Global Optimization for Integrated Design and Control of Computationally Expensive Process Models

*† ‡ § An efficient and high-fidelity design approach is proposed by combining global and local optimization methods for wing planform and surface design. For enhanced design results, aerodynamic shape optimization process is carried out via 2-stage with different optimization strategy. In the first stage, global optimization techniques are applied to planform design with a few geometric design variables. In the second stage, local optimization techniques are used for wing surface design with a lot of design variables to maintain a sufficient design space with high DOF (Degree of Freedom) geometric change. For global optimization, meta-modeling techniques such as RS (Response Surface) and Kriging methods are used in conjunction with Genetic Algorithm (GA). For local optimization, a discrete adjoint variable method is used. By the successive combination of global and local optimization techniques, drag minimization is performed for a multi-body aircraft configuration while maintaining the baseline lift and the wing weight at the same time. Through the design process, performances of the test models are remarkably improved in comparison with the single stage design approach. The capability of proposed design framework including wing planform design variables can be evaluated by the drag decomposition method which can provide improvement of induced drag and wave drag, respectively.

A Tabu search-based algorithm for mixed-integer nonlinear problems and its application to integrated process and control system design

Neural networks are often characterized as highly nonlinear systems of fairly large amount of parameters (in order of 103 - 104 ). This fact makes the optimization of parameters to be a nontrivial problem. But the astonishing moment is that the local optimization technique is widely used and yields reliable convergence in many cases. Obviously, the optimization of neu­ ral networks is high-dimensional, multi-extremal problem, so, as usual, the global optimization methods would be applied in this case. On the basis of Perceptron-like'i unit (which is the building block for the most architectures of neural networks) we analyze why the local optimization technique is so successful in the field of neural networks. The result is that a linear approximation of the neural network can be sufficient to evaluate the start point for the local optimization procedure in the nonlinear regime. This result can help in developing faster and more robust algorithms for the optimization of neural network parameters.

In this paper, we first establish some sufficient and some necessary global optimality conditions for quadratic integer programming problems. Then we present a new local optimization method for quadratic integer programming problems according to its necessary global optimality conditions. A new global optimization method is proposed by combining its sufficient global optimality conditions, local optimization method and an auxiliary function. The numerical examples are also presented to show that the proposed optimization methods for quadratic integer programming problems are very efficient and stable.

We present a framework for the construction of vascular systems based on optimality principles of theoretical physiology. Given the position and flow distribution of end points of a vascular system, we construct the topology and positions of internal nodes to complete the vascular system in a realistic manner. Optimization is driven by intravascular volume minimization with constraints derived from physiological principles. Direct optimization of a vascular system, including topological changes, is used instead of simulating vessel growth. A good initial topology is found by extracting key information from a previously optimized model with less detail. This technique is used iteratively in a multi-level approach to create a globally optimized vascular system. Most of this work was completed at Fraunhofer MeVis during the summer of 2004.

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>

A reconsideration on the resolution of phase stability analysis using stochastic global optimization methods: Proposal of a reliable set of benchmark problems

Computational models and inverse dynamic optimization methods are used to predict in-vivo spinal loading. Spinal force is conventionally predicted using the constant loading path method, which is based on the concept that the physiological directions of the spine loads follow the same path of the spinal curve. However, the global convergence optimization method, in which the instantaneous center of rotation of the joint should be also predicted, is necessary for accurate prediction of joint forces of the human body. In this study, we investigate the joint forces, instantaneous centers of rotation, and muscle forces of the human lumbar spine using both global convergence optimization method and constant loading path method during flexion, upright standing, and extension postures. The joint forces predicted using the constant loading path method were 130%, 234%, and 253% greater than those predicted using the global convergence optimization method for the three postures. The instantaneous centers of rotation predicted using the global convergence optimization method were segment level-dependent and moved anteriorly in the flexion and posteriorly in the extension, whereas those predicted using the constant loading path method moved posteriorly in both the flexion and extension. The data indicated that compared to the global convergence optimization method, the constant loading path method introduces additional constraints to the spinal joint model, and thus, it results in greater joint and muscle forces.

