Class Of Optimization Problems Research Articles

Process control of chemical dynamic system based on multi-strategy mayfly optimization algorithm

Dynamic optimization problems exist widely in chemical industry, and its operational variables change with the evolution of both space and time. Therefore, dynamic optimization problems have important research significance and challenges. To solve this problem, a multi-strategy mayfly optimization algorithm (MMOA) combined with control variable parameterization method(CVP) is proposed in this paper. MMOA introduces three improvements on the basis of the original algorithm, namely, circle chaos crossover strategy, center wandering strategy and boundary correction strategy. The hybrid strategy can better balance the exploration and exploitation ability of the algorithm. Based on MATLAB simulation environment, MMOA was evaluated. The experimental results show that MMOA has excellent performance in solving precision, convergence speed and stability for the benchmark function. For the six classical chemical dynamic optimization problems, MMOA obtained the performance indexes of 0.61071, 0.4776, 0.57486, 0.73768, 0.11861 and 0.13307, respectively. Compared with the data in the previous literature, MMOA can obtain more accurate control trajectory and better performance indicators. It provides an effective way to solve the dynamic optimization problem.

Covariance matrix adaptation evolution strategy based on ensemble of mutations for parking navigation and maneuver of autonomous vehicles

Previous works have demonstrated the use of human-inspired frameworks to address autonomous vehicle parking. In such frameworks, autonomous vehicle parking is formulated as a two-stage optimization problem where the first stage involves the generation of initial waypoints which are consequently used to reduce the autonomous parking task into a local motion planning task. In the second stage, the local motion planning is solved as a classical optimization problem. However, the overall algorithmic performance in the second stage depends on the planning horizon used to formulate the local planning task. In this work, we show first that the state-of-the-art algorithms often employed in the second stage fail with an increase in the planning horizon. Motivated by the need for algorithms that are scalable to different planning horizons, this work proposes Covariance Matrix Adaptation Evolution Strategy (CMA-ES) with an ensemble of mutations for the second stage. Specifically, the proposed algorithm features an ensemble of Gaussian- and Cauchy-based mutations to facilitate an efficient blend of both exploitation and exploration that is crucial for both short and long horizons. Furthermore, to handle the associated constraints, Superiority of Feasible (SF) constraint handling technique is incorporated into the proposed algorithm. Performance analysis based on 24 parking missions conducted with planning horizons of 2 and 4 shows that the proposed algorithm is scalable to different planning horizons compared with three commonly employed state-of-the-art algorithms.

Exact and Heuristic Solution Techniques for Mixed-Integer Quantile Minimization Problems

We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important real-world problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop valid inequalities and present an overlapping alternating direction method. Moreover, we discuss an adaptive scenario clustering method for which we prove that it terminates after a finite number of iterations with a global optimal solution. We study the computational impact of all presented techniques and finally show that their combination leads to an overall method that can solve the maintenance planning problem on large-scale real-world instances provided by the ROADEF/EURO challenge 2020 1 and that they also lead to significant improvements when solving a quantile-version of the classic portfolio optimization problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [CRC TRR 154], Fonds De La Recherche Scientifique [PDR T0098.18], and Bundesministerium für Bildung und Forschung. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0105 .

Chaotic Binarization Schemes for Solving Combinatorial Optimization Problems Using Continuous Metaheuristics

Chaotic maps are sources of randomness formed by a set of rules and chaotic variables. They have been incorporated into metaheuristics because they improve the balance of exploration and exploitation, and with this, they allow one to obtain better results. In the present work, chaotic maps are used to modify the behavior of the binarization rules that allow continuous metaheuristics to solve binary combinatorial optimization problems. In particular, seven different chaotic maps, three different binarization rules, and three continuous metaheuristics are used, which are the Sine Cosine Algorithm, Grey Wolf Optimizer, and Whale Optimization Algorithm. A classic combinatorial optimization problem is solved: the 0-1 Knapsack Problem. Experimental results indicate that chaotic maps have an impact on the binarization rule, leading to better results. Specifically, experiments incorporating the standard binarization rule and the complement binarization rule performed better than experiments incorporating the elitist binarization rule. The experiment with the best results was STD_TENT, which uses the standard binarization rule and the tent chaotic map.

