Multi-objective steady state optimization of biochemical reaction networks using a constrained genetic algorithm

A hybrid genetic algorithm-based method to solve constrained multi-objective optimization problems is proposed. Considering operation around a steady state of a dynamical system, the task of the algorithm consists on finding a set of optimal, but constrained solutions. The method is exemplified on a (bio)chemical reaction network in Saccharomyces cerevisiae. In the steady state the model reduces to a system of non-linear equations which must be solved by a search method. This iterative search was integrated into a genetic algorithm in order to look up for optimal steady states. The basic idea is to use individuals of the genetic algorithm as starting points for the search algorithm. The optimization goal was to simultaneously maximize ethanol production and reduce metabolic burden. Two alternative kinetic approaches are compared to Michaelis Menten-type kinetics: a S-System and a generalized mass action model, both based on Power-Law kinetics.     

Constraint handling optimal PI control of open-loop unstable process: Analytical approach

This paper proposes the closed-form analytical design of proportional-integral (PI) controller parameters for the optimal control of an open-loop unstable first order process subject to operational constraints. The main idea of the design process is not only to minimize the control performance index, but also to cope with the constraints in the process variable, controller output, and its rate of change. To derive an analytical design formula, the constrained optimal control problem in the time domain was transformed to an unconstrained optimization in a parameter space associated with closed-loop dynamics. By taking advantage of the proposed analytical approach, a convenient shortcut algorithm was also provided for finding the optimal PI parameters quickly, based on the graphical analysis for the optimal solution of the corresponding optimization problem in the parameter space. The resulting optimal PI controller guarantees the globally optimal closed-loop response and handles the operational constraints precisely.     

Optimal robust optimization approximation for chance constrained optimization problem

Chance constraints are useful for modeling solution reliability in optimization under uncertainty. In general, solving chance constrained optimization problems is challenging and the existing methods for solving a chance constrained optimization problem largely rely on solving an approximation problem. Among the various approximation methods, robust optimization can provide safe and tractable analytical approximation. In this paper, we address the question of what is the optimal (least conservative) robust optimization approximation for the chance constrained optimization problems. A novel algorithm is proposed to find the smallest possible uncertainty set size that leads to the optimal robust optimization approximation. The proposed method first identifies the maximum set size that leads to feasible robust optimization problems and then identifies the best set size that leads to the desired probability of constraint satisfaction. Effectiveness of the proposed algorithm is demonstrated through a portfolio optimization problem, a production planning and a process scheduling problem.     

A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm: Applications to particulate processes and discrete dynamic systems

The solution of optimal control problems (OCPs) becomes a challenging task when the analyzed system includes non-convex, non-differentiable, or equation-free models in the set of constraints. To solve OCPs under such conditions, a new procedure, LARES-PR, is proposed. The procedure is based on integrating the LARES algorithm with a generalized representation of the control function. LARES is a global stochastic optimization algorithm based on the artificial chemical process paradigm. The generalized representation of the control function consists of variable-length segments, which permits the use of a combination of different types of finite elements (linear, quadratic, etc.) and/or specialized functions. The functional form and corresponding parameters are determined element-wise by solving a combinatorial optimization problem. The element size is also determined as part of the solution of the optimization problem, using a novel two-step encoding strategy. These building blocks result in an algorithm that is flexible and robust in solving optimal control problems. Furthermore, implementation is very simple.The algorithm's performance is studied with a challenging set of benchmark problems. Then LARES-PR is utilized to solve optimal control problems of systems described by population balance equations, including crystallization, nano-particle formation by nucleation/coalescence mechanism, and competitive reactions in a disperse system modeled by the Monte Carlo method. The algorithm is also applied to solving the DICE model of global warming, a complex discrete-time model.     

