A Parallel Granular Sieving Algorithm for Global Optimization

The challenge of determining the best design in a multimodal design space with multiple local optimal solutions often challenges the best available optimization techniques. By casting the objective function of the optimization problem in the form of a Non-Uniform Rational B-spline (NURBs) metamodel, known as a HyPerModel, significant optimization advantages can be achieved, including the ability to efficiently find the global metamodel optimum solution with less computational expense than traditional approaches. This optimization strategy, defined by the HyPerOp algorithm, uses the underlying structure of a HyPerModel to intelligently select starting points for optimization runs and to identify regions of the design space that do not contain locations for the global metamodel optimum location. This paper describes the application of the HyPerOp algorithm to mixed integer programming problems and demonstrates its use with two example applications. The algorithm works with design spaces composed of continuous and integer design variables and provides a complementary approach for improved optimization capabilities.

We consider a class of problems where a given number p of facility locations must be selected from a set of s potential locations so as to optimize a pre-determined fitness function. There exist outstanding location problems in this class, such as the p-median, max-covering, MAXCAP, and MAXPROFIT problems. These location problems may contain more than a single global optimal solution. Obtaining multiple global optimal solutions allows us to consider other characteristics in the process of selecting the preferred solution. In [B. Pelegrín, J. Redondo, P. Fernández, I. García, and P. Ortigosa, GASUB: Finding global optima to discrete location problems by a genetic-like algorithm, J. Glob. Optim. 38 (2007), pp. 249–264], a new ‘multimodal’ algorithm (gasub) was presented for solving the previously mentioned location problems. In the paper, the algorithm was fine-tuned to obtain a single global optimum (with 100% success) as fast as possible, and it was compared with two widely used techniques, that is, the standard optimizer Xpress-MP [Xpress-MP, Dashoptimization, 2004.] and the Multi Start Heuristic. In this paper, we propose a new set of parameter values so that gasub can explore the space more deeply and consequently find several alternative global optimal solutions. Moreover, two coarse-grain parallelizations of gasub are presented, cggasub and cggasub_mo. cggasub is able to reduce the computational time of gasub where solutions of the same quality are generated, while cggasub_mo finds more alternative global optima than the sequential version and spending similar computational time in the process. gasub, with the new parameter setting and also the parallel algorithms, will be evaluated using a comprehensive computational set of experiments. Additionally, some criteria can also be used to select one of the known global optimal solutions when several alternatives optimize the corresponding objective function.

We discuss the choice of the estimation of the optimal solution when random search methods are applied to solve discrete stochastic optimization problems. At the present time, such optimization methods usually estimate the optimal solution using either the feasible solution the method is currently exploring or the feasible solution visited most often so far by the method. We propose using all the observed objective function values generated as the random search method moves around the feasible region seeking an optimal solution to obtain increasingly more precise estimates of the objective function values at the different points in the feasible region. At any given time, the feasible solution that has the best estimated objective function value (largest one for maximization problems; the smallest one for minimization problems) is used as the estimate of the optimal solution. We discuss the advantages of using this approach for estimating the optimal solution and present numerical results showing that modifying an existing random search method to use tnhis approach for estimating the optimal soluation appears to yield improved performance. We also present sereval rate of convergence results for random search methods using our approach for estimating the optimal solution. One these random search methods is a new variant of the stochastic comparison method; in addition to specifying the rate of convergence of this method, we prove that it is guaranteed to converge almost surely to the set of global optimal solutions and present a result that demonstrates that this method is likely to perform well in practice.

A hybrid global optimization method: The multi-dimensional case

Obtaining guaranteed lower bounds for problems with unknown algebraic form has been a major challenge in derivative-free optimization. In this work, we present a deterministic global optimization method for black-box problems where the derivatives are not available or it is computationally expensive to obtain. However, a global upper bound on the diagonal Hessian elements is known. An edge-concave underestimator (Hasan in J Glob Optim 71:735–752, 2018) can be then constructed with vertex polyhedral solution. Evaluating this underestimator only at the vertices leads to a valid lower bound on the original black-box problem. We have implemented this lower bounding technique within a branch-and-bound framework and assessed its computational performance in locating $$\epsilon $$-global optimal solution for several box-constrained, nonconvex black-box functions.

It is hard to simultaneously improve the peak efficiency (η *) and the width of the high-efficiency region ( Gη) for a hydrodynamic torque converter. A combination of comprehensive CFD simulation and multi-objective optimization was pretested. The elaborate CFD simulation calculation included a reasonable mesh layout, a robust algorithm and a correct turbulence model, whose results were also experimentally verified. In our study, the Kriging surrogate model was first used to construct a nonlinear relationship between the inlet and outlet angle and the economic performance index of the hydrodynamic torque converter. To ensure that the accuracy of the surrogate model meet the requirements, we also used 10 sets of sample points to verify the accuracy of our surrogate model. The accuracy is found to meet the requirements, which shows that the accuracy of the constructed surrogate model is relatively high. We choose to apply the second-generation non-dominant sorting genetic algorithm (NSGA-II) to solve our problem. After solving the Pareto frontier solution set, we obtain a set of global optimal solutions on the Pareto frontier solution set. The optimization results show that the η * is increased by 2.49% and that the Gη is increased by 14.23%. We extracted the flow field structure near the turbine region, characterized the difference between original and optimal model from the flow field perspective, and demonstrated the accuracy of our optimization results. Finally, we used CFD to verify our optimization results, further illustrating the accuracy of the optimization results prediction. Literature research indicates that a large amount of experiments to optimize the η * and the Gη of the hydrodynamic torque converter will bring huge trial cost and time cost. We conclude from our research that the proposed calculation method can solve such problems well.

Global optimization of Savonius-type vertical axis wind turbine with multiple circular-arc blades using validated 3D CFD model

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.

*† ‡ § 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.

Neste artigo são discutidos aspectos relacionados à tomada de decisão dentro de um processo hierárquico de planejamento da produção. Neste contexto, foi desenvolvido um modelo agregado para um problema de planejamento estocástico, com restrições de chances nas variáveis de decisão, que serve como meta de produção a ser atingida nos níveis inferiores da hierarquia. Solução ótima global para este tipo de problema seqüencial é muito difícil, a menos em casos de pequena dimensão via algoritmo de programação dinâmica estocástica. Assim, como alternativa, investigou-se a utilização de uma heurística subótima denominada open-loop feedback controller (OLFC), que é muito simples de ser implementada computacionalmente. Como exemplo de aplicação, um problema mono-produto de planejamento agregado da produção foi formulado e a política subótima OLFC foi comparada à solução ótima usando o algoritmo de programação dinâmica estocástica. Além dos aspectos numéricos, algumas propriedades são analisadas, como a quase adaptabilidade do procedimento OLFC.

Global optimization of MSF seawater desalination processes

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

This paper considers the locally and globally optimal solutions of the global optimisation problem whose objective function is a max‐plus vector‐valued function and constraint function is a real affine function. The formulas about global optimisation in the existing work are further explained and simplified. The necessary and sufficient conditions for the existence and uniqueness of locally optimal solutions of a single objective are established, which are then used to deduce these of globally optimal solutions. Furthermore, the general formulas of locally and globally optimal solutions are presented, respectively. The local optimisation is then applied to solve the load distribution problem of distributed systems with no globally optimal solution, and the optimal allocation scheme is proposed to complete the overall task at the earliest time. The proposed method is constructive, and the obtained results are illustrated by the numerical examples.

Virulence Optimization Algorithm

In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.