Multi-start Strategy Research Articles

Optimal design-for-control of self-cleaning water distribution networks using a convex multi-start algorithm

The provision of self-cleaning velocities has been shown to reduce the risk of discolouration in water distribution networks (WDNs). Despite these findings, control implementations continue to be focused primarily on pressure and leakage management. This paper considers the control of diurnal flow velocities to maximize the self-cleaning capacity (SCC) of WDNs. We formulate a new optimal design-for-control problem where locations and operational settings of pressure control and automatic flushing valves are jointly optimized. The problem formulation includes a nonconvex objective function, nonconvex hydraulic conservation law constraints, and binary variables for modelling valve placement, resulting in a nonconvex mixed integer nonlinear programming (MINLP) optimization problem. Considering the challenges with solving nonconvex MINLP problems, we propose a heuristic algorithm which combines convex relaxations (with domain reduction), a randomization technique, and a multi-start strategy to compute feasible solutions. We evaluate the proposed algorithm on case study networks with varying size and degrees of complexity, including a large-scale operational network in the UK. The convex multi-start algorithm is shown to be a more robust solution method compared to an off-the-shelf genetic algorithm, finding good-quality feasible solutions to all design-for-control numerical experiments. Moreover, we demonstrate the implemented multi-start strategy to be a fast and scalable method for computing feasible solutions to the nonlinear SCC control problem. The proposed method extends the control capabilities and benefits of dynamically adaptive networks to improve water quality in WDNs.

A stepped tabu search method for the clique partitioning problem

Given an undirected graph, a clique is a subset of vertices in which the induced subgraph is complete; that is, all pairs of vertices of this subset are adjacent. Clique problems in graphs are very important due to their numerous applications. One of these problems is the clique partitioning problem (CPP), which consists of dividing the set of vertices of a graph into the smallest number of cliques possible. The CPP is an NP-hard problem with many application fields (timetabling, manufacturing, scheduling, telecommunications, etc.). Despite its great applicability, few recent studies have focused on proposing specific resolution methods for the CPP. This article presents a resolution method that combines multistart strategies with tabu search. The most novel characteristic of our method is that it allows unfeasible solutions to be visited, which facilitates exploration of the solution space. The computational tests show that our method performs better than previous methods proposed for this problem. In fact, our method strictly improves the results of these methods in most of the instances considered while requiring less computation time.

Hierarchical model predictive vibration control for high‐rise structures using semi‐active tuned liquid column damper

Vibration control for high-rise structures has become more important as buildings become more lightweight. Tuned liquid column dampers are an established approach for this problem, as they do not consume high amounts of energy while achieving high damping forces. The natural frequency of tuned liquid column dampers (TLCDs) can be changed semi-actively by modifying their column geometry. As shown by previous studies, this approach increases the control performance of the damper and allows a larger application field. However, this modification simultaneously provokes strong nonlinearities, which need to be addressed for controller design. In this contribution a model-based controller for such TLCDs is proposed, which explicitly accounts for the nonlinear dynamics of the coupled system consisting of a multi-storey structure with a semi-active TLCD. To account for various system dynamics of the structure, a hierarchical nonlinear model predictive controller (NMPC) in combination with a multi-start strategy is formulated. In addition, an extrapolation algorithm is developed to determine an approximation of the future trajectory of the structure's induced excitation load. The excitation trajectory is then incorporated in the NMPC framework. For validation, a frequency response analysis and a case study are conducted on a 20-storey benchmark structure, which is equipped with a semi-active TLCD. The study is carried out in simulation, and the results are compared with passive vibration control. With the proposed hierarchical control, a significant improvement is shown in the reduction of structural vibrations over the relevant spectrum of excitation frequencies up to a mean of 50%.