A modified tunneling function method for non-smooth global optimization and its application in artificial neural network

Surrogate-based method can dramatically reduce the number of expensive function evaluations in real-world multi- objective optimization problems (MOP). When the number of objectives is small, using surrogate models combined with expected hypervolume improvement (EHVI) infill sampling criterion (ISC) has been proved to be efficient to provide a set of solutions with good diversity and good proximity to the Pareto front (PF) in aerodynamic shape optimization. However, traditional hypervolume-based infilling strategies use only one kind of ISC to generate one or multiple sample points, the advantages of various kinds of ISC cannot be comprehensively utilized and the parallelization is not easy to implement. This paper proposes a combined multi-point infilling strategy based on Kriging models and develops an efficient global multi-objective constrained optimization method (EGMOCO) to solve multi- objective aerodynamic shape optimization with complex constraints. Multiple sample points are generated by using four ISC considering hypervolume at each iteration and then evaluated in parallel. Firstly, the performance of EGMOCO is compared with that of single criterion EHVI strategy on six benchmarks within the same computational budget to prove its effectiveness, and then EGMOCO is implemented in an aerodynamic shape optimization problem with complex constraints. The result shows that EGMOCO has good performance in balancing local exploitation and global exploration with faster convergence rate and high robustness, the whole PF can be fully explored in limited evaluations and the constraint handling is effective especially for real-world problems with complex and nonlinear constraints, the comprehensive aerodynamic performance of the airfoil is greatly improved. It can be confirmed that Kriging-based multi-objective optimization method combined with multi-point infilling strategy performs better than single infilling criterion EHVI, since different sample infilling criteria can complement with each other, both local exploitation and global exploration can be considered and well balanced.

The Wilson equation for the excess Gibbs energy has found wide use in successfully representing the behavior of polar and nonpolar multicomponent mixtures with only binary parameters but was incapable of predicting more than one liquid phase. The UNIFAC and ASOG group contribution methods do not have this limitation and can predict the presence of multiple liquid phases. The most important area of application of all these equations is in the prediction of phase equilibria. The calculation of phase equilibria involves two important problems: (1) the minimization of the Gibbs free energy and (2) the tangent plane stability criterion. Problem (2), which can be implemented as the minimization of the tangent plane distance function, has found wide application in aiding the search for the global minimum of the Gibbs free energy. However, a drawback of all previous approaches is that they could not provide theoretical guarantees that the true equilibrium solution will be obtained. The goal of the work is to find the equilibrium solution corresponding to the global minimum of the Gibbs free energy. A proof that the Wilson equation leads to a convex formulation for the minimization of the Gibbs energy is provided so that a local optimization technique will always converge to a global minimum. In addition, new expressions are derived for the molar Gibbs free energy function when the UNIFAC, ASOG, and modified Wilson equations are employed. These expressions are then transformed so that application of a branch and bound based global optimization algorithm originally due to Falk and Soland (1969) is possible. This allows global solutions to be obtained for both the minimization of the Gibbs free energy and the minimization of the tangent plane distance function. The algorithm is implemented in C as part of the package GLOPEQ, global optimization for the phase equilibrium problem (McDonald and Floudas, 1994d). Results for several examples are presented