Exploratory Dividend Optimization with Entropy Regularization

This study investigates the dividend optimization problem in the entropy regularization framework in the continuous-time reinforcement learning setting. The exploratory HJB is established, and the optimal exploratory dividend policy is a truncated exponential distribution. We show that, for suitable choices of the maximal dividend-paying rate and the temperature parameter, the value function of the exploratory dividend optimization problem can be significantly different from the value function in the classical dividend optimization problem. In particular, the value function of the exploratory dividend optimization problem can be classified into three cases based on its monotonicity. Additionally, numerical examples are presented to show the effect of the temperature parameter on the solution. Our results suggest that insurance companies can adopt new exploratory dividend payout strategies in unknown market environments.

An efficient solution space exploring and descent method for packing equal spheres in a sphere

The problem of packing equal spheres in a spherical container is a classic global optimization problem, which has attracted enormous studies in academia and found various applications in industry. This problem is computationally very challenging, and many efforts focus on small-scale instances with the number of spherical items less than 200 in the literature. In this work, we propose an efficient local search heuristic algorithm named solution space exploring and descent for solving this problem, which can quantify the solution’s quality to determine the number of exploring actions and quickly discover a high-quality configuration. Besides, we propose an adaptive neighbor item maintenance method to speed up the convergence of the continuous optimization process and reduce the time consumption. Computational experiments on a large number of benchmark instances with 5≤n≤400 spherical items show that our algorithm significantly outperforms the state-of-the-art algorithm. Specifically, our algorithm improves 274 best-known results and matches 84 best-known results out of the 396 well-known benchmark instances.

Chance-Constrained Multiple-Choice Knapsack Problem: Model, Algorithms, and Applications.

The multiple-choice knapsack problem (MCKP) is a classic NP-hard combinatorial optimization problem. Motivated by several significant real-world applications, this work investigates a novel variant of MCKP called the chance-constrained MCKP (CCMCKP), where item weights are random variables. In particular, we focus on the practical scenario of CCMCKP, in which the probability distributions of random weights are unknown and only sample data is available. We first present the problem formulation of CCMCKP and then establish the two benchmark sets. The first set contains synthetic instances, while the second set is designed to simulate a real-world application scenario of a telecommunication company. To solve CCMCKP, we propose a data-driven adaptive local search (DDALS) algorithm. Compared to existing stochastic optimization and distributionally robust optimization methods, the main novelty of DDALS lies in its data-driven solution evaluation approach, which does not make any assumptions about the underlying distributions and is highly effective even when faced with a high intensity of the chance constraint and a limited amount of sample data. Experimental results demonstrate the superiority of DDALS over the baselines on both the benchmarks. Finally, DDALS can serve as the baseline for future research, and the benchmark sets are open-sourced to further promote research on this challenging problem.

Deep learning for mean field optimal transport

Mean field control (MFC) problems have been introduced to study social optima in very large populations of strategic agents. The main idea is to consider an infinite population and to simplify the analysis by using a mean field approximation. These problems can also be viewed as optimal control problems for McKean-Vlasov dynamics. They have found applications in a wide range of fields, from economics and finance to social sciences and engineering. Usually, the goal for the agents is to minimize a total cost which consists in the integral of a running cost plus a terminal cost. In this work, we consider MFC problems in which there is no terminal cost but, instead, the terminal distribution is prescribed as in optimal transport problem. By analogy with MFC, we call such problems mean field optimal transport problems (or MFOT for short) since they can be viewed as a generalization of classical optimal transport problems when mean field interactions occur in the dynamics or the running cost function. We propose three numerical methods based on neural networks. The first one is based on directly learning an optimal control. The second one amounts to solve a forward-backward PDE system characterizing the solution. The third one relies on a primal-dual approach. We illustrate these methods with numerical experiments conducted on two families of examples.

Deep Graph Reinforcement Learning for Solving Multicut Problem.