Multi-Fidelity Model-Based Size Optimization of Electric Machines

The multi-modal problem and high computational cost represent challenges in the optimization of electric machines owing to their highly nonlinear electromagnetic response. To overcome these challenges, this paper proposes a multi-fidelity model-based sequential optimization method in which both low- and high-fidelity models are employed in two phases. In phase 1, the reluctance network (RN) is adopted as the low-fidelity model and mainly contributes to alleviating the abovementioned challenges. To overcome the low accuracy of the RN, the optimal design is obtained using a finite element model (FEM) in phase 2. The multi-start strategy and gradient-based algorithm are utilized instead of a heuristic algorithm in all phases to avoid excessive calculations. This multi-fidelity model concept presents a novelty compared to previous research that focused only on algorithm development. The effectiveness of the proposed method is validated with two examples, consisting of the TEAM workshop problem 25 and the torque ripple minimization of an interior permanent magnet synchronous motor. The optimal designs and computational time resulting from the proposed method are compared with those of the conventional method, where only a FEM is used during optimization. The results show that the proposed method is remarkable in finding superior optimal designs, while ignoring unimportant local optima. Additionally, it can save up to 90% of the computational time required by the conventional method.

Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems

Characterisation and quantification of nonlinearities in the engineering structures include selecting and fitting a good mathematical model to a set of experimental vibration data with significant nonlinear features. These tasks involve solving an optimisation problem where it is difficult to choose a priori the best optimisation technique. This paper presents a systematic comparison of ten optimisation methods used to select the best nonlinear model and estimate its parameters through nonlinear system identification. The model selection framework fits the structure’s equation of motions using time-domain dynamic response data and takes into account couplings due to the presence of the nonlinearities. Three benchmark problems are used to evaluate the performance of two families of optimisation methods: (i) deterministic local searches and (ii) global optimisation metaheuristics. Furthermore, hybrid local–global optimisation methods are examined. All benchmark problems include a free play nonlinearity commonly found in mechanical structures. Multiple performance criteria are considered based on computational efficiency and robustness, that is, finding the best nonlinear model. Results show that hybrid methods, that is, the multi-start strategy with local gradient-based Levenberg–Marquardt method and the particle swarm with Levenberg–Marquardt method, lead to a successful selection of nonlinear models and an accurate estimation of their parameters within acceptable computational times.

DEFT-FUNNEL: an open-source global optimization solver for constrained grey-box and black-box problems

The fast-growing need for grey-box and black-box optimization methods for constrained global optimization problems in fields such as medicine, chemistry, engineering and artificial intelligence, has led to the development of new efficient algorithms for finding the best possible solution. In this work, we present DEFT-FUNNEL, an open-source global optimization algorithm for general constrained grey-box and black-box problems that belongs to the class of trust-region sequential quadratic optimization algorithms. Polynomial interpolation models are used as surrogates for the black-box functions and a clustering-based multistart strategy is applied for searching for the global minima. Numerical experiments show that DEFT-FUNNEL compares favorably with state-of-the-art methods on two sets of benchmark problems: one set containing problems where every function is a black box and another set with problems where some of the functions and their derivatives are known to the solver. The code as well as the test sets used for experiments are available at the Github repository http://github.com/phrsampaio/deft-funnel.

Multistart solution-based tabu search for the Set-Union Knapsack Problem

The NP-hard Set-Union Knapsack Problem is a general model able to formulate a number of practical problems. As a variant of the popular knapsack problem, SUKP is to find a subset of candidate items (an item is composed of several distinct weighted elements) such that a profit function is maximized while a knapsack capacity constraint is satisfied. We investigate for the first time a multistart solution-based tabu search algorithm for solving the problem. The proposed algorithm combines a solution-based tabu search procedure with a multistart strategy to ensure an effective examination of candidate solutions. We report computational results on 60 benchmark instances from the literature, including new best results (improved lower bounds) for 7 large instances. We show additional experiments to shed lights on the roles of the key composing ingredients of the algorithm. The code of the algorithm will be publicly available.