Simultaneous solution approach to model‐based experimental design

A model‐based experimental design is formulated and solved as a large‐scale NLP problem. The key idea of the proposed approach is the extension of model equations with sensitivity equations forming an extended sensitivities‐state equation system. The resulting system is then totally discretized and simultaneously solved as constraints of the NLP problem. Thereby, higher derivatives of the parameter sensitivities with respect to the control variables are directly calculated and exact. This is an advantage in comparison with proposed sequential approaches to model‐based experimental design so far, where these derivatives have to be additionally integrated throughout the optimization steps. This can end up in a high‐computational load especially for systems with many control variables. Furthermore, an advanced sampling strategy is proposed which combines the strength of the optimal sampling design and the variation of the collocation element lengths while fully using the entire optimization space of the simultaneous formulation. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4169–4183, 2013     

Optimization of injection molding design. Part II: Molding conditions optimization

The quality of an injection molded part is affected by many factors. These include geometric parameters associated with the mold design and the cooling system design as well as process parameters such as the molding conditions during the filling phase. In the companion paper, the problem of automatic optimization of gate location was addressed. In this paper, a methodology for molding condition optimization is presented. The optimization problem can be broken into three parts. An approximate feasible molding space (AFMS) is first determined to constrain the search space for the optimization algorithm. Quality is quantified as a function of flow simulation outputs and constitutes the objective function that must be minimized. The resulting optimization is solved by iterative search in the constrained space based on numerical optimization algorithms. The proposed methodology is not dependent on any particular simulation package and may be applied for any thermoplastic material and any complex mold geometry.     

Generalized initialization for the dynamic simulation and optimization of grade transition processes using two-dimensional collocation

Optimization modeling tools are essential to determine optimal design specifications and operation conditions of polymerization processes, especially when quality indices based on molecular weight distributions (MWDs) must be enforced. This study proposes a generalized MWD-based optimization strategy using orthogonal collocation in two dimensions, which can capture the dynamic features of MWDs accurately. To enable the strategy, this study considers generalized initialization methods for large-scale simulation and optimization. Here, a homotopy method based on intermediate solutions is adopted to generate initial values for general steady-state simulation models, starting from an arbitrary known solution for any steady-state simulation model. For dynamic simulation models, the response of a first-order linear system is adopted to initialize the state variables. Case studies show the effectiveness of this procedure to enable systematic, reliable, and efficient solution of the optimization problem.     

化工过程预测控制的在线优化实现机制

多层结构的预测控制已逐渐成为工业过程控制领域的主流控制方案。在此控制架构基础上,根据操作工或工艺优化所给定期望值的不同,将稳态优化问题具体化为两种基本情况,并对此提出基于复合目标函数的优化问题,可针对不同过程要求退化为线性、二次或二者兼有的优化问题形式。为保证最优目标的可行性并在一定程度上避免关键变量饱和,对不可行的期望值适当调整。将所得最优目标增量化处理后送入模型预测控制动态控制层,确保了上下层之间变量传递的一致性。包含约束的全混槽反应器系统仿真实例表明,流程的优化实现层可针对不同的过程要求有效给出最优目标以便动态控制,说明了该优化流程的可行性。     

Optimum design of integrated liquid recovery plants by variable population size genetic algorithm

Increase in the price of energy sources as well as economic problems have caused cryogenic natural gas plants to become more complex and efficient. After selecting the process configuration, the flow rate, pressure, and temperature of the process fluid streams are determining factors which should be tuned in order to find the optimum condition. Products specification and operating costs of the plant are two significant parameters which should be considered in an optimal design. Moreover, process design limitations contribute to the problem being more difficult. This paper shows how the optimal operating point in an integrated NGL recovery plant can be found through solving a complex constrained optimization problem. A Variable Population size Genetic Algorithm (VPGA) was used for optimization. As well, the role of VPGA algorithm parameters in solving the process design problems is investigated in this study. The analysis showed that the VPGA method has better performance compared to the general GA methods. The plant‐wide net profit increases 12493360 $/year only by changing the selected operating conditions to its optimal value.     

Numerical solution approaches for robust nonlinear optimal control problems

Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that leads to a bi-level min–max optimization problem. We propose and investigate a numerical method to solve this min–max optimization problem exactly in the case that the underlying maximization problem always has its solution on the boundary of the uncertainty set. This is an adoption of the local reduction approach used to solve generalized semi-infinite programs. The approach formulates an equilibrium constraint employing first order derivatives of both the uncertainty set and the user defined constraints. We propose two different ways for computation of these derivatives, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show the equivalence of the proposed approach to a method based on geometric considerations that was recently developed by some of the authors. We show how to generalize the techniques to optimal control problems. The robust dynamic optimization of a batch distillation illustrates that both techniques are numerically efficient and able to overcome the inexactness of another recently proposed numerical approach to address uncertainty in optimal control problems.     

