Class Of Optimization Problems Research Articles

A fast layered path planning algorithm for job shop scheduling problem

Job shop scheduling problem (JSP) is a classical system resource optimisation problem and also an NP hard problem. The search algorithm based on Akers obstacle graph model is an effective algorithm to solve JSP, which first removes part of jobs from the original schedule, then constructs obstacle graph and finds the shortest path from the graph, and finally reinserts the jobs according to the shortest path decoding method to get the new schedule. Although the new scheduling can achieve good results, it is time-consuming to find the shortest path. Therefore, it is necessary to further study how to quickly plan the shortest path. This study presents a fast layered path search algorithm for solving the obstacle graph of job shop scheduling. The algorithm designs a node expansion method and a delay distance formula. The obstacles generated by different machines in the obstacle graph are layered. When the nodes expand, the extended nodes are compared with the parent layer nodes to quickly avoid closely arranged obstacles, and multiple child nodes are generated at one time through node expansion to improve the node expansion ability. At the same time, node expansion method and delay distance formula can be well integrated with A* algorithm. Finally, the test verifies that the algorithm can spend less time to find the shortest path.

A Parallel DNA Algorithm for Solving the Quota Traveling Salesman Problem Based on Biocomputing Model.

The quota traveling salesman problem (QTSP) is a variant of the traveling salesman problem (TSP), which is a classical optimization problem. In the QTSP, the salesman visits some of the n cities to meet a given sales quota Q while having minimized travel costs. In this paper, we develop a DNA algorithm based on Adleman-Lipton model to solve the quota traveling salesman problem. Its time complexity is O(n2+Q), which is a significant improvement over previous algorithms with exponential complexity. A coding scheme of element information is pointed out, and a reasonable biological algorithm is raised by using limited conditions, whose feasibility is verified by simulation experiments. The innovation of this study is to propose a polynomial time complexity algorithm to solve the QTSP. This advantage will become more obvious as the problem scale increases compared with the algorithm of exponential computational complexity. The proposed DNA algorithm also has the significant advantages of having a large storage capacity and consuming less energy during the operation. With the maturity of DNA manipulation technology, DNA computing, as one of the parallel biological computing methods, has the potential to solve more complex NP-hard problems.

A Hierarchy of Standard Polynomial Programming Formulations for the Maximum Clique Problem

The maximum clique problem is a classical optimization problem which finds many important applications. Some of the most theoretically interesting and computationally successful approaches to this problem are based on the Motzkin--Straus formulation, expressing the clique number in terms of the maximal value of a standard quadratic program. This intriguing relationship has also motivated significant developments in quadratic optimization, copositive programming, and complexity in nonlinear optimization. Inspired by such developments, this paper establishes a hierarchy of polynomial programming formulations (\textbfP$^k$), $k\in\{2,\ldots, \omega\}$ for the maximum clique problem, relating the clique number $\omega$ of a given graph to the global maximal value of a multilinear polynomial function $f_k(x)$ of degree $k$ over the standard simplex in $\mathbb{R}^{|V|}$. The case of $k=2$ corresponds to the Motzkin--Straus formulation, and the hierarchical feature is in the fact that the set of local maxima of (\textbfP$^{k+1}$) is a subset of the set of local maxima of (\textbfP$^{k}$), $k\in\{2,\ldots, \omega-1\}$. In particular, every local maximum of (\textbfP$^{\omega}$) is global.

Mutational Slime Mould Algorithm for Gene Selection.