Probabilistic definition of the clinical target volume—implications for tumor control probability modeling and optimization

Evidence has been presented that moving beyond the binary definition of clinical target volume (CTV) towards a probabilistic CTV can result in better treatment plans. The probabilistic CTV takes the likelihood of disease spread outside of the gross tumor into account. An open question is: how to optimize tumor control probability (TCP) based on the probabilistic CTV. We derive expressions for TCP under the assumptions of voxel independence and dependence. For the dependent case, we make the assumption that tumors grow outward from the gross tumor volume. We maximize the (non-convex) TCP under convex dose constraints for all models. For small numbers of voxels, and when a dose-influence matrix is not used, we use exhaustive search or Lagrange multiplier theory to compute optimal dose distributions. For larger cases we present (1) a multi-start strategy using linear programming with a random cost vector to provide random feasible starting solutions, followed by a local search, and (2) a heuristic strategy that greedily selects which subvolumes to dose, and then for each subvolume assignment runs a convex approximation of the optimization problem. The optimal dose distributions are in general different for the independent and dependent models even though the probabilities of each voxel being tumorous are set to the same in both cases. We observe phase transitions, where a subvolume is either dosed to a high level, or it gets ‘sacrificed’ by not dosing it at all. The greedy strategy often yields solutions indistinguishable from the multi-start solutions, but for the 2D case involving organs-at-risk and the dependent TCP model, discrepancies of around 5% (absolute) for TCP are observed. For realistic geometries, although correlated voxels is a more reasonable assumption, the correlation function is in general unknown. We demonstrate a tractable heuristic that works very well for the independent models and reasonably well for the dependent models. All data are provided.

Optimized Layout of Spherical Objects in a Polyhedral Domain

Introduction. The article studies the problem of arranging spherical objects in a bounded polyhedral domain in order to maximize the packing factor. The spherical objects have variable placement parameters and variable radii within the given upper and lower bounds. The constraints on the allowable distance between each pair of spherical objects are taken into account. The phi-function technique is used for analytical description of the placement constraints, involving object non-overlapping and containment conditions. The problem is considered as a nonlinear programming problem. The feasible region is described by a system of inequalities with differentiable functions. To find the local maximum of the problem the decomposition algorithm is used. We employ the strategy of active set of inequalities for reducing the computational complexity of the algorithm. IPOPT solver for solving nonlinear programming subproblems is used. The multistart strategy allows selecting the best local maximum point. Numerical results and the appropriate graphic illustration are given. The purpose of the article is presenting a mathematical model and developing a solution algorithm for arranging spherical objects in a polyhedral region with the maximum packing factor. It allows obtaining a locally optimal solution in a reasonable time. Results. A new formulation of the problem of arranging spherical objects in a polyhedral domain is considered, where both the placement parameters and the radii of the spherical objects are variable. A mathematical model in the form of nonlinear programming problem is derived. A solution approach based on the decomposition algorithm and multistart strategy is developed. The numerical results combined with the graphical illustration are given. Conclusions. The proposed approach allows modeling optimized layouts of spherical objects into a polyhedral domain. Keywords: layout, spherical objects, polyhedral domain, phi-function.

Batch Bayesian optimization via adaptive local search

Bayesianoptimization (BO) provides an efficient tool for solving the black-box global optimization problems. Under situations where multiple points can be evaluated simultaneously, batch Bayesian optimization has been a popular extension by taking full use of the computational and experimental resources. In this paper, an adaptive local search strategy is investigated to select batch points for Bayesian optimization. First, multi-start strategy and gradient-based optimization method are combined to maximize the acquisition function. Then, an automatic cluster approach (e.g., X-means) is applied to adaptively identify the acquisition function’s local maxima from the gradient-based optimization results. Third, the Bayesian stopping criterion is utilized to guarantee all the local maxima can be obtained theoretically. Moreover, the lower bound confidence criterion and frontend truncation operation are employed to select the most promising local maxima as batch points. Extensive evaluations on various synthetic functions and two hyperparameter tuning problems for deep learning models are utilized to verify the proposed method.

Optimized packing multidimensional hyperspheres: a unified approach.

In this paper an optimized multidimensional hyperspheres packing problem (HPP) is considered for a bounded container. Additional constraints, such as prohibited zones in the container or minimal allowable distances between spheres can also be taken into account. Containers bounded by hyper- (spheres, cylinders, planes) are considered. Placement constraints (non-intersection, containment and distant conditions) are formulated using the phi-function technique. A mathematical model of HPP is constructed and analyzed. In terms of the general typology for cutting & packing problems, two classes of HPP are considered: open dimension problem (ODP) and knapsack problem (KP). Various solution strategies for HPP are considered depending on: a) objective function type, b) problem dimension, c) metric characteristics of hyperspheres (congruence, radii distribution and values), d) container's shape; e) prohibited zones in the container and/or minimal allowable distances. A solution approach is proposed based on multistart strategies, nonlinear programming techniques, greedy and branch-and-bound algorithms, statistical optimization and homothetic transformations, as well as decomposition techniques. A general methodology to solve HPP is suggested. Computational results for benchmark and new instances are presented.

