Extension of multiunit global optimisation to three‐input systems

Finding the global optimum of an objective function has been of interest in many disciplines. Recently, a global optimisation technique based on multiunit extremum seeking has been proposed for scalar and two‐input systems. The idea of multiunit extremum‐seeking is to control the gradient evaluated using finite difference between two identical units operating with an offset. For scalar systems, it was shown that the global optimum could be obtained by reducing the offset to zero. For two‐input systems, the univariate global optimisation is performed on the circumference of a circle of reducing radius. In this study, the concept is extended to three‐input systems where the circle of varying radius sits on a shrinking sphere. The key contribution lies in formulating the rotation required to keep the best point found in the search domain. The theoretical concepts are illustrated on the global optimisation of several examples. Comparison results with other competitive methods show that the proposed technique performs well in terms of number of function evaluations and accuracy. © 2011 Canadian Society for Chemical Engineering     

Optimization-based design of reactive distillation columns using a memetic algorithm

The design optimization of reactive distillation columns (RDC) is characterized by complex nonlinear constraints, nonlinear cost functions, and the presence of many local optima. The standard approach is to use MINLP solvers that work on a superstructure formulation where structural decisions are represented by discrete variables and lead to an exponential increase in the computational effort. The mathematical programming (MP) methods which solve the continuous sub-problems provide only one local optimum which depends strongly on the initialization. In this contribution a memetic algorithm (MA) is introduced and applied to the global optimization of four different formulations of a computational demanding real-world design problem. An evolution strategy addresses the global optimization of the design decisions, while continuous sub-problems are efficiently solved by a robust MP solver. The MA is compared to MINLP techniques. It is the only algorithm that finds the global solution in reasonable times for all model formulations.     

Model-free real-time optimization of process systems using safe Bayesian optimization

Conventional real-time optimization (RTO) requires detailed process models, which may be challenging or expensive to obtain. Model-free RTO methods are an attractive alternative to circumvent the challenge of developing accurate models. Most model-free RTO methods are based on estimating the steady-state cost gradient with respect to the decision variables and driving the estimated gradient to zero using integral action. However, accurate gradient estimation requires clear time scale separation from the plant dynamics, such that the dynamic plant can be assumed to be a static map. For processes with long settling times, this can lead to prohibitively slow convergence to the optimum. To avoid the need to estimate the cost gradients from the measurement, this article uses Bayesian optimization, which is a zeroth order black-box optimization framework. In particular, this article uses a safe Bayesian optimization based on interior point methods to ensure that the setpoints computed by the model-free steady-state RTO layer are guaranteed to be feasible with high probability (i.e., the safety-critical constraints will not be violated at steady-state). The proposed method can thus be seen as a model-free variant of the conventional two-step steady-state RTO framework (with steady-state detection), which is demonstrated on a benchmark Williams-Otto reactor example.     

Joint optimization of number of wells,well locations and controls using a gradient-based algorithm

This paper presents a detailed algorithm for solving the general well-placement optimization problem in which the number of wells, their locations and rates are simultaneously optimized with an efficient gradient-based algorithm. The proposed well-placement optimization algorithm begins by placing a large number of wells in the reservoir, where, the well rates are the optimization variables. During iterations of the algorithm, most of the wells are eliminated by setting their rates to zero. The remaining wells and their controls determine the optimal number of wells, their optimum locations and rates. The well-placement algorithm consists of two optimization stages. In the initialization stage, the appropriate total reservoir production rate (or the total injection rate) for the set of to-be-optimized producers (or injectors) is estimated by maximizing the net-present-value for the specified operational life of the reservoir. In the second stage, a modified net-present-value functional which also considers the drilling cost of the wells is maximized subject to the a total rate constraint determined in the initialization stage. Both stages of the algorithm use gradient projection to enforce the linear and bound constraints, where the required gradients are computed with the adjoint method. The bottomhole pressure constraints on the wells are enforced using a practical approach. The applicability and robustness of our well-placement algorithm is discussed through several example problems.     

Optimization strategies for the design and synthesis of distributed wastewater treatment networks

This paper applies an optimization strategy for the design of a distributed wastewater network where multicomponent streams are considered. The streams are to be processed by different technologies for reducing the concentration of several contaminants to meet environmental regulations. The model gives rise to a nonconvex nonlinear and a heuristic search procedure is applied to find the global optimum or a good upper bound of the global optimum. The procedure is based on the successive solution of a relaxed linear model and the original nonconvex nonlinear problem and on the use of several objective functions in the relaxed model. Two examples are presented to illustrate that method.     