A large volume of high-dimensional genetic data has been produced in modern medicine and biology fields. Data-driven decision-making is particularly crucial to clinical practice and relevant procedures. However, high-dimensional data in these fields increase the processing complexity and scale. Identifying representative genes and reducing the data’s dimensions is often challenging. The purpose of gene selection is to eliminate irrelevant or redundant features to reduce the computational cost and improve classification accuracy. The wrapper gene selection model is based on a feature set, which can reduce the number of features and improve classification accuracy. This paper proposes a wrapper gene selection method based on the slime mould algorithm (SMA) to solve this problem. SMA is a new algorithm with a lot of application space in the feature selection field. This paper improves the original SMA by combining the Cauchy mutation mechanism with the crossover mutation strategy based on differential evolution (DE). Then, the transfer function converts the continuous optimizer into a binary version to solve the gene selection problem. Firstly, the continuous version of the method, ISMA, is tested on 33 classical continuous optimization problems. Then, the effect of the discrete version, or BISMA, was thoroughly studied by comparing it with other gene selection methods on 14 gene expression datasets. Experimental results show that the continuous version of the algorithm achieves an optimal balance between local exploitation and global search capabilities, and the discrete version of the algorithm has the highest accuracy when selecting the least number of genes.

Route Stability in the Uncertain Capacitated Arc Routing Problem

Power line inspections in a microgrid can be modeled as the uncertain capacitated arc routing problem, which is a classic combinatorial optimization problem. As an evolutionary computation method, genetic programming is used as a hyper-heuristic method to automatically evolve routing policies that can make real-time decisions in an uncertain environment. Most existing research on genetic programming hyper-heuristic for the uncertain capacitated arc routing problem only focuses on optimizing the total cost of solutions. As a result, the actual routes directed by the routing policies evolved by genetic programming hyper-heuristic are usually not stable, i.e., the routes have large fluctuations in different uncertain environments. However, for marketing or considering the drivers’ and customers’ perspectives, the routes should not be changed too often or too much. Addressing this problem, this study first proposes a method to estimate the similarity between two routes and then extends it for evaluating the stability of the routes in uncertain environments. A novel genetic programming hyper-heuristic, which considers two objectives, i.e., the solution quality (total cost) and the stability of routes, was designed. Experimental studies demonstrate that the proposed genetic programming is hyper-heuristic with stability in consideration and can obtain more stable solutions than the traditional algorithm, without deteriorating the total cost. The approach provided in this study can be easily extended to solving other combinatorial optimization problems in the microgrid.

A memetic algorithm based on edge-state learning for max-cut

Max-cut is one of the most classic NP-hard combinatorial optimization problems. The symmetry nature of it leads to special difficulty in extracting meaningful configuration information for learning; none of the state-of-the-art algorithms has employed any learning operators. This paper proposes an original learning method for max-cut, namely post-flip edge-state learning (PF-ESL). Different from previous algorithms, PF-ESL regards edge-states (cut or not cut) rather than vertex-positions as the critical information of a configuration, and extracts their statistics over a population for learning. It is based on following observations. 1) Edges are the only factors considered by the objective function. 2) Edge-states keep invariant when rotating a local configuration to its symmetry position, but vertex-positions do not. These suggest that edge-states contain more meaningful information about a configuration than vertex-positions do. It is impossible to set the state of an edge without influencing some other edges’ states due to their dependencies. Therefore, instead of setting edge-states directly, PF-ESL samples the flips on vertices. Flips on vertices are sampled according to their capacities in increasing the similarity on edge-states between the given solution and a population. PF-ESL is employed in an EDA (Estimation of Distribution Algorithm) perturbation operator and a path-relinking operator. Experimental results show that our algorithm is competitive, and show that edge-state learning is value-added for both the two operators.The main contributions of this paper are as follows. Firstly, previous state-of-the-art evolutionary algorithms for max-cut focus on vertex positions in their evolutionary operation, this paper proposes a new and more reasonable perspective suggesting that edge-states are the critical information of divided graphs rather than vertex positions, and introduces a novel method to measure and utilize their similarities based on it. Such a perspective is fundamental to learning based algorithms design for max-cut and other graph partitioning problems, and can shed lights on future researches. Furthermore, since max-cut is one of the most classic and fundamental NP hard problems, many real-world problems involve dividing graph data into different parts to optimize certain functions, this new perspective may inspire related or similar problems. Secondly, besides the original edge-states based perspective, and the post-flip edge-states learning (PFESL) operator based on it, our memetic algorithm also incorporates a novel evolutionary framework which alternates between EDA based Iterated Tabu search (ITS) and path relinking based genetic algorithm. Finally, the proposed algorithm provides competitive results on two mostly used benchmark sets and improves the best-known results of 6 most challenging instances.