The multicut problem, also known as correlation clustering, is a classic combinatorial optimization problem that aims to optimize graph partitioning given only node (dis)similarities on edges. It serves as an elegant generalization for several graph partitioning problems and has found successful applications in various areas such as data mining and computer vision. However, the multicut problem with an exponentially large number of cycle constraints proves to be NP-hard, and existing solvers either suffer from exponential complexity or often give unsatisfactory solutions due to inflexible heuristics driven by hand-designed mechanisms. In this article, we propose a deep graph reinforcement learning method to solve the multicut problem within a combinatorial decision framework involving sequential edge contractions. The customized subgraph neural network adapts to the dynamically edge-contracted graph environment by extracting bilevel connected features from both contracted and original graphs. Our method can learn to infer feasible multicut solutions end-to-end toward optimization of the multicut objective in a data-driven manner. More specifically, by exploring the decision space adaptively, it implicitly gains heuristic knowledge from topological patterns of instances and thereby generates more targeted heuristics overcoming the short-sightedness inherent in the hand-designed ones. During testing, the learned heuristics iteratively contract graphs to construct high-quality solutions within polynomial time. Extensive experiments on synthetic and real-world multicut instances show the superiority of our method over existing combinatorial solvers, while also maintaining a certain level of out-of-distribution generalization ability.

Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey.

Optimal Transport (OT) is a mathematical framework that first emerged in the eighteenth century and has led to a plethora of methods for answering many theoretical and applied questions. The last decade has been a witness to the remarkable contributions of this classical optimization problem to machine learning. This paper is about where and how optimal transport is used in machine learning with a focus on the question of scalable optimal transport. We provide a comprehensive survey of optimal transport while ensuring an accessible presentation as permitted by the nature of the topic and the context. First, we explain the optimal transport background and introduce different flavors (i.e. mathematical formulations), properties, and notable applications. We then address the fundamental question of how to scale optimal transport to cope with the current demands of big and high dimensional data. We conduct a systematic analysis of the methods used in the literature for scaling OT and present the findings in a unified taxonomy. We conclude with presenting some open challenges and discussing potential future research directions. A live repository of related OT research papers is maintained in https://github.com/abdelwahed/OT_for_big_data.git.

An improved slope-based adaptive control vector parameterization method for dynamic programming

Control vector parameterization method is the most commonly used numerical computation method in solving dynamic programming problems. However, to balance the expected trajectory and the computational cost, this method faces difficulties in dividing the optimal discretization time grid. In this paper, we propose an effective control approach for non-uniform adaptive grid division. By analyzing the slope change trend of the control parameter, time nodes are adaptively refined by merging time grids with gentle slopes to remove unnecessary time nodes and adding time nodes to time grids with steep slopes to improve function approximation accuracy. Eventually, we obtain an adaptive time grid division method under which the trajectory of discretization control parameter is more approximate to the optimal control trajectory. Under the condition of intuitive slope information, the proposed method can achieve better performance with fewer optimization parameters and shorter computation time. Finally, the proposed method is applied to solve a classic optimal control problem and the obtained results are compared with the traditional control vector parameterization method. By comparison through the example, our proposed method can overcome the contradiction between approximation accuracy and computational cost in control vector parameterization method and improve the optimization efficiency.

Walrus optimizer: A novel nature-inspired metaheuristic algorithm

Metaheuristic algorithms are intelligent optimization approaches that lead the searching procedure through utilizing exploitation and exploration. The increasing complexity of real-world optimization problem has prompted the development of more metaheuristic algorithms. Hence, this work proposes a novel swarm intelligence algorithm, Walrus optimizer (WO). It is inspired by the behaviors of walruses that choose to migrate, breed, roost, feed, gather and escape by receiving key signals (danger signals and safety signals). To test the capability of the proposed algorithm, 23 standard functions and the benchmark suite from the IEEE (Institute of Electrical and Electronics Engineers) Congress on Evolutionary Computation (CEC) 2021 are used. In addition, to evaluate the practicability of the proposed algorithm to solve various real-world optimization problems, 6 standard classical engineering optimization problems are examined and compared. For statistical purposes, 100 independent optimization runs are conducted to determine the statistical measurements, including the mean, standard deviation, and the computation time of the program, by considering a predefined stopping criterion. Some well-known statistical analyses are also used for comparative purposes, including the Friedman and Wilcoxon analysis. The results demonstrate that the proposed algorithm can provide special stability features and very competitive performance in dealing with high-dimensional benchmarks and real-world problems. The proposal of WO promotes the continuous development and application expansion of artificial intelligence, improves the efficiency of optimization calculation, and provides powerful tools for solving complex problems in the real world. The source code of WO is publicly availabe at https://ww2.mathworks.cn/matlabcentral/fileexchange/154702-walrus-optimizer-wo.

