for designing porous composites with multi-domain microstructures. First, to address the difficulty of predicting local stress due to varying of microstructural type throughout the optimisation process, a continuous and differentiable stress measure is proposed to effectively approximate the local stress. Second, an inverse homogenisation method based on isogeometric analysis (IGA) is developed to improve the accuracy of stress prediction, and then it is integrated into a CMTO which is developed based on the discrete material optimisation (DMO) interpolation scheme. Third, a stress constraint which is differentiable with respect to both macro and micro design variables is proposed to enable the stress-constrained concurrent optimisation of the macrostructural configuration, microstructural configuration and distribution. Fourth, a novel post-processing approach is established to achieve smooth while volume preserving contour of unit cells with layouts. Finally, two benchmark design examples, namely l-bracket and Crack problems, are implemented using the presented CMTO under a global stress constraint to demonstrate the effectiveness of the proposed method. The result indicates that the proposed method can effectively decrease the stress concentration via three design manners, i.e., the macrostructural configuration, microstructural configuration and distribution. Also, an “interface-enlarging” phenomenon was interestingly but reasonably found in those cases when subjected to stress-constraint considerations.

On separability of semigroups: application to models of physics and discrete optimization

Nilpotent semigroups are used in various fields of physics to model processes like system decay in quantum mechanics, state transitions in statistical mechanics, and simplifying dynamical systems. They also assist in optimizing control processes. The set of nilpotent matrices forms a semigroup and plays a key role in Jordan decomposition, which has many physical applications. In quantum mechanics, the Jordan form helps analyze linear operators on Hilbert spaces, especially with systems involving eigenvalue multiplicities. In vibration analysis, it aids in studying normal modes of mechanical systems with degenerate frequencies. Additionally, Jordan decomposition simplifies control theory for linear time-invariant systems, stability analysis in dynamical systems, and the behavior of coupled oscillators, as well as solving systems of linear differential equations. Certainly, the semigroup theory has even more applications in theoretical computer science: suffice it to say, for example, that some authors identify semigroups and finite automata. In this paper, we address the problem of P-separability of semigroups with respect to three key predicates: equality-separability, divisibility-separability, and subsemigroup-separability, achieved through homomorphisms into a nilpotent semigroup. We present some significant results that advance the understanding of these concepts.