Solid isotropic material with thickness penalization – A 2.5D method for structural topology optimization

SIMP is the most common topology optimization scheme to minimize the material utilization of a structural component using element densities as a design variable. This paper presents a new methodology, SIMTP, Solid Isotropic Material with Thickness Penalization, introducing a nodal thickness variable to the plane stress element. SIMTP uses a 2.5D element, and it is developed considering the varying thickness as a 3D problem and planar transformation as a 2D problem. The developed 2.5D element is used for projecting the 2D strain energy onto a 3D space. Classical optimization problems like Cantilever, MBB, and L-, beams are solved using 2.5D SIMTP, and results are compared with SIMP. The implementation is simple, directly representing the thickness, and nodal design variables enhance the resolution. Checkerboarding or other topology-related issues are not noticed during the process. Also, 2.5D SIMTP can yield desired results with fewer elements, thereby reducing the computational efforts. Besides, a 211-line MATLAB implementation of 2.5D SIMTP with examples is provided at the end of this paper.

Zeroth-Order Optimization for Composite Problems with Functional Constraints

In many real-world problems, first-order (FO) derivative evaluations are too expensive or even inaccessible. For solving these problems, zeroth-order (ZO) methods that only need function evaluations are often more efficient than FO methods or sometimes the only options. In this paper, we propose a novel zeroth-order inexact augmented Lagrangian method (ZO-iALM) to solve black-box optimization problems, which involve a composite (i.e., smooth+nonsmooth) objective and functional constraints. This appears to be the first work that develops an iALM-based ZO method for functional constrained optimization and meanwhile achieves query complexity results matching the best-known FO complexity results up to a factor of variable dimension. With an extensive experimental study, we show the effectiveness of our method. The applications of our method span from classical optimization problems to practical machine learning examples such as resource allocation in sensor networks and adversarial example generation.

Quantum approximate optimization algorithm with adaptive bias fields

The quantum approximate optimization algorithm (QAOA) transforms a simple many-qubit wavefunction into one which encodes a solution to a difficult classical optimization problem. It does this by optimizing the schedule according to which two unitary operators are alternately applied to the qubits. In this paper, the QAOA is modified by updating the operators themselves to include local fields, using information from the measured wavefunction at the end of one iteration step to improve the operators at later steps. It is shown by numerical simulation on MaxCut problems that, for a fixed accuracy, this procedure decreases the runtime of QAOA very substantially. This improvement appears to increase with the problem size. Our method requires essentially the same number of quantum gates per optimization step as the standard QAOA, and no additional measurements. This modified algorithm enhances the prospects for quantum advantage for certain optimization problems.

Technical Perspective

The optimal assignment problem is a classic combinatorial optimization problem. Given a set of n agents A, a set T of m tasks, and an n×m cost matrix C, the objective is to find the matching between A and T, which minimizes or maximizes an aggregate cost of the assigned agent-task pairs. In its standard definition, n = m and we are looking for the 1-to-1 matching with the minimum total cost. From a graph theory perspective, this is a weighted bipartite graph matching problem. A classic algorithm for solving the assignment problem is the Hungarian algorithm (a.k.a. Kuhn-Munkres algorithm) [3], which bears a O(n3) computational cost (assuming that n = m); this is the best run-time of any strongly polynomial algorithm for this problem. There are many variants of the assignment problem, which differ in the optimization objective (i.e., minimize/maximize an aggregate cost, achieve a stable matching, maximize the number of agents matched which their top preferences, etc.) and in whether there are constraints on the number of matches for each agent or task.