Smooth penalty function method for rigorous optimization of distillation processes

Integer decisions on stage numbers and feed locations, and global optimality are still challenging for rigorous optimization of distillation processes. In the present article, we propose a smooth penalty function method to address both these problems. The proposed method is based on the relaxation of the integer decision problem into continuous nonlinear programming (NLP) problem by adopting the bypass efficiency model developed by Dowling and Biegler. A smooth penalty term (SPT) is proposed and added to the total annual cost (TAC) function to form a new objective function, namely, the smooth penalty function. Using the new objective function, the problem is initially solved with negative weight coefficients for the SPTs regarding each column section to get an optimum near the global optimum of the SPT. Then, starting from this solution, the problem is solved again iteratively by increasing the values of the weight coefficients until all the stage numbers become integers. The performance of the method is validated by an illustrating problem and in three case studies, including a reactive distillation optimization problem.     

Logarithm-transform piecewise linearization method for the optimization of fasoline blending processes

Gasoline blending is a key process in a petroleum refinery, as it can yield 60%–70% of a typical refinery's total revenue. This process not only exhibits non-convex nonlinear blending behavior due to the complicated blend mechanism of various component feedstocks with different quality properties, but also involves global optimum searching among numerous blending recipes. Since blend products are required to meet a series of quality requirements and highly-sensitive to the proportion changes of blending feedstocks, global optimization methods for NLP problems are often difficult to be applied because of heavy computational burdens. Thus, piecewise linearization methods are naturally proposed to provide an approximate global optimum solution by adding binary variables into the models and converting the original NLP problems into MILP ones. In this paper, Logarithmtransform piecewise linearization(LTPL) method, an improved piecewise linearization, is proposed. In this method a logarithm transform is applied to convert multi-variable multi-degree constraints into a series of single-variable constraints. As a result, the number of 0–1 variables is greatly reduced. In the final part of this paper, an industrial case study is conducted to demonstrate the effectiveness of LTPL method. In principle, this method would be useful for blending problems with complicated empirical or theoretical models.     

Dependence of the Error in the Optimal Solution of Perturbation‐Based Extremum Seeking Methods on the Excitation Frequency

In perturbation‐based extremum‐seeking methods, an excitation signal is added to the input, and the gradient, computed from the correlation between the input and output variations, is forced to zero. The main drawback of the method is that the speed of convergence, which is linked to the dither frequency, is slow due to the low value of dither frequency typically chosen. Increasing the excitation frequency may cause instability, but that could be corrected by phase compensation. In this paper, it is shown that an additional problem exists, i.e., the distance between the optimum and solution reached by the perturbation method is proportional to the square of the frequency of excitation and does not go to zero even when the amplitude of the excitation goes to zero. However, for Wiener/Hammerstein approximations, the error will indeed go to zero with the excitation amplitude. Simulation results on a distributed reaction system are used to illustrate the concepts presented in this work.     

基于微粒群优化算法的不确定性调和调度

Blending is an important unit operation in process industry. Blending scheduling is nonlinear optimization problem with constraints. It is difficult to obtain optimum solution by other general optimization methods. Particle swarm optimization （PSO） algorithm is developed for nonlinear optimization problems with both continuous and discrete variables. In order to obtain a global optimum solution quickly, PSO algorithm is applied to solve the problem of blending scheduling under uncertainty. The calculation results based on an example of gasoline blending agree satisfactory with the ideal values, which illustrates that the PSO algorithm is valid and effective in solving the blending scheduling problem.     

一种改进的混合遗传算法及在化工过程动态优化中的应用(英文)

The solutions of dynamic optimization problems are usually very difficult due to their highly nonlinear and multidimensional nature. Genetic algorithm (GA) has been proved to be a feasible method when the gradient is difficult to calculate. Its advantage is that the control profiles at all time stages are optimized simultaneously, but its convergence is very slow in the later period of evolution and it is easily trapped in the local optimum. In this study, a hybrid improved genetic algorithm (HIGA) for solving dynamic optimization problems is proposed to overcome these defects. Simplex method (SM) is used to perform the local search in the neighborhood of the optimal solution. By using SM, the ideal searching direction of global optimal solution could be found as soon as possible and the convergence speed of the algorithm is improved. The hybrid algorithm presents some improvements, such as protecting the best individual, accepting immigrations, as well as employing adaptive crossover and Gaussian mutation operators. The efficiency of the proposed algorithm is demonstrated by solving several dynamic optimization problems. At last, HIGA is applied to the optimal production of secreted protein in a fed batch reactor and the optimal feed-rate found by HIGA is effective and relatively stable.     

