Class Of Optimization Problems Research Articles

Optimal solving of a binary knapsack problem on a D-Wave quantum machine and its implementation in production systems

AbstractThe efficient management of complex production systems is a challenge in today’s logistics. In the field of intelligent and sustainable logistics, the optimization of production batches, especially in the context of a rapidly changing product range, requires fast and precise computational solutions. This paper explores the potential of quantum computers for solving these problems. Traditional computational methods are often limited when it comes to optimizing complex logistics systems. In response to these challenges, the paper proposes the use of a hybrid algorithm that combines quantum technologies with classical computational methods. Such integration allows the computational power of both types of technologies to be harnessed, leading to faster and more efficient identification of optimal solutions. In this work, we consider the knapsack problem, a classic NP-hard optimization problem that is commonly used to verify the effectiveness of new algorithm construction methods. The algorithm presented is based on the Branch and Bound method and aims to ensure solution optimality in the context of the non-determinism of quantum computers. Within the algorithm, computations are performed alternately on a classical processor and a quantum processor. In addition, the lower and upper bounds of the objective function are computed in constant time using the D-Wave quantum machine.

On conflict-free cuts: Algorithms and complexity

One way to define the Matching Cut problem is: Given a graph G, is there an edge-cut M of G such that M is an independent set in the line graph of G? We propose the more general Conflict-Free Cut problem: Together with the graph G, we are given a so-called conflict graph Gˆ on the edges of G, and we ask for an edge-cutset M of G that is independent in Gˆ. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is NP-complete even when the maximum degree of G is 5 and Gˆ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of G as a parameter, and we show W[1]-hardness even when G has a feedback vertex set of size one, and the clique cover number of Gˆ is the parameter. Since the clique cover number of Gˆ is an upper bound on the independence number of Gˆ and thus the solution size, this implies W[1]-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT.

Кількісне оцінювання функціональної ефективності мережевих структур за аналізом чутливості оптимізаційної моделі до зовнішніх впливів

The real network organizational structure is constantly under the influence of external influences, which can cause unexpected consequences due to the cascading spread of relevant disturbances directed at critically important parameters. Parametric sensitivity analysis is the latest computational procedure, the results of which provide the researcher with an idea and quantitative assessment of the appropriate response of the structure in the form of an analytical platform as a composition and interaction of scenario analysis, cascade process development and optimization modeling technology. The main task is timely adaptation to expected changes in business analytics, oriented to optimal resources distribution. The generalized minimum cost problem is a typical example of widespread problems of optimal distribution and use of limited, in particular, energy resources, of any nature and purpose, using the scientific management and analysis of decisions, in its network model attention is paid to the potentials of nodes. The maximum flow problem is an example of the active and effective application of methods and models of flows optimization, where the processes and objects used a network organization, the energy systems and complexes investigated, where arc parameters are critical in its model. In the article, for these classic optimization problems, mathematical and spreadsheet models were built on concrete examples. Direct and dual mathematical programming problems were solved, the following stages were carried out - the organization of a computer experiment and the construction of an analytical platform for evaluating the behavior of the network structure under the influence of external influences and disturbances as their consequences. To take into account the specific conditions of real objects with a network structure at the stage of modification, the model supplemented with appropriate restrictions and correction of input data sets. The obtained results should be useful for planning and management personnel and top-management for discussion and decision-making regarding, for example, the optimal placement of energy equipment, the weighted distribution of energy flows in the "source-sink" system or the appointment of executors for work, automated project management using project networks, etc. Keywords: optimization modeling, maximum network flow, minimum cost flows, network organizational structures, decision making, spreadsheet modeling and analytics.

Two-Stage Probe-Based Search Optimization Algorithm for the Traveling Salesman Problems

As a classical combinatorial optimization problem, the traveling salesman problem (TSP) has been extensively investigated in the fields of Artificial Intelligence and Operations Research. Due to being NP-complete, it is still rather challenging to solve both effectively and efficiently. Because of its high theoretical significance and wide practical applications, great effort has been undertaken to solve it from the point of view of intelligent search. In this paper, we propose a two-stage probe-based search optimization algorithm for solving both symmetric and asymmetric TSPs through the stages of route development and a self-escape mechanism. Specifically, in the first stage, a reasonable proportion threshold filter of potential basis probes or partial routes is set up at each step during the complete route development process. In this way, the poor basis probes with longer routes are filtered out automatically. Moreover, four local augmentation operators are further employed to improve these potential basis probes at each step. In the second stage, a self-escape mechanism or operation is further implemented on the obtained complete routes to prevent the probe-based search from being trapped in a locally optimal solution. The experimental results on a collection of benchmark TSP datasets demonstrate that our proposed algorithm is more effective than other state-of-the-art optimization algorithms. In fact, it achieves the best-known TSP benchmark solutions in many datasets, while, in certain cases, it even generates solutions that are better than the best-known TSP benchmark solutions.

Quantum approximate optimization via learning-based adaptive optimization

Combinatorial optimization problems are ubiquitous and computationally hard to solve in general. Quantum approximate optimization algorithm (QAOA), one of the most representative quantum-classical hybrid algorithms, is designed to solve combinatorial optimization problems by transforming the discrete optimization problem into a classical optimization problem over continuous circuit parameters. QAOA objective landscape is notorious for pervasive local minima, and its viability significantly relies on the efficacy of the classical optimizer. In this work, we design double adaptive-region Bayesian optimization (DARBO) for QAOA. Our numerical results demonstrate that the algorithm greatly outperforms conventional optimizers in terms of speed, accuracy, and stability. We also address the issues of measurement efficiency and the suppression of quantum noise by conducting the full optimization loop on a superconducting quantum processor as a proof of concept. This work helps to unlock the full power of QAOA and paves the way toward achieving quantum advantage in practical classical tasks.

A branch, bound and remember algorithm for maximizing the production rate in the simple assembly line balancing problem

The simple assembly line balancing problem is a classical optimization problem. It aims at efficiently assigning elementary operations to the workstations of a paced assembly line to optimize some efficiency measure. In this paper, we deal with the type-2 version of the problem in which the efficiency measure is the maximization of throughput with a given number of workstations. This problem has been scarcely studied compared to its type-1 counterpart, in which the throughput is fixed and the objective is to minimize the number of workstations. This has led to the development of indirect methods using type-1 exact solution methods to tackle the type-2 case.In this paper, we provide lower bounds, logical tests, and dominance rules for the type-2 problem and adapt the state-of-the-art method for the type-1 problem to develop a direct approach for solving the type-2 problem with a direct approach. Computational results show the effectiveness of the direct method, which can outperform recent type-1-based indirect approaches. The proposed method is able to close all instances from the classical instance set and solves to optimality over 90% of the large (100 tasks) and 76.5% of the very large (1000 tasks) instances derived from the current reference set for its type-1 counterpart. The average gaps between the upper and lower bounds on the optimal cycle time are also tight and are less than 0.13% (0.4%) for large (very large) instances. Finally, our results show that direct and indirect methods are comparable in terms of solution quality, but the direct method outperforms its indirect counterpart in terms of running time and optimality gaps. Moreover, we identify that instances averaging two to three tasks per workstation and featuring large variability between task times pose significant challenges to solve to optimality.

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.

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.