Extension of Piyavskii's Algorithm to Continuous Global Optimization

Many real–world problems are modeled as global continuous optimization problems with a nonlinear objective function. Stochastic methods are used to solve these problems approximately, when solving them exactly is impractical. In this class of methods, swarm intelligence (SI) presents metaheuristics that exploit a population of interacting agents able to self–organize, such as ant colony optimization (ACO), particle swarm optimization (PSO), and artificial bee colony (ABC). This paper presents a new SI-based method for solving continuous optimization problems. The new algorithm, called Swarm Random Walk (SwarmRW), is based on a random walk of a swarm of potential solutions. SwarmRW is validated on test functions and compared to PSO and ABC. Results show improved performance on most of the test functions.KeywordsContinuous OptimizationSwarm IntelligenceMeta- heuristic

Many real-life optimization problems often face an increased rank of nonsmoothness (many local minima) which could prevent a search algorithm from moving toward the global solution. Evolution-based algorithms try to deal with this issue. The algorithm proposed in this paper is called GAAPI and is a hybridization between two optimization techniques: a special class of ant colony optimization for continuous domains entitled API and a genetic algorithm (GA). The algorithm adopts the downhill behavior of API (a key characteristic of optimization algorithms) and the good spreading in the solution space of the GA. A probabilistic approach and an empirical comparison study are presented to prove the convergence of the proposed method in solving different classes of complex global continuous optimization problems. Numerical results are reported and compared to the existing results in the literature to validate the feasibility and the effectiveness of the proposed method. The proposed algorithm is shown to be effective and efficient for most of the test functions.

Deterministic branch-and-bound algorithms for continuous global optimization often visit a large number of boxes in the neighborhood of a global minimizer, resulting in the so-called cluster problem (Du and Kearfott in J Glob Optim 5(3):253---265, 1994). This article extends previous analyses of the cluster problem in unconstrained global optimization (Du and Kearfott 1994; Wechsung et al. in J Glob Optim 58(3):429---438, 2014) to the constrained setting based on a recently-developed notion of convergence order for convex relaxation-based lower bounding schemes. It is shown that clustering can occur both on nearly-optimal and nearly-feasible regions in the vicinity of a global minimizer. In contrast to the case of unconstrained optimization, where at least second-order convergent schemes of relaxations are required to mitigate the cluster problem when the minimizer sits at a point of differentiability of the objective function, it is shown that first-order convergent lower bounding schemes for constrained problems may mitigate the cluster problem under certain conditions. Additionally, conditions under which second-order convergent lower bounding schemes are sufficient to mitigate the cluster problem around a global minimizer are developed. Conditions on the convergence order prefactor that are sufficient to altogether eliminate the cluster problem are also provided. This analysis reduces to previous analyses of the cluster problem for unconstrained optimization under suitable assumptions.

Memetic computation (MC) has emerged recently as a new paradigm of efficient algorithms for solving the hardest optimization problems. On the other hand, artificial bees colony (ABC) algorithms demonstrate good performances when solving continuous and combinatorial optimization problems. This study tries to use these technologies under the same roof. As a result, a memetic ABC (MABC) algorithm has been developed that is hybridized with two local search heuristics: the Nelder-Mead algorithm (NMA) and the random walk with direction exploitation (RWDE). The former is attended more towards exploration, while the latter more towards exploitation of the search space. The stochastic adaptation rule was employed in order to control the balancing between exploration and exploitation. This MABC algorithm was applied to a Special suite on Large Scale Continuous Global Optimization at the 2012 IEEE Congress on Evolutionary Computation. The obtained results the MABC are comparable with the results of DECC-G, DECC-G*, and MLCC.

In this article, we introduce a new global optimization method to the field of multiview 3D reconstruction. While global minimization has been proposed in a discrete formulation in form of the maxflow-mincut framework, we suggest the use of a continuous convex relaxation scheme. Specifically, we propose to cast the problem of 3D shape reconstruction as one of minimizing a spatially continuous convex functional. In qualitative and quantitative evaluation we demonstrate several advantages of the proposed continuous formulation over the discrete graph cut solution. Firstly, geometric properties such as weighted boundary length and surface area are represented in a numerically consistent manner: The continuous convex relaxation assures that the algorithm does not suffer from metrication errors in the sense that the reconstruction converges to the continuous solution as the spatial resolution is increased. Moreover, memory requirements are reduced, allowing for globally optimal reconstructions at higher resolutions. We study three different energy models for multiview reconstruction, which are based on a common variational template unifying regional volumetric terms and on-surface photoconsistency. The three models use data measurements at increasing levels of sophistication. While the first two approaches are based on a classical silhouette-based volume subdivision, the third one relies on stereo information to define regional costs. Furthermore, this scheme is exploited to compute a precise photoconsistency measure as opposed to the classical estimation. All three models are compared on standard data sets demonstrating their advantages and shortcomings. For the third one, which gives the most accurate results, a more exhaustive qualitative and quantitative evaluation is presented.