A new optimization model and a customized solution method for natural gas production network design and operation

This article proposes to tackle integrated design and operation of natural gas production networks under uncertainty, using a new two‐stage stochastic programming model, a novel reformulation strategy, and a customized global optimization method. The new model addresses material balances for multiple key gas components, pressure flow relationships in gas wells and pipelines, and compressor performance. This model is a large‐scale nonconvex mixed‐integer nonlinear programming problem that cannot be practically solved by existing global optimization solvers or decomposition‐based optimization methods. With the new reformulation strategy, the reformulated model has a better decomposable structure, and then a new decomposition‐based global optimization method is developed for efficient global optimization. In the case study of an industrial naturals production system, it is shown that the proposed modeling and optimization methods enable efficient solution, and the proposed optimization method is faster than a state‐of‐the‐art decomposition method by at least an order of magnitude. © 2016 American Institute of Chemical Engineers AIChE J, 63: 933–948, 2017     

自适应混合粒子群约束优化算法及其在软测量模型参数估计中的应用

对于软测量模型参数估计问题, 针对传统梯度法求解非线性最小二乘模型时依赖初值、需要追加趋势分析进行验证和无法直接求解复杂问题的缺陷, 提出将参数估计化为约束优化问题, 使用混合优化算法求解的新思路。为此提出一种自适应混合粒子群约束优化算法(AHPSO-C)。在AHPSO-C算法中, 为平衡全局搜索(混沌粒子群)和局部搜索(内点法), 引入自适应内点法最大函数评价次数更新策略。对12个经典测试函数的仿真结果表明, AHPSO-C是求解约束优化问题的一种有效算法。将算法用于淤浆法高密度聚乙烯(HDPE)串级反应过程中熔融指数软测量模型参数估计, 验证了方法的可行性与优越性。     

Linear method for the design of shell and tube heat exchangers using the Bell–Delaware method

In this article, we present a rigorous reformulation of the Bell–Delaware model for the design optimization of shell and tube heat exchanger to obtain a linear model. We extend a previously presented methodology1,2 of rigorously reformulate the mixed-integer nonlinear programing Kern model and we add disjunctions to automatically choose the different correlations to calculate heat transfer coefficients and pressure drop under different flow regimes. The linear character of the formulation allows the identification of the global optimum, even using conventional optimization algorithms. The proposed mixed-integer linear programming formulation with the Bell–Delaware method is able to identify feasible solutions for the design of heat exchangers at a lower cost than those obtained through conventional design formulations in the literature. Comparisons with the Kern method also indicate an average 22% difference (usually lower) in area.     

基于全局优化改进混沌粒子群遗传算法的物料平衡数据校正

结合遗传算法（GA）和粒子群算法（PSO）的优点以及混沌运动的特性,提出了加入混沌扰动的混沌粒子群遗传算法（DCPSO-GA）,并使用5个高维非线性测试函数考察全局优化混合算法的性能。DCPSO-GA解决了在寻优搜索时出现的停滞现象,扩大了全局优化的搜索空间,丰富了粒子的多样性,且不需要函数梯度信息。测试结果证明,针对本文的5个测试函数DCPSO-GA能找到全局最优解,其收敛速度很快,大大减少了计算量。而且,经过与其他相关算法比较可知,当总的目标函数调用次数较接近或更少时,改进算法不论在计算精度还是收敛速度上,均有很大的提高。并将DCPSO-GA算法应用到重油裂解参数估计和预测中,测试结果证明,其提高了参数估计和预测的准确性,降低了误差,能有效找到全局最优解,收敛速度快,大大减少计算量。     

An augmented Lagrangian interior-point approach for large-scale NLP problems on graphics processing units

The demand for fast solution of nonlinear optimization problems, coupled with the emergence of new concurrent computing architectures, drives the need for parallel algorithms to solve challenging nonlinear programming (NLP) problems. In this paper, we propose an augmented Lagrangian interior-point approach for general NLP problems that solves in parallel on a Graphics processing unit (GPU). The algorithm is iterative at three levels. The first level replaces the original problem by a sequence of bound-constrained optimization problems using an augmented Lagrangian method. Each of these bound-constrained problems is solved using a nonlinear interior-point method. Inside the interior-point method, the barrier sub-problems are solved using a variation of Newton's method, where the linear system is solved using a preconditioned conjugate gradient (PCG) method, which is implemented efficiently on a GPU in parallel. This algorithm shows an order of magnitude speedup on several test problems from the COPS test set.     

