Optimal parameter selection for nonlinear multistage systems with time-delays

In this paper, we consider a novel dynamic optimization problem for nonlinear multistage systems with time-delays. Such systems evolve over multiple stages, with the dynamics in each stage depending on both the current state of the system and the state at delayed times. The optimization problem involves choosing the values of the time-delays, as well as the values of additional parameters that influence the system dynamics, to minimize a given cost functional. We first show that the partial derivatives of the system state with respect to the time-delays and system parameters can be computed by solving a set of auxiliary dynamic systems in conjunction with the governing multistage system. On this basis, a gradient-based optimization algorithm is proposed to determine the optimal values of the delays and system parameters. Finally, two example problems, one of which involves parameter identification for a realistic fed-batch fermentation process, are solved to demonstrate the algorithm’s effectiveness.     

Parameter identification in multistage population dynamics model

We present a numerical analysis to solve a parameter identification problem. We identify the demographical parameters of a multistage population dynamics model (Ainseba et al., 2011 [12]). Our nonlinear optimization problem with constraints is solved by a Quasi-Newton method. The convergence proof of this numerical method is performed here. Some numerical applications of it are also given at the end of the paper.     

A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming

This paper presents a new and high performance solution method for multistage stochastic convex programming. Stochastic programming is a quantitative tool developed in the field of optimization to cope with the problem of decision-making under uncertainty. Among others, stochastic programming has found many applications in finance, such as asset-liability and bond-portfolio management. However, many stochastic programming applications still remain computationally intractable because of their overwhelming dimensionality. In this paper we propose a new decomposition algorithm for multistage stochastic programming with a convex objective and stochastic recourse matrices, based on the path-following interior point method combined with the homogeneous self-dual embedding technique. Our preliminary numerical experiments show that this approach is very promising in many ways for solving generic multistage stochastic programming, including its superiority in terms of numerical efficiency, as well as the flexibility in testing and analyzing the model.Research supported by Hong Kong RGC Earmarked Grant CUHK4233/01E.     

A dynamic programming approach to adjustable robust optimization

In this paper we consider the adjustable robust approach to multistage optimization, for which we derive dynamic programming equations. We also discuss this from the point of view of risk averse stochastic programming. We consider as an example a robust formulation of the classical inventory model and show that, like for the risk neutral case, a basestock policy is optimal.     

Design of optimal utility of wireless rechargeable sensor networks via joint spatiotemporal scheduling

In wireless rechargeable sensor networks, how to optimize energy resources for maximizing the sensor data is a challenging problem. In this paper, mobile charging vehicle scheduling, sensor charging time splitting and rate control with battery capacity constraints are considered together to maximize network utility. However, they are considered independently in exist works even though these problems are interdependent. In order to improve network performance through collaborative optimization of three problems, a joint optimization problem is formulated firstly. Then, a multistage approach is developed to jointly optimize the three subproblems iteratively. Furthermore, an accelerated distributed algorithm is integrated to improve the convergence speed of rate control. The results of extended experiments demonstrate that proposed approach can obtain higher network utility and charging efficiency compared to other charging scheduling methods.     

多目标多层次系统多维模糊决策理论模型

分析了模糊优选理论模型,指出了其对大系统决策的局限性,利用模糊模式识别理论与模糊关系优选理论,建立了多目标多层次系统多维模糊决策理论模型,进一步丰富了系统模糊决策理论。     

Computational strategies for non-convex multistage MINLP models with decision-dependent uncertainty and gradual uncertainty resolution

In many planning problems under uncertainty the uncertainties are decision-dependent and resolve gradually depending on the decisions made. In this paper, we address a generic non-convex MINLP model for such planning problems where the uncertain parameters are assumed to follow discrete distributions and the decisions are made on a discrete time horizon. In order to account for the decision-dependent uncertainties and gradual uncertainty resolution, we propose a multistage stochastic programming model in which the non-anticipativity constraints in the model are not prespecified but change as a function of the decisions made. Furthermore, planning problems consist of several scenario subproblems where each subproblem is modeled as a nonconvex mixed-integer nonlinear program. We propose a solution strategy that combines global optimization and outer-approximation in order to optimize the planning decisions. We apply this generic problem structure and the proposed solution algorithm to several planning problems to illustrate the efficiency of the proposed method with respect to the method that uses only global optimization.     

Multistage portfolio optimization with multivariate dominance constraints

Computational Management Science - In this article we focus on multistage portfolio optimization problem with usage of multivariate stochastic dominance constraints. The first part of the work is...     

微生物批式流加发酵多阶段最优控制的最优性条件

在微生物批式流加发酵生产1,3一丙二醇(1,3-PD)过程中,关键是如何控制甘油和碱的流加速度.本文将流加速度看成一个随时间变化的控制函数,提出一个带控制的多阶段动力系统描述批式发酵过程,并证明了系统的一些性质.以终端时刻1,3-PD的生产强度最大为性能指标,以上述动力系统和连续状态不等式为约束条件建立了最优控制模型,最后利用不可微优化理论得到了最优控制问题的最优性条件,并证明了最优性条件和最优性函数零点的等价性.     