This survey covers the state of the art of techniques for solving general-purpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably find all solutions (if there are finitely many). The core of the material is presented in sufficient detail that the survey may serve as a text for teaching constrained global optimization. After giving motivations for and important examples of applications of global optimization, a precise problem definition is given, and a general form of the traditional first-order necessary conditions for a solution. Then more than a dozen software packages for complete global search are described. A quick review of incomplete methods for bound-constrained problems and recipes for their use in the constrained case follows; an explicit example is discussed, introducing the main techniques used within branch and bound techniques. Sections on interval arithmetic, constrained propagation and local optimization are followed by a discussion of how to avoid the cluster problem. Then a discussion of important problem transformations follows, in particular of linear, convex, and semilinear (= mixed integer linear) relaxations that are important for handling larger problems. Next, reliability issues – centring on rounding error handling and testing methodologies – are discussed, and the COCONUT framework for the integration of the different techniques is introduced. A list of challenges facing the field in the near future concludes the survey.

An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should be portable, easy to learn, use, and maintain, and sufficiently fast for some production work. The motivation, design principles, uses, and capabilities for this environment are outlined. The environment includes an interval data type, a symbolic form of automatic differentiation to obtain an internal representation for functions, a special technique to allow conditional branches with operator overloading and interval computations, and generic routines to give interval and noninterval function and derivative information. Some of these generic routines use a special version of the backward mode of automatic differentiation. The package also includes dynamic data structures for exhaustive search algorithms.

This paper presents the Constructive Cooperative Coevolutionary (mathrm {C}^3) algorithm, applied to continuous large-scale global optimisation problems. The novelty of mathrm {C}^3 is that it utilises a multi-start architecture and incorporates the Cooperative Coevolutionary algorithm. The considered optimisation problem is decomposed into subproblems. An embedded optimisation algorithm optimises the subproblems separately while exchanging information to co-adapt the solutions for the subproblems. Further, mathrm {C}^3 includes a novel constructive heuristic that generates different feasible solutions for the entire problem and thereby expedites the search. In this work, two different versions of mathrm {C}^3 are evaluated on high-dimensional benchmark problems, including the CEC’2013 test suite for large-scale global optimisation. mathrm {C}^3 is compared with several state-of-the-art algorithms, which shows that mathrm {C}^3 is among the most competitive algorithms. mathrm {C}^3 outperforms the other algorithms for most partially separable functions and overlapping functions. This shows that mathrm {C}^3 is an effective algorithm for large-scale global optimisation. This paper demonstrates the enhanced performance by using constructive heuristics for generating initial feasible solutions for Cooperative Coevolutionary algorithms in a multi-start framework.

An evolutionary programming algorithm for continuous global optimization

Nowadays, there are various optimization problems that exact mathematical methods are not applicable. Metaheuristics are considered as efficient approaches for finding the solutions. Yet there are many real-world problems that consist of different properties. For instance, financial portfolio optimization may contain many dimensions for different sets of assets, which suggests the need of a more adaptive metaheuristic method for tackling such problems. However, few existing metaheuristics can achieve robust performance across these variable problems even though they may obtain impressive results in specific benchmark problems. In this paper, a metaheuristic named the Adaptive Multi-Population Optimization (AMPO) is proposed for continuous optimization. The algorithm hybridizes yet modifies several useful operations like mutation and memory retention from evolutionary algorithms and swarm intelligence (SI) techniques in a multi-population manner. Furthermore, the diverse control on multiple populations, solution cloning and reset operation are designed. Compared with other metaheuristics, the AMPO can attain an adaptive balance between the capabilities of exploration and exploitation for various optimization problems. To demonstrate its effectiveness, the AMPO is evaluated on 28 well-known benchmark functions. Also, the parameter sensitivity analysis and search behavior study are conducted. Finally, the AMPO is validated on its applicability through a portfolio optimization problem as a challenging example of real-world applications. The benchmark results show that the AMPO achieves a better performance than those of nine state-of-the-art metaheuristics including the IEEE CEC winning algorithms, recent SI and multi-population/hybrid metaheuristics. Besides, the AMPO can consistently produce a good performance in portfolio optimization.

Computer analysis and simulation based design optimization requires more computationally efficient global optimization tools. In this work, a new global optimization algorithm based on design experiments, region elimination and response surface model, namely Approximated Unimodal Region Elimination Method (AUREM), is introduced. The approach divides the field of interest into several unimodal regions using design experiment data; identify and rank the regions that most likely contain the global minimum; form a response surface model with additional design experiment data over the most promising region; identify its minimum, remove this processed region, and move to the next most promising region. By avoiding redundant searches, the approach identifies the global optimum with reduced number of objective function evaluations and computation effort. The new algorithm was tested using a variety of benchmark global optimization problems and compared with several widely used global optimization algorithms. The experiments results present comparable search accuracy and superior computation efficiency, making the new algorithm an ideal tool for computer analysis and simulation black-box based global design optimization.