Parallel Computing Technologies for Solving Optimization Problems of Geometric Design

This paper describes the application of parallel computing technologies in systems with shared and distributed memory for solving optimization problems of geometric design. The first technology is based on the maximin properties of phi-functions for composite objects, and the second technology uses the multistart strategy and methods for minimizing nonsmooth functions. This allowed to several times reduce time expenditures for searching for locally optimum placements of 2D and 3D objects and to obtain better results as for the objective function value.

Adaptive Clustering via Symmetric Nonnegative Matrix Factorization of the Similarity Matrix

The problem of clustering, that is, the partitioning of data into groups of similar objects, is a key step for many data-mining problems. The algorithm we propose for clustering is based on the symmetric nonnegative matrix factorization (SymNMF) of a similarity matrix. The algorithm is first presented for the case of a prescribed number k of clusters, then it is extended to the case of a not a priori given k. A heuristic approach improving the standard multistart strategy is proposed and validated by the experimentation.

Optimization of the Multi-Start Strategy of a Direct-Search Algorithm for the Calibration of Rainfall–Runoff Models for Water-Resource Assessment

Calibration of conceptual rainfall–runoff models (CRRM) for water-resource assessment (WRA) is a complicated task that contributes to the reliability of results obtained from catchments. In recent decades, the application of automatic calibration techniques has been frequently used because of the increasing complexity of models and the considerable time savings gained at this phase. In this work, the traditional Rosenbrock (RNB) algorithm is combined with a random sampling method and the Latin hypercube (LH) to optimize a multi-start strategy and test the efficiency in the calibration of CRRMs. Three models (the French rural-engineering-with-four-daily-parameters (GR4J) model, the Swedish Hydrological Office Water-balance Department (HBV) model and the Sacramento Soil Moisture Accounting (SAC-SMA) model) are selected for WRA at nine headwaters in Spain in zones prone to long and severe droughts. To assess the results, the University of Arizona’s shuffled complex evolution (SCE-UA) algorithm was selected as a benchmark, because, until now, it has been one of the most robust techniques used to solve calibration problems with rainfall–runoff models. This comparison shows that the traditional algorithm can find optimal solutions at least as good as the SCE-UA algorithm. In fact, with the calibration of the SAC-SMA model, the results are significantly different: The RNB algorithm found better solutions than the SCE-UA for all basins. Finally, the combination created between the LH and RNB methods is detailed thoroughly, and a sensitivity analysis of its parameters is used to define the set of optimal values for its efficient performance.

Three multi-start data-driven evolutionary heuristics for the vehicle routing problem with multiple time windows

This paper considers the vehicle routing problem with multiple time windows. It introduces a general framework for three evolutionary heuristics that use three global multi-start strategies: ruin and recreate, genetic cross-over of best parents, and random restart. The proposed heuristics make use of information extracted from routes to guide customized data-driven local search operators. The paper reports comparative computational results for the three heuristics on benchmark instances and identifies the best one. It also shows more than 16% of average cost improvement over current practice on a set of real-life instances, with some solution costs improved by more than 30%.

Development of Differing Extent Mesh Adaptive Direct Search Applied for Optimal Design of Spoke-Type PMSM

Optimal design of electric machines using finite-element analysis (FEA) necessitates too much computation time to maintain high accuracy. To reduce the computation time and obtain a global optimum in multi-modal problems, this paper presents differing extent mesh adaptive direct search (DEMADS). The proposed algorithm is classified into three groups according to the poll searching frame and mesh size for assigning a global and local searching. In particular, DEMADS shares information among solutions based on a parallel multi-start strategy for earlier stopping by deviation and checking revisits. The effectiveness of DEMADS has been validated and evaluated as the function call number and the convergence number at a global optimum using benchmarking function. Finally, it is applied to an optimization problem that minimize torque ripple of a spoke-type permanent magnet synchronous machine via FEA.

