Multivariate Global Optimization Research Articles

New technique for solving multivariate global optimization

In this paper, we propose an algorithm based on branch and bound method to underestimate the objective function and reductive transformation which is transformed the all multivariable functions on univariable functions. We also demonstrate several quadratic lower bound functions are proposed which they are better/preferable than the others well-known in literature. We obtain that our experimental results are more effective when we face different nonconvex functions.

A new solution approach for non-convex combined heat and power economic dispatch problem considering power loss

The combined heat and power economic dispatch (CHPED) problem is a non-convex multivariate global optimization problem. The objective of the problem is to reduce total production costs while imposing a variety of constraints and meeting the demand for power and heat. Three recently presented metaheuristic approaches, Slime Mould Algorithm (SMA), COOT algorithm and Marine Predators Algorithm (MPA), are applied for solving CHPED problem. Studies dealing with the CHPED problem in the literature often do not consider valve points effect, prohibited operation zones for power-only units, feasible region constraints of combined heat and power units, all at once. Furthermore, power losses are neglected especially in large-scale problems. In this study, the CHPED problem is solved by considering all operational constraints including active power transmission losses. Three separate case studies with dimensions of 11 units, 48 units, and 96 units were used in the tests under various limitations. The experimental results revealed that MPA outperformed not only SMA, and COOT but also the algorithms proposed previously in the literature.

Multivariate optimization of the electrochemical degradation for COD and TN removal from wastewater: An inverse computation machine learning approach

An inverse computation machine learning (ML) framework that couples genetic algorithm (GA) to feedforward neural network was developed to optimize the operating parameters for maximizing electrochemical removal of chemical oxygen demand (COD) and total nitrogen (TN) from wastewater. Conventional neural networks (NN) are generally unable to implement inverse computation from a desired abatement to feasible input conditions, and NN trained from small-scale datasets tends to be unreliable for multivariate simulation. To address this issue, we employed GA for tuning the weights and biases of the network to enhance the stability and generalization performance of NN, the optimization of multiple operating conditions was performed by the GA strategy of iterative evolution and global search.In this work, we investigated and analyzed the performance of multivariate optimization approaches such as orthogonal design, response surface methodology (RSM), traditional NN, and the developed neuro-genetic model (GANN). Significance and error analysis demonstrate that GANN exhibits superior stability and generalization ability with the highest R2 of 0.946 (COD), 0.874 (TN), and the lowest RMSE of 0.022 (COD), 0.028 (TN). The introduction of GA is significant for improving the prediction accuracy of NN derived from small-scale datasets. The average error of the GANN trained by 25 data points is lower than that of the RSM model derived from 54 data points. According to the validation outcomes, the scheme suggested by GANN achieves the best COD (93.6%) and TN (62.8%) degradation efficiencies, which are 6.1% and 9.9% higher than the optimal values in the original dataset, respectively. The proposed multivariate global optimization strategy can be extended to other cases, and the computational framework of GANN also contributes to the progress of more reliable ML models.

Optimization of maximum power density output for proton exchange membrane fuel cell based on a data-driven surrogate model

Operating conditions are of great significance for proton exchange membrane fuel cells (PEMFCs) and directly determine the output performance of PEMFC. In this paper, an optimization framework is proposed, which is combining a neural network data-driven surrogate model and a stochastic optimization algorithm to achieve multi-variable global optimization, and optimizes the operating conditions for enhancing the power density of the PEMFC. A numerical model of the PEMFC is constructed as the source of the database, and the database is used to train a data-driven surrogate model based on radial basis functions (RBF), which is a typical feed-forward neural network. Then the surrogate model is fed into a particle swarm optimization (PSO) algorithm to obtain the optimal solution for the best combination of operating conditions. Results show that the surrogate model can accurately predict the output voltage of the PEMFC model, where the squared correction factor (R-square) and the mean percentage error of the test set are 0.99638 and 4.4686% respectively. And the optimization framework using a combination of data-driven surrogate model and stochastic optimization algorithm can obtain the maximum power density of 0.6097 W cm−2, with a relative error of only 1.5321% to the PEMFC model results. The result shows that the framework can effectively handle the multivariate optimization of complex systems.

A Design Method for Low-Pressure Venturi Nozzles

The purpose of this work is to provide empirical design models for low-pressure, subsonic Venturi nozzles. Experimentally validated simulations were used to determine the effect of nozzle geometry and operating conditions on the suction ratio (ratio of suction mass flow rate to motive mass flow rate) of low-pressure, subsonic Venturi nozzles, over a wide range of geometries and operating conditions, through a parametric study. The results of the parametric study were used to develop seven empirical models, each with a different range of applicability or calculating a different indicator of nozzle performance (i.e., suction ratio, momentum ratio, or dynamic pressure ratio), of the Venturi nozzles using a constrained multi-variable global optimization method. Of the seven empirical models, the best models were found to be those for low- (less than one) and high-suction ratios (greater than one), with mean absolute percentage errors of 5% and 18%, respectively. These empirical models provide a design tool for subsonic, low-pressure Venturi nozzles that is more than an order of magnitude more accurate than a governing equation approach or conventional flow head calculations. These newly-developed empirical models can be applied for initial nozzle design when precise suction ratios are required.