Optimum brightness level and simplified characterization of CRT color monitors

The nonlinear relationship between luminance and DAC count could be characterized with the simplified model, if optimum brightness level is set. In this study, we propose a technique to set the optimum level of brightness, in which offsets for RGB channels can be assumed to zero, and determine the gamma coefficients from log–log data without nonlinear optimization. The optimum brightness level could be found by measuring a few tones of neutral for the combination of 3 levels of brightness and 2 levels of contrast. This technique has two advantages. It does not require measurements for 0 DAC count, and does not require nonlinear optimization in finding the gamma coefficient of the display system. Two CRT monitors by different manufacturers have been tested. As the result, all monitors could be set to their optimum state with a different combination of brightness and contrast. In that state, the gamma coefficient for each channel could be determined from two measuring data and the tone reproduction characteristics of the RGB channel could be characterized with the simplified equation, neglecting offset and gain. The accuracy of characterization was better than 0.5 ΔE*_ab for 125 colors for a monitor having good channel independence. © 2000 John Wiley & Sons, Inc. Col Res Appl, 25, 408–415, 2000     

Shell and tube heat exchanger design using mixed‐integer linear programming

The design of heat exchangers, especially shell and tube heat exchangers was originally proposed as a trial and error procedure where guesses of the heat transfer coefficient were made and then verified after the design was finished. This traditional approach is highly dependent of the experience of a skilled engineer and it usually results in oversizing. Later, optimization techniques were proposed for the automatic generation of the best design alternative. Among these methods, there are heuristic and stochastic approaches as well as mathematical programming. In all cases, the models are mixed integer non‐linear and non‐convex. In the case of mathematical programming solution procedures, all the solution approaches were likely to be trapped in a local optimum solution, unless global optimization is used. In addition, it is very well‐known that local solvers need good initial values or sometimes they do not even find a feasible solution. In this article, we propose to use a robust mixed integer global optimization procedure to obtain the optimal design. Our model is linear thanks to the use of standardized and discrete geometric values of the heat exchanger main mechanical components and a reformulation of integer nonlinear expressions without losing any rigor. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1907–1922, 2017     

Global optimization based on subspaces elimination: Applications to generalized pooling and water management problems

A global optimization strategy based on the partition of the feasible region in boxed subspaces defined by the partition of specific variables into intervals is described. Using a valid lower bound model, we create a master problem that determines several subspaces where the global optimum may exist, disregarding the others. Each subspace is then explored using a global optimization methodology of choice. The purpose of the method is to speed up the search for a global solution by taking advantage of the fact that tighter lower bounds can be generated within each subspace. We illustrate the method using the generalized pooling problem and a water management problem, which is a bilinear problem that has proven to be difficult to solve using other methods. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2336–2345, 2012     

Real-time optimization for a laboratory-scale flotation column

In this paper, a supervisory layer with real-time optimization (RTO) has been implemented in an experimental laboratory-scale flotation column for copper concentration. A two-stage and modifier adaptation (MA) methodology for RTO has been compared under structural, experimental and dynamic uncertainty. In addition, a gradient-free alternative for MA, called nested modifier optimization, has been proposed and tested. The results show that the KKT updates of the MA approach allow the process optimum to be determined under uncertain scenarios, unlike the two-stage approach. From the perspective of gradient modifiers, the performance of the nested methodology is comparable to the dual approach because previous past values are used to update the modifiers without requiring the gradient estimation step. In addition, the interaction of RTO with the regulatory layer must be considered to propose an optimal implementation.     

An efficient constraint handling method with integrated differential evolution for numerical and engineering optimization

Constrained optimization problems are very important as they are encountered in many engineering applications. Equality constraints in them are challenging to handle due to tiny feasible region. Additionally, global optimization is required for finding global optimum when the objective function and constraints are nonlinear. Stochastic global optimization methods can handle non-differentiable and multi-modal objective functions. In this paper, a new constraint handling method for use with such methods is proposed for solving equality and/or inequality constrained problems. It incorporates adaptive relaxation of constraints and the feasibility approach for selection. The recent integrated differential evolution (IDE) with the proposed constraint handling technique is tested for solving benchmark problems with constraints, and then applied to many chemical engineering application problems with equality and inequality constraints. The results show that the proposed constraint handling method with IDE (C-IDE) is reliable and efficient for solving constrained optimization problems, even with equality constraints.