Including preventive maintenance activities in an unrelated parallel machine environment with dependent setup times

Abstract The assumption that machines are always available during the production horizon may not be true in real industrial settings. One of the main causes of machine unavailability is the failure of equipment and an efficient way to reduce the failure frequency is through preventive maintenance activities. From that the convenience of including preventive maintenance into the production planning process. In this paper we study a new scheduling problem on unrelated parallel machines that considers, in conjunction, preventive maintenance activities and setup times depending on the sequence and the machine. We present a mathematical formulation for this problem and derive valid inequalities to improve its performance, allowing to the model to obtain optimal solutions for small-medium instances. In addition, we design an efficient metaheuristic algorithm based on a multi-start strategy for solving larger instances. We carry out an extensive computational experimentation to investigate the scope of the model, to validate the influence of the valid inequalities and to assess the performance of the proposed algorithm.

Laminate composite magnetoelectric multiferroics optimized by global derivative-free optimization method

The magnetoelectric multiferroics where magnetism and ferroelectricity coexist in one material have recently attracted renewed interest due to its potential applications in novel functional devices. Natural multiferroic single-phase compounds are rare and an alternative approach to obtain a magnetoelectric (ME) effect is through multilayered composites of a ferroelectric and a ferromagnetic material. An applied electric field creates a piezoelectric strain in the ferroelectric, which produces a corresponding strain in the ferromagnetic material and a subsequent change in magnetization. Various efforts to improve the value of ME coupling coefficient α have been made by modifying preparation techniques of the samples, by the proper choice of materials or different structures and by choosing different thickness of the samples. In this study, we have applied numerical optimization for arriving at the solution for maximum ME coupling coefficient α of a laminar ME composite by making use of the anisotropy of the ferroelectric phase. We have used a global derivative-free optimization method based in directional direct search coupled with specific multistart strategies for setting up the optimization problem. The effective ME couping coefficients αij∼ are computed using the asymptotic homogenization method. Optimal composite microstructure with a range of the constituent ferroelectric single-crystal configurations that enhances the overall α is identified. Optimal composite would have the [001]-axis of the ferroelectric phase oriented out-of-plane of the lamina. Yet the elasticity of the composite is found to be anisotropic at the optimal orientations of the ferroelectric phase. Stress-mediated enhancement of the ME coupling is demonstrated using the analysis of the inplane elastic stiffness of the composite.

MultiGLODS: global and local multiobjective optimization using direct search

The optimization of multimodal functions is a challenging task, in particular when derivatives are not available for use. Recently, in a directional direct search framework, a clever multistart strategy was proposed for global derivative-free optimization of single objective functions. The goal of the current work is to generalize this approach to the computation of global Pareto fronts for multiobjective multimodal derivative-free optimization problems. The proposed algorithm alternates between initializing new searches, using a multistart strategy, and exploring promising subregions, resorting to directional direct search. Components of the objective function are not aggregated and new points are accepted using the concept of Pareto dominance. The initialized searches are not all conducted until the end, merging when they start to be close to each other. The convergence of the method is analyzed under the common assumptions of directional direct search. Numerical experiments show its ability to generate approximations to the different Pareto fronts of a given problem.

A robust and cooperative parallel tabu search algorithm for the maximum vertex weight clique problem

Abstract The maximum vertex weight clique problem (MVWCP) is a challenging NP-Hard combinatorial optimization problem that searches for a clique with maximum total sum of vertices’ weights. In this study, we propose a robust and cooperative parallel tabu search algorithm (PTC) for the MVWCP. Our proposed algorithm uses a dedicated tabu search algorithm with a multistart strategy for the diversification of search space on a parallel computation environment. An effective seeding mechanism is developed with respect to the rank of the processors to choose diversified starting points for a better exploration of the search space. Classical add, swap and drop operators of tabu search are improved with parallel computation and a combined neighborhood approach. The PTC algorithm is evaluated on a set of 120 problem instances from DIMACS-W and BHOSLIB-W benchmarks. Computational results show that the PTC algorithm competes with state-of-the-art heuristic algorithms by reporting average best (optimal) result hit ratios up to 99.0%.