Deterministic and stochastic algorithms for resolving the flow fields in ducts and networks using energy minimization

Several deterministic and stochastic multi-variable global optimization algorithms (Conjugate Gradient, Nelder–Mead, Quasi-Newton and global) are investigated in conjunction with energy minimization principle to resolve the pressure and volumetric flow rate fields in single ducts and networks of interconnected ducts. The algorithms are tested with seven types of fluid: Newtonian, power law, Bingham, Herschel–Bulkley, Ellis, Ree–Eyring and Casson. The results obtained from all those algorithms for all these types of fluid agree very well with the analytically derived solutions as obtained from the traditional methods which are based on the conservation principles and fluid constitutive relations. The results confirm and generalize the findings of our previous investigations that the energy minimization principle is at the heart of the flow dynamics systems. The investigation also enriches the methods of computational fluid dynamics for solving the flow fields in tubes and networks for various types of Newtonian and non-Newtonian fluids.

New quadratic lower bound for multivariate functions in global optimization

The method investigated in this paper is concerned with the multivariate global optimization with box constraints. A new quadratic lower bound in a branch and bound framework is proposed. For a continuous, twice differentiable function f, the new lower bound is given by a difference of the linear interpolant of f and a quadratic concave function. The proposed BB algorithm using this new lower bound is easy to implement and often provides high quality bounds. The performances of the proposed algorithm are compared with those of two others branch and bound algorithms, the first uses a linear lower bound and the second a quadratic lower bound. Computational results conducted on several test problems show the efficiency of the proposed algorithm.

NEW UNDERESTIMATOR FOR MULTIVARIATE GLOBAL OPTIMIZATION WITH BOX CONSTRAINTS

The paper is concerned with the multivariate global optimization with box constraints. A new underestimator is investigated for twice contin- uously differentiable function on a box which is an extension of the approach developed in (5) for univariate global optimization.

On Covering Methods for D.C. Optimization

Covering methods constitute a broad class of algorithms for solving multivariate Global Optimization problems. In this note we show that, when the objective f is d.c. and a d.c. decomposition for f is known, the computational burden usually suffered by multivariate covering methods is significantly reduced. With this we extend to the (non-differentiable) d.c. case the covering method of Breiman and Cutler, showing that it is a particular case of the standard outer approximation approach. Our computational experience shows that this generalization yields not only more flexibility but also faster convergence than the covering method of Breiman-Cutler.

A Multivariate Global Optimization Using Linear Bounding Functions

Recently linear bounding functions (LBFs) were proposed and used to find e-global minima. This paper presents an LBF-based algorithm for multivariate global optimization problems. The algorithm consists of three phases. In the global phase, big subregions not containing a solution are quickly eliminated and those which possibly contain the solution are detected. An efficient scheme for the local phase is developed using our previous local minimization algorithm, which is globally convergent with superlinear/quadratic rate and does not require evaluation of gradients and Hessian matrices. To ensure that the found minimizers are indeed the global solutions or save computation effort, a third phase called the verification phase is often needed. Under adequate conditions the algorithm finds the e-global solution(s) within finite steps. Numerical testing results illustrate how the algorithm works, and demonstrate its potential and feasibility.

The cluster problem in multivariate global optimization

We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the “midpoint test,” but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multi-dimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension.

Nested annealing: a provable improvement to simulated annealing

Simulated annealing is a family of randomized algorithms for solving multivariate global optimization problems. Empirical results from the application of simulated annealing algorithms to certain hard problems including certain types of NP-complete problems demonstrate that these algorithms yield better results than known heuristic algorithms. But for the worst case input, the time bound can be exponential. In this paper, for the first time, we show how to improve the performance of simulated annealing algorithms by exploiting some special properties of the cost function to be optimized. In particular, the cost functions we consider are small-separable, with parameter s( n). We develop an algorithm we call “Nested Annealing” which is a simple modification of simulated annealing where we assign different temperatures to different regions. Simulated annealing can be shown to have expected run time 2 Ω( n) whereas our improved algorithm has expected performance 2 Os( n) . Thus for example, in many vision and VLSI layout problem, for which s(n=O( n ) , our time bound is 2 O( n ) rather than 2 Ω(n) .