Quadratic interpolation boosted black widow spider-inspired optimization algorithm with wavelet mutation

Meta-heuristic algorithms are effective in solving complex optimization problems with advantages of flexibility for coding, robustness and global optimization capability. An enhanced Black Widow Optimization called QIWBWO algorithm with three improvement strategies is proposed in this paper. At the beginning of search, the theory of good points set is used to obtain the better initial population, which helps the algorithm to quickly determine the correct search direction. Then, quadratic interpolation strategy is used to improve the solution accuracy and accelerate the convergence. Meanwhile, to avoid the algorithm falling into a local optimum, wavelet mutation is introduced to improve population diversity and helps the algorithm to search the global optimum rather than local optimums. The proposed BWO algorithm is compared with other different kinds of meta-heuristic algorithms on 25 traditional benchmark functions and CEC2017 competition suite. The statistical results show the improved BWO algorithm delivers better performance in accuracy, stability and convergence rate. Finally, QIWBWO also obtains the best results on four classical optimization problems in engineering application, which verifies its practicality and effectiveness. The source code of QIWBWO is publicly available in the supplementary material related to this article.

Improving Variable Orderings of Approximate Decision Diagrams Using Reinforcement Learning

Prescriptive analytics provides organizations with scalable solutions for large-scale, automated decision making. At the core of prescriptive analytics methodology is optimization, a field devoted to the study of algorithms that solve complex decision-making problems. Optimization algorithms rely heavily on generic methods for identifying tight bounds, which provide both solutions to problems and optimality guarantees. In the last decade, decision diagrams (DDs) have demonstrated significant advantages in obtaining bounds compared with the standard linear relaxation commonly used by commercial solvers. However, the quality of the bounds computed by DDs depends heavily on the variable ordering chosen for the construction. Besides, the problem of finding an ordering that optimizes a given metric is generally NP-hard. This paper studies how machine learning, specifically deep reinforcement learning (DRL), can be used to improve bounds provided by DDs, in particular through learning a good variable ordering. The introduced DRL models improve primal and dual bounds, even over standard linear programming relaxations, and are integrated in a full-fledged branch-and-bound algorithm. This paper, therefore, provides a novel mechanism for utilizing machine learning to tighten bounds, adding to recent research on using machine learning to obtain high-quality heuristic solutions and, for the first time, using machine learning to improve relaxation bounds through a generic bounding method. We apply the methods on a classic optimization problem, the maximum independent set, and demonstrate through computational testing that optimization bounds can be significantly improved through DRL. We provide the code to replicate the results obtained on the maximum independent set. Summary of Contribution: This paper studies the use of reinforcement learning to compute a variable ordering of decision diagram-based approximations for discrete optimization problems. This is among the first works to propose the use of machine learning to improve upon generic bounding methods for discrete optimization problems, thereby establishing a critical bridge between optimization and learning.

The 1-D bin packing problem optimisation using bees algorithm

The bin packing problem is a classic combinatorial optimisation problem that is widely used in various applications such as assembly line balancing, scheduling, and time-tabling. Metaheuristic algorithms can provide solutions to these problems faster than exact methods. Bees Algorithm, a metaheuristic algorithm inspired by the foraging activity of bees, is known for its performance in solving optimisation problems. To our best knowledge, this is the first use of the Bees Algorithm to solve a bin packing problem. In this paper, we use the basic Bees Algorithm to demonstrate near-optimal solutions and measure the accuracy of solutions to the one-dimensional bin packing problem. The algorithm procedure and parameter settings are set following the previous research. Benchmark datasets are used in the experiment, and accuracy is measured. The results indicate that the basic Bees Algorithm for bin packing problems and previous research on travelling salesman problems produce similar accuracy results.