二元提馏式间歇精馏的优化操作与最小汽化总量

理想操作条件下二元提馏式间歇精馏优化操作的汽化总量与最小汽化总量的计算是约束函数优化问题。本文采用罚函数法,将此约束函数优化转变为无约束函数优化,并采用固定双步长因子梯度法数值求解该函数的极值。计算表明:固定双步长因子梯度法具有良好的收敛性,同时,降低分段数较多时,数值截断误差积累对计算结果的影响。二元提馏式间歇精馏优化操作较恒残液组成操作的能耗低的原因如下:在理论板数相对较少(接近二元提馏式间歇精馏恒残液组成操作所需的最少理论板)时,优化操作通过控制再沸比提高了能耗效率;在理论板数相对较多时,优化操作通过控制再沸比,在保证过程的能耗效率较高的同时,可尽可能快地将物料移出系统,减少了精馏过程中塔顶贮槽内液体的混合熵产。通过对计算结果的归纳与外推,得到了理想操作条件下理论板数为无穷多时二元提馏式间歇精馏优化操作再沸比的变化方式以及最小汽化总量的计算公式。     

Robust supply chain performance via Model Predictive Control

This paper presents a novel robust Model Predictive Control (MPC) method for real-time supply chain optimization under uncertainties. This method optimizes the closed-loop economic performance of supply chain systems and addresses different sources of uncertainties located external to and within the feedback loop. The future system behavior is predicted by a closed-loop model, which includes both the open-loop system model and a controller model described by an optimization problem. The robust MPC formulation involves the solution of a constrained, bi-level stochastic optimization problem, which is transformed into a tractable problem involving a limited number of deterministic conic optimization problems solved reliably using an interior point method. The robust controller is applied to a real industrial multi-echelon supply chain optimization problem, and its performance is shown to reduce stock-outs without excessive inventories.     

Linear model predictive control for transport‐reaction processes

The article deals with systematic development of linear model predictive control algorithms for linear transport‐reaction models emerging from chemical engineering practice. The finite‐horizon constrained optimal control problems are addressed for the systems varying from the convection dominated models described by hyperbolic partial differential equations (PDEs) to the diffusion models described by parabolic PDEs. The novelty of the design procedure lies in the fact that spatial discretization and/or any other type of spatial approximation of the process model plant is not considered and the system is completely captured with the proposed Cayley‐Tustin transformation, which maps a plant model from a continuous to a discrete state space setting. The issues of optimality and constrained stabilization are addressed within the controller design setting leading to the finite constrained quadratic regulator problem, which is easily realized and is no more computationally intensive than the existing algorithms. The methodology is demonstrated for examples of hyperbolic/parabolic PDEs. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2644–2659, 2017     

Analysis of the local well-posedness of optimization-constrained differential equations by local optimality conditions

Optimization-constrained differential equations (OCDE) are a class of mathematical problems where differential equations are constrained by an embedded algebraic optimization problem. We analyze the well-posedness of the local solutions of OCDE based on local optimality. By assuming linear independence constraint qualification and applying the Karush-Kuhn-Tucker optimality conditions, an OCDE is transformed into a complementarity system (CS). Under second-order sufficient condition we show that (a) if strict complementary condition (SCC) holds, the local solution of OCDE is well-posed, which corresponds to a mode of the derived CS; (b) at points where SCC is violated, a local solution of OCDE exists by sequentially connecting the local solutions of two selected modes of the derived CS. We propose an event-based algorithm to numerically solve OCDE. We illustrate the approach and algorithm for microbial cultivation, single flash unit and contrived numerical examples.     

Optimization issues of the broke management system in papermaking

This paper presents an optimization strategy for the design and operation of a broke management system in a papermaking process. A stochastic model based on a two-state Markov process is presented for the broke system and a multiobjective and bi-level stochastic optimization model is developed featuring (i) a multiobjective operational subproblem for the optimization of the broke dosage and (ii) a multiobjective design problem formulation. An efficient optimization strategy is proposed for the operational subproblem along with a simulation based Pareto optimal solution for the design problem, and illustrated with a detailed case study.     