A unified exact approach for a broad class of vehicle routing problems with simultaneous pickup and delivery

The vehicle routing problem (VRP) with simultaneous pickup and delivery (VRPSPD) is a classical combinatorial optimization problem in which one aims at determining least-cost routes, starting and ending at the depot, while simultaneously meeting the pickup and delivery demands of geographically dispersed customers in a single visit, and without violating the capacity of the vehicle. In this work, we propose a unified branch-cut-and-price (BCP) algorithm to solve ten variants of the problem, considering not only the standard version but also those including additional attributes such as a heterogeneous fleet of vehicles, time windows, route duration, multiple depots, and location decisions. The resulting problem that generalizes all of these variants can be formulated as a heterogeneous location routing problem with simultaneous pickup and delivery and time windows (HLRPSPDTW), for which we propose a compact formulation to serve as the basis for an extended formulation associated with the BCP approach. The proposed exact method includes many of the features developed in recent years, such as a bucket graph-based labeling algorithm, rank-1 cuts with limited memory, and dynamic ng-sets. Extensive computational experiments were performed on more than 550 benchmark instances and our algorithm was capable of obtaining a number of new optimal solutions, as well as improved lower bounds, for all variants addressed in this paper.

A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique

In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.

Optimal instant discounts of multiple ride options at a ride-hailing aggregator

Recently, ride-hailing aggregators have emerged to help passengers find the best ride options. It aggregates the results from different service providers, allowing sorting and filtering all in a single app. It is not unusual for an aggregator to offer instant discounts on the prices returned from the service providers to increase its Gross Transaction Value (GTV) or Total Transaction Volume (TTV). In this research, we study this optimal instant discount problem, which can be viewed as a new variant of the classical multi-product price optimization problem. We first use the Nested Logit model to predict the probability a passenger would complete the trip with each ride option. We then formulate the instant discount problem as a nonlinear optimization model with a budget constraint and a group of discount bound constraints. To solve this model, we construct a surrogate relaxation formulation with strong duality. We develop a Lagrangian-dual-based approach to decompose this problem into a series of subproblems, and then design heuristic methods to give feasible solutions. For both cases when the GTV or the TTV is maximized, we quantify the optimality gap, give its asymptotic properties, and establish conditions under which it becomes zero and tight. Finally, we use real data from Meituan, a leading ride-hailing aggregator in China, to validate the proposed approach. Results show that compared to the baseline methods, we can improve the GTV by 1.293%, improve the TTV by 0.475%, and decrease the magnitude of the optimality gap.

A shape parameter insensitive CRBFs‐collocation for solving nonlinear optimal control problems

AbstractIn this work, we introduce a Coupled Radial Basis Functions collocation (CRBFs‐Coll) approach for solving optimal control problems. CRBFs are real‐valued Radial Basis Functions (RBFs) augmented with a conical spline. They show insensitivity to the shape parameter resulting in robust behavior in function approximation. The method is applied to classical nonlinear optimal control problems: Zermelo's problem, a Duffing oscillator with various boundary conditions, and a nonlinear pendulum on a cart problem. The nonlinear optimal control problem is solved by deriving the necessary conditions for optimality followed by collocation using CRBFs as basis functions for approximating the two‐point boundary value problem (TPBVP). The resulting system of nonlinear algebraic equations (NAEs) is then solved using a standard solver. Unlike existing methods that rely on nodal distributions that are denser at the boundaries to obtain a solution, the present CRBFs‐Coll approach is capable of solving the optimal control problem on uniform nodes without the need for interpolation. The present CRBFs‐Coll approach is shown to be simple to implement and provides accurate results over an equidistant nodal distribution while maintaining a continuous representation of the control and states.