A discrete cuckoo search algorithm for traveling salesman problem and its application in cutting path optimization

The Traveling Salesman Problem (TSP) is a classic combinatorial optimization problem with various industrial applications. This paper proposes a discrete cuckoo search algorithm based on random walk and cluster analysis to solve the traveling salesman problem (TSP). Although the original cuckoo search algorithm is not suitable to solve the TSP, this algorithm is modified to solve the TSP. Lévy flights and random preference walk in the original cuckoo search algorithm are replaced with novel tools, including the local adjustment operator and discrete random walk. The former is utilized to preserve superior solutions found by the algorithm, while the latter is employed to maintain the diversity of the population. For large-scale TSP problems, the k-means algorithm is first utilized to divide cities into k categories for optimization, while random algorithms are adopted to combine them. A simple 2-opt operator is utilized as the local optimization operator to accelerate the algorithm's convergence rate. Multiple groups of standard TSPLIB datasets are chosen to compare the proposed method with state-of-the-art optimization algorithms that also use the 2-opt/k-opt optimization operator. According to the experimental results, the stability and accuracy of the proposed algorithm are superior to the state-of-the-art optimization algorithms. Regarding the solution accuracy, the algorithm improves 0.39%, 0.29%, 0.84%, 0.4%, 0.07% and 2.22% in the optimal solution over discrete cuckoo search (DCS), discrete bat algorithm (DBA), discrete sine–cosine algorithm (DSCA), random-key cuckoo search (RKCS) and discrete spider monkey optimization (DSMO) on the standard test cases, and 0.58%, 0.19%, 0.85%, 0.43%, 0.36% and 5.83% in the average optimal solution. Regarding stability, the variance of the optimal solutions of RKCS, DSMO, and novel discrete differential evolution (NDEE) for 30 independent runs is 38.5, 52.6, and 59.4 times higher than that of the algorithm in this paper. Finally, the proposed algorithm optimizes air knife migration during glass cutting. Accordingly, the path for air knife migration is reduced by 8556 mm.

A Subexponential Parameterized Algorithm for Directed Subset Traveling Salesman Problem on Planar Graphs

There are numerous examples of the so-called ``square root phenomenon'' in the field of parameterized algorithms: many of the most fundamental graph problems, parameterized by some natural parameter $k$, become significantly simpler when restricted to planar graphs and in particular the best possible running time is exponential in $O(\sqrt{k})$ instead of $O(k)$ (modulo standard complexity assumptions). We consider a classic optimization problem Subset Traveling Salesman, where we are asked to visit all the terminals $T$ by a minimum-weight closed walk. We investigate the parameterized complexity of this problem in planar graphs, where the number $k=|T|$ of terminals is regarded as the parameter. We show that Subset TSP can be solved in time $2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}$ even on edge-weighted directed planar graphs. This improves upon the algorithm of Klein and Marx [SODA 2014] with the same running time that worked only on undirected planar graphs with polynomially large integer weights.

Optimal Trajectory Design Accounting for Robust Stability of Path-Following Controller

This study presents an optimal control framework that designs an optimal trajectory while additionally accounting for the stability of a path-deviation error controller that is dependent on the states provided by the optimal trajectory. Thus, instead of designing the controller independent of the trajectory, the stability is enforced as part of the optimization constraints in connection with the optimal trajectory it should follow. This allows to trade off controller and trajectory optimally and thus permits an improved exploitation of the system capabilities. To achieve this goal, the classic trajectory optimization problem formulation is augmented by stability constraints and feedback gains, with the latter being used as optimization parameters. These stabilize the path-following error dynamics by means of gain scheduling and enforce desired response characteristics. Here, the response is shaped, for once, by enforcing limits on the position of the real and imaginary parts of the eigenvalues as well as by limiting their sensitivity. Especially the latter is a way of robustifying the feedback controller in the time domain. The applicability of the proposed method is shown in an aviation-related scenario by solving a missile guidance problem.

Bioinspired Bare Bones Mayfly Algorithm for Large-Scale Spherical Minimum Spanning Tree.