Multi-vessel formation shipping demonstrates significant potential for enhancing maritime transportation efficiency and economy. However, existing route planning systems inadequately address the unique challenges of formations, where traditional methods fail to integrate global optimality, local dynamic obstacle avoidance, and formation coordination into a cohesive system. Global planning often neglects multi-ship collaborative constraints, while local methods disregard vessel maneuvering characteristics and formation stability. This paper proposes GLFM, a three-layer hierarchical framework (global optimization–local adjustment-formation collaboration module) for intelligent route planning of transport ship formations. GLFM integrates an improved multi-objective A* algorithm for global path optimization under dynamic meteorological and oceanographic (METOC) conditions and International Maritime Organization (IMO) safety regulations, with an enhanced Artificial Potential Field (APF) method incorporating ship safety domains for dynamic local obstacle avoidance. Formation, structural stability, and coordination are achieved through an improved leader–follower approach. Simulation results demonstrate that GLFM-generated trajectories significantly outperform conventional routes, reducing average risk level by 38.46% and voyage duration by 12.15%, while maintaining zero speed and period violation rates. Effective obstacle avoidance is achieved, with the leader vessel navigating optimized global waypoints and followers maintaining formation structure. The GLFM framework successfully balances global optimality with local responsiveness, enhances formation transportation efficiency and safety, and provides a comprehensive solution for intelligent route optimization in multi-constrained marine convoy operations.

In this work, optimization methodologies of the kinematics of flapping wings in forward and hover flights are considered. Particularly, local and global optimization algorithms are combined with the unsteady vortex lattice method (UVLM) to determine the most efficient kinematics. In the first problem, the kinematic optimization of a three-dimensional flapping wing in forward flight with active shape morphing is aimed at maximizing the propulsive efficiency under lift and thrust constraints. Results show that due to the quasi-concavity of the objective function associated with the flapping kinematics employed in forward flight, local gradient-based optimizers perform very well in terms of accuracy and computational cost. In the second problem, the objective is to identify the optimized kinematics of a twodimensional hovering wing that minimize the aerodynamic power under lift constraint. The results show that using a hybrid optimization algorithm (combination of local and global schemes) accelerates the convergence to optimal points and avoids being trapped at local minimum points. While the efficiency of using a local optimizer, a global optimizer, or a hybrid of the two depends on the nature of the problem and associated design space, it is determined that hybrid optimization is best suited for determining local minima.

Abstract: Inversion of self-potential anomaly for 2-D inclined sheets of infinite horizontal extent has been studied. Least-square inversion and very fast simulated annealing global optimization has been used to model the five parameters of self potential anomaly. The method of least square and very fast simulated annealing global optimization method is compared and analyzed. Very fast simulated annealing can model the noisy and field data of self potential anomaly very precisely than linear inversion technique. However, time taken by very fast simulated annealing inversion is larger than linearized inversion. The comparative analysis has been done on synthetic data (noise free and noisy) and two field data from Bavarian woods anomaly, Germany and Surda anomaly, India to show the efficacy of both the methods. The estimated parameters were compared with those from previous studies using various global optimization algorithms, mainly neural network, genetic algorithm and particle swarm optimization on the same field data sets. It can be concluded that the global optimization algorithms considered in this study were able to yield compatible solutions with those from least-square methods. The present global optimization method is in good agreement with the other global optimization methods in terms of results and computation time.

<div class="section abstract"><div class="htmlview paragraph">Plenum system of automotive vehicles draws atmospheric air using HVAC blower motor through air inlet grille at the bottom of the windshield and uses lower plenum for rainwater drainage after which air flow enters into awning and sugar scoop for evaporator air flow requirement. Awning and sugar scoop component is used in plenum air flow system to ensure the air flow with the required velocity that would meet the performance metric of total pressure drop of plenum air flow system. The optimal design of plenum component is essential since blower motor energy needs to be drawn from the expensive battery pack and any wastage of energy would lead to reduction in range of the Electric Vehicle. The conventional design process utilizes incremental design iteration, engineering intuition and requires significant effort from engineers to meet performance targets within the specified design space. This also may or may not ensure an optimal design.</div><div class="htmlview paragraph">In this work, a Machine learning tool with extreme gradient boost algorithm is used for development of a robust meta model that is used for quick prediction of awning and sugar scoop performance parameters such as total pressure drop and velocity of air prediction. On this response surface, application of a meta heuristic [<span class="xref">1</span>] based global optimization method is demonstrated which uses simplex homological global optimization (SHGO) and differential evolutionary (DE) method for optimum awning and sugar scoop design. Simplex homological global optimization (SHGO) is the concept of homology group growth which is used to solve a global and derivative free optimization problem of arbitrarily high dimensions. Similarly Differential evolution (DE) is a population-based metaheuristic search algorithm [<span class="xref">2</span>] that optimizes a problem which can quickly explore very large design spaces. Using these global optimization algorithms, it is observed that the optimized design of the awning and sugar scoop does result in significant reduction of total pressure drop that helps improve the required performance substantially compared to baseline design parameters within the given design space.</div></div>