A mini-batch algorithm for large-scale learning problems with adaptive step size

The step size selection in stochastic optimization methods is a crucial aspect that holds significant importance in theoretical analysis and practical applications. We propose two stochastic optimization methods based on the competitive Barzilai-Borwein (BB) step size in both the inner and outer loops of the mini-batch semi-stochastic gradient descent (mS2GD) algorithm. The competitive BB step size is automatically updated using the latest and most accurate secant equation. We introduce two algorithms: mS2GD-CBB, which updates the step size in the outer loop, and mS2GD-RCBB, which updates the step size in the inner loop. We evaluate the performance of the proposed algorithms on the classical optimization problem and compare them against existing methods. Experimental results demonstrate that the methods exhibit favorable convergence properties and can effectively handle the challenges of big data arising from the fields of signal processing, statistics, and machine learning. The methods offer enhanced adaptability, flexibility, and exploration capabilities, making them promising approaches for a wide range of optimization problems.

A Privacy-Preserving Service Framework for Traveling Salesman Problem-Based Neural Combinatorial Optimization Network

The traveling salesman problem (TSP) is one of the classic combinatorial optimization problems, which can be widely used in intelligent transportation and logistics field. Neural network has shown great potential in combinatorial optimization tasks. However, it faces privacy leakage when a TSP neural combinatorial optimization network and user's data are directly outsourced to a cloud platform to provide and request service. In order to address the issue, this paper proposes a privacy-preserving service framework for a TSP-based neural combinatorial optimization network called PPSF. We first protect the service provider's model parameters and users' data by different split methods in multiple cloud servers, providing a secure outsourced mode for the participators in the PPSF framework. Then, in the secure outsourced mode, a series of secure computation protocols are designed for the cloud servers to support performing the secure computing in each service task. Moreover, it can also protect the process that the cloud servers respond to users after data processing and achieve private service result recovery. Finally, we prove that the proposed framework can realize privacy protection for TSP-based combinatorial optimization service and verify its utility and efficiency by experiments.

Diabatic quantum annealing for the frustrated ring model

Quantum annealing (QA) is a continuous-time heuristic quantum algorithm for solving or approximately solving classical optimization problems. The algorithm uses a schedule to interpolate between a driver Hamiltonian with an easy-to-prepare ground state and a problem Hamiltonian whose ground state encodes solutions to an optimization problem. The standard implementation relies on the evolution being adiabatic: keeping the system in the instantaneous ground state with high probability and requiring a time scale inversely related to the minimum energy gap between the instantaneous ground and excited states. However, adiabatic evolution can lead to evolution times that scale exponentially with the system size, even for computationally simple problems. Here, we study whether non-adiabatic evolutions with optimized annealing schedules can bypass this exponential slowdown for one such class of problems called the frustrated ring model. For sufficiently optimized annealing schedules and system sizes of up to 39 qubits, we provide numerical evidence that we can avoid the exponential slowdown. Our work highlights the potential of highly-controllable QA to circumvent bottlenecks associated with the standard implementation of QA.

Global optimization and structural analysis of Coulomb and logarithmic potentials on the unit sphere using a population-based heuristic approach

Global optimization of high-dimensional non-convex functions arises in many important applications and researches. In this paper, we focus on the minimum energy configuration problem on the unit sphere, whose goal is to determine the minimum-energy configuration of N particles interacting via the standard Coulomb potential or the discrete logarithmic potential on the unit sphere. This is a classic global optimization problem with many important applications in mathematics, physics, biology and chemistry, and is one of the 18 great mathematical problems for the twenty-first century listed by the Fields Medal winner (Smale, 1998). To determine the minimum-energy configuration for these two potential functions, we propose a population-based global optimization algorithm that combines a two-phase local optimization method with early-stopping strategy, a random perturbation operator, a distance-based population update strategy using a novel distance function to measure the difference between solutions, and a population rebuilding method. Computational results show that the proposed algorithm is very competitive compared to the best known results in the literature. In particular, it improves the best-known result for one instance of the classic Thomson problem (i.e., N=360) widely studied in the literature. The configurations of instances with up to N=500 particles are systematically optimized for the first time by the proposed algorithm for the discrete logarithmic potential. The comparative study shows that the putative optimal configurations have a high similarity for the Coulomb and logarithmic potentials and that the defects play an important role in reducing the energy of configuration. The energies of particles on the putative optimal configurations are analyzed for the studied logarithmic potential.