Mayfly algorithm (MA) is a bioinspired algorithm based on population proposed in recent years and has been applied to many engineering problems successfully. However, it has too many parameters, which makes it difficult to set and adjust a set of appropriate parameters for different problems. In order to avoid adjusting parameters, a bioinspired bare bones mayfly algorithm (BBMA) is proposed. The BBMA adopts Gaussian distribution and Lévy flight, which improves the convergence speed and accuracy of the algorithm and makes better exploration and exploitation of the search region. The minimum spanning tree (MST) problem is a classic combinatorial optimization problem. This study provides a mathematical model for solving a variant of the MST problem, in which all points and solutions are on a sphere. Finally, the BBMA is used to solve the large-scale spherical MST problems. By comparing and analyzing the results of BBMA and other swarm intelligence algorithms in sixteen scales, the experimental results illustrate that the proposed algorithm is superior to other algorithms for the MST problems on a sphere.

Digitized-counterdiabatic quantum approximate optimization algorithm

The quantum approximate optimization algorithm (QAOA) has proved to be an effective classical-quantum algorithm serving multiple purposes, from solving combinatorial optimization problems to finding the ground state of many-body quantum systems. Since QAOA is an ansatz-dependent algorithm, there is always a need to design ansatz for better optimization. To this end, we propose a digitized version of QAOA enhanced via the use of shortcuts to adiabaticity. Specifically, we use a counterdiabatic (CD) driving term to design a better ansatz, along with the Hamiltonian and mixing terms, enhancing the global performance. We apply our digitized-counterdiabatic QAOA to Ising models, classical optimization problems, and the P-spin model, demonstrating that it outperforms standard QAOA in all cases we study.

A New Cooperative Recourse Strategy for Emergency Material Allocation in Uncertain Environments

Emergency material allocation is an important issue in the urgent handling of public health emergencies. This article models the relief allocation and transportation route planning as an uncertain capacitated arc routing problem, which is a classic combinatorial optimization problem that considers stochastic factors such as uncertain demand and travel time in the service. The stochastic of demand leads to the route failure that the vehicle cannot serve the tasks successfully unexpected. Most existing research uses the independent recourse strategy. That is, each vehicle takes a back-and-forth trip separately when its remaining capacity cannot meet the actual demand of the task. This leads to a considerable recourse cost. However, a few studies have considered vehicular cooperation to deal with route failure, which is beneficial for pooling the capacity of multiple vehicles. In this paper, we propose a new recourse strategy called OneFAll that lets one vehicle take charge of all the failed tasks. In this case, other vehicles can finish the service once they are full. We develop the genetic programming hyper-heuristic with the OneFAll recourse strategy for solving the uncertain capacitated arc routing problem. The experimental studies show that our proposed method outperforms the existing genetic programming hyper-heuristic with the independent recourse strategy to the uncertain capacitated arc routing problem for the ugdb and uval benchmark instances. Moreover, our strategy outperforms the recourse strategy that failed tasks are returned to the unassigned task set for any vehicle to complete. This reflects that there exists resource waste if all vehicles are involved to repair the failed routes.

An improved brain storm optimization algorithm with new solution generation strategies for classification

In recent years, brain storm optimization (BSO) algorithm has received much attention in solving classical optimization problems and is used to implement evolutionary classification models. However, in practical applications, large-scale datasets complicate the structure of the classification model, which can have a great impact on the classification performance. In the optimization process, the traditional single-strategy BSO cannot preserve the information of dominant solution well, and its generation strategy is inefficient in solving various complex practical problems. To solve this problem, we introduce feature selection to improve the optimization model structure. Meanwhile, in order to enhance the search capability of BSO, three new generation strategy are embedded in the BSO algorithm in this paper. With the three generation methods of global optimal, local optimal and nearest neighbor, the information of the dominant solution can be better preserved and the search efficiency can be improved. The performance of the proposed generation strategy in solving classification problems is demonstrated on ten datasets with different sizes and dimensions. The experimental results reveal that the new generation strategy can enhance the performance of BSO algorithm for solving classification problems.