We consider here the average deviation as the most important objective when designing numerical techniques and algorithms. We call that a Bayesian approach. We start by describing the Bayesian approach to the continu­ ous global optimization. Then we show how to apply the results to the adaptation of parameters of randomized techniques of optimiza­ tion. We assume that there exists a simple function which roughly predicts the consequences of decisions. We call it heuristics. We define the probability of a decision by a randomized decision func­ tion depending on heuristics. We fix this decision function, except for some parameters that we call the decision parameters. We repeat the randomized decision procedure several times given the decision parameters and regard the best outcome as a result. We optimize the decision parameters to make the search more efficient. Thus we replace the original optimization problem by an auxiliary problem of continuous stochastic optimization. We solve the auxiliary problem by the Bayesian methods of global op­ timization. Therefore we call the approach as the Bayesian one. We discuss the advantages and disadvantages of the Bayesian approach. We describe the applications to some of discrete pro-

Features of several well-known global and local optimization methods were discussed and their robustness and efficiency were tested on several artificial test functions in Matlab environment. The tested local methods were the interior-point, the quasi-Newton method, Nelder–Mead simplex, the pattern search, the NEWUOA and the BOBYQA methods. The global methods were the genetic algorithm (GA), the simulated annealing (SA), the particle swarm optimization (PSO), and the covariance matrix adaptation evolutionary strategy (CMA-ES) methods (see subsections 2.2 and 2.3 for their details). Furthermore, a novel global optimization method, called FOCusing robusT Optimization with Uncertainty-based Sampling (FOCTOPUS), which proved to be very efficient in the optimization of constrained and highly correlated parameters of combustion kinetic models, was also tested. The test functions were selected in such a way that they had a variety of features: uni-modal and multi-modal, differentiable and non-differentiable, separable and non-separable, low dimensional and high dimensional. The following test functions were used: the 20D Alpine function, the 4D Ackley function, the Cross-in-tray 2D function, the Hartmann-6D function, the Holder table 2D function, the 5D Rastrigin function, the 5D Rosenbrock function, the 4D modified Rosenbrock function, the 20D Zakharov function and a typical 2D multi-modal function. The general conclusion here is that, the global methods performed well on the multi-modal and high-dimensional test functions while the local methods were superior in the case of low-dimensional and uni-modal test functions. For the highly multi-modal test functions, the GA was better than all the other methods. The FOCTOPUS method proved to be inferior to GA for most of the test functions, thus its application cannot be generally recommended.

Global optimisation problems in high-dimensional and infinite-dimensional spaces arise in various real-world applications such as engineering, economics and finance, geophysics, biology, machine learning, optimal control, etc. Among stochastic approaches to global optimisation, biology-inspired methods are currently very popular in the literature. Bio-inspired approaches imitate natural ecological and evolutionary processes and are reported to be efficient in a large number of practical study cases. However, many bio-inspired methods can possess some vital drawbacks. For example, due to their semi-empirical nature, convergence to the globally optimal solution cannot always be guaranteed. Another major obstacle is that the existing methods often struggle with higher dimensionality of the space of parameters, which results in a slow convergence. Moreover, it is often difficult to adjust the dimensionality of the space of parameters in the corresponding computer code for a practical realisation of the optimisation method. Here, we present a bio-inspired global stochastic optimisation method applicable in Hilbert function spaces. The proposed method is an evolutionary algorithm inspired by the Darwin’s famous idea of survival of the fittest and is, therefore, referred to as the ‘Survival of the Fittest Algorithm’ (SoFA). Mathematically, the convergence of SoFA is a consequence of the fundamental property of localisation of probabilistic measure in a Hilbert space and we rigorously prove the convergence of the introduced algorithm for a generic class of functionals. The approach is simple in terms of practical coding. As an insightful, real-world problem, we apply our method to find the globally optimal trajectory for the daily vertical migration of zooplankton in the ocean and lakes, this phenomenon is considered to be the largest synchronised movement of biomass on Earth. We maximise fitness in a function space derived from a von-Foerster stage-structured population model with biologically realistic parameters. We show that for problems of fitness maximisation in high-dimensional spaces, SoFA provides better performance as compared to some other stochastic global optimisation algorithms. We highlight the links between the new optimisation algorithm and natural selection processes in ecosystems occurring within a population via gradual exclusion of competitive con-specific strains.