A Simplified Numerical Approach to Feedback Controller Design

A generalized parameter optimization method for computing feedback controller parameters is proposed. The method utilizes the downhill simplex method (DSM), a pattern search algorithm, to determine the optimal parameters that minimize an objective function or performance index. The system model, expressed in terms of state‐space equations is integrated with respect to time at each DSM iteration in order to determine the states. A fourth‐order Runge‐Kutta scheme is used for integrating the state equations. A penalty function approach is used for problems with inequality constraints on the state variables or controls. Though relatively inefficient in terms of the number of function evaluations, DSM requires only that the user provide the model equations, and not their derivatives. Additionally, the DSM code is very compact. Thus, a small and straightforward program allows for controller parameter determination for a variety of state‐space and classical PID feedback control design problems.     

Optimal control of linear and nonlinear systems by two-level optimization

The decomposition of a system into simpler subsystems, followed by the optimization of the individual subsystems, and then co-ordinating the subsystem optimal solutions to yield the optimal control policy for the original system, is considered. Lasdon's co-ordination algorithm, in which the subsystems are regarded as being completely independent, is improved by including the interdependence of the adjacent subsystems. For linear systems with a quadratic performance index, this procedure yields the optimal control policy in a single iteration from an arbitrary initial guess of the decomposition parameters. For nonlinear systems, the method includes linearizing the state equations, the calculation of the optimal control for the linearized system, and the adjustment of the corresponding decomposition parameters of the original problem until convergence is achieved. For the optimization of a nonlinear continuous stirred tank reactor such a procedure requires considerably more computation time than the standard techniques, and the optimum policies are less accurate.     

不确定条件下的湿法炼锌除铜过程机会约束优化控制

除铜过程是湿法炼锌净化工艺中的重要步骤,受生产环境多变、矿源多样、机理复杂等因素的影响,除铜过程存在不确定性,影响生产的稳定性和可靠性。针对除铜过程中入口溶液流量、底流返回量和入口铜离子浓度的不确定性,造成出口铜离子浓度不稳定的问题,研究不确定条件下的除铜过程机会约束优化控制方法。首先分析了除铜过程的不确定性,利用统计学方法分析不确定参数的分布特性,引入了机会约束的思想,将不确定条件下的除铜过程优化问题建模为机会约束优化问题。然后采用可行域映射方法,将机会约束优化问题转化为非线性规划问题。最后,使用序列二次规划求解该非线性规划问题。Monte Carlo仿真验证了该方法的有效性,可以提高系统的鲁棒性。     

Constrained robust Bayesian optimization of expensive noisy black-box functions with guaranteed regret bounds

Many real-world design problems involve optimization of expensive black-box functions. Bayesian optimization (BO) is a promising approach for solving such challenging problems using probabilistic surrogate models to systematically tradeoff between exploitation and exploration of the design space. Although BO is often applied to unconstrained problems, it has recently been extended to the constrained setting. Current constrained BO methods, however, cannot identify solutions that are robust to unavoidable uncertainties. In this article, we propose a robust constrained BO method, constrained adversarially robust Bayesian optimization (CARBO), that addresses this challenge by jointly modeling the effect of the design variables and uncertainties on the unknown functions. Using exact penalty functions, we establish a bound on the number of CARBO iterations required to find a near-global robust solution and provide a rigorous proof of convergence. The advantages of CARBO are demonstrated on two case studies including a non-convex benchmark problem and a realistic bubble column reactor design problem.     

Simultaneous optimization and solution of systems described by differential/algebraic equations

Engineering approaches to the solution of constrained variational problems often involve converting the problem into a nonlinear programming (NLP) problem and solving it using current NLP methods. These methods usually use a sequential optimization and solution strategy. We propose a method, using piecewise constant functions for the independent variables, that combines the technologies of quasi-Newton optimization algorithms and global spline collocation to simultaneously optimize and integrate systems described by differential/algebraic equations. A computer implementable algorithm is discussed and three test problems are solved. The algorithm allows the solution of a more general class of optimization problems than previous methods employing this strategy.