基于效用比的多工序并联制造过程能力分析

多工序制造过程通常包含串联和并联两种结构,具有串并联混合结构的多工序制造过程是实践中最为常见的形式,而不同模式的并联结构其上游工序质量波动对下游工序及总过程能力的影响不同。针对多工序制造过程并联结构特点,本文从波动减少的角度重点对多工序并联制造过程中并行、分散和收敛三种基本模式进行过程能力分析,研究多工序制造过程中各子过程波动对整体过程能力的影响,并根据各子过程质量波动减少的“困难度”和“效用比”评价质量改进的效果,给出多工序并联过程能力改进策略选择依据。通过实例表明,本方法能较好地识别各工序质量波动减少对本工序过程能力和总过程能力的不同影响,确定质量改进的优先顺序,实现多工序制造过程的经济性质量改进和优化。     

Multistage Markov Chain Modeling of the Genetic Algorithm and Convergence Results

The genetic algorithm (GA) has been widely used to solve combinatorial global optimization problems. Despite the successes that GA encounters in practical applications, there exist few precise results on its behavior. In this article, we formulate a fully rigorous mathematical modeling of GA as a multistage Markov chain and derive convergence results. Variations that include the simulated annealing algorithm and the GA with superindividual are considered.     

优化路问题的代数方法—论动态规划（Ⅱ）

本文用同一思路求解多阶段有向图中三种优化路问题:最优路、N阶最优路及多指标Pareto优化路问题,它们都服从嘉量原理,都用同一个代数公式表达它们的嘉量,并可在同一种表格中进行计算,只是所在半域不同,以本文的方法讨论动态规划中一些离散决定型典型应用问题,其提法、建模思路以及求解过程都有可观的扩大与改善。     

Stochastic programming approach to optimization under uncertainty

In this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems. We argue that two stage (say linear) stochastic programming problems can be solved with a reasonable accuracy by Monte Carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages. We discuss an extension of coherent risk measures to a multistage setting and, in particular, dynamic programming equations for such problems.      

Subregular recourse in nonlinear multistage stochastic optimization

We consider nonlinear multistage stochastic optimization problems in the spaces of integrable functions. We allow for nonlinear dynamics and general objective functionals, including dynamic risk measures. We study causal operators describing the dynamics of the system and derive the Clarke subdifferential for a penalty function involving such operators. Then we introduce the concept of subregular recourse in nonlinear multistage stochastic optimization and establish subregularity of the resulting systems in two formulations: with built-in nonanticipativity and with explicit nonanticipativity constraints. Finally, we derive optimality conditions for both formulations and study their relations.

     

An optimistic decision-making in fuzzy environment

Bellman and Zadeh have originated three systems of multistage decision processes in a fuzzy environment: deterministic, stochastic and fuzzy systems. In this article, we consider an optimization problem with an optimistic criterion on a fuzzy system. By making use of minimization–maximization expectation in a fuzzy environment, we derive a recursive equation for the fuzzy decision process through invariant imbedding approach. By illustrating a three-state, two-decision and two-stage model, we give an optimal solution through dynamic programming. The optimal solution is also verified by the method of multistage fuzzy decision tree-table.     

Mixed-integer minimax dynamic optimization for structure identification of glycerol metabolic network

Cell metabolism is a dynamic regulation process, in which its network structure and/or regulatory mechanisms can change constantly over time due to internal and external perturbations. This paper models glycerol metabolism in continuous fermentation as a nonlinear mixed-integer dynamic system by defining the time-varying metabolic network structure as an integer-valued function. To identify the dynamic network structure and kinetic parameters, we establish a mixed-integer minimax dynamic optimization problem with concentration robustness as its objective functional. By direct multiple shooting strategy and a decomposition approach consisting of convexification, relaxation and rounding strategy, the optimization problem is transformed into a large-scale approximate multistage parameter optimization problem. It is then solved using a competitive particle swarm optimization algorithm. We also show that the relaxation problem yields the best lower bound for the optimization problem, and its solution can be arbitrarily approximated by the solution obtained from rounding strategy. Numerical results indicate that the proposed mixed-integer dynamic system can better describe cellular self-regulation and response to intermediate metabolite inhibitions in continuous fermentation of glycerol. These numerical results show that the proposed numerical methods are effective in solving the large-scale mixed-integer dynamic optimization problems.     

A general framework for multistage mean-variance post-tax optimization

An investor’s decisions affect the way taxes are paid in a general portfolio investment, modifying the net redemption value and the yearly optimal portfolio distribution. We investigate the role of these decisions on multistage mean-variance portfolio allocation model. A number of risky assets grouped in wrappers with special taxation rules is integrated in a multistage financial portfolio optimization problem. The uncertainty on the returns of assets is specified as a scenario tree generated by simulation/clustering based approach. We show the impact of decisions in the yearly reallocation of the investments for three typical cases with an annual fixed withdrawal in a fixed horizon that utilizes completely the option of taper relief offered by banks in UK. Our computational framework can be used as a tool for testing decisions in this context.     

Multi-stage stochastic optimization applied to energy planning

This paper presents a methodology for the solution of multistage stochastic optimization problems, based on the approximation of the expected-cost-to-go functions of stochastic dynamic programming by piecewise linear functions. No state discretization is necessary, and the combinatorial explosion with the number of states (the well known curse of dimensionality of dynamic programming) is avoided. The piecewise functions are obtained from the dual solutions of the optimization problem at each stage and correspond to Benders cuts in a stochastic, multistage decomposition framework. A case study of optimal stochastic scheduling for a 39-reservoir system is presented and discussed.     

On Optimality Conditions for Some Nonsmooth Optimization Problems over <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Spaces

The paper deals with the minimization of an integral functional over an L^p space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on several concepts of nonsmooth analysis. In particular, we employ the generalized differential calculus of Mordukhovich and the fuzzy calculus of proximal subgradients. The results are specialized to nonsmooth two-stage and multistage stochastic programs.The authors express their gratitude to Boris Mordukhovich (Detroit) for his extensive support during this research and to Marian Fabian (Prague) and Alexander Kruger (Ballarat) for valuable discussions. They are indebted also to two anonymous referees for helpful suggestions.The research of this author was partly supported by Grant 1075005 of the Czech Academy of SciencesThe research of this author was supported by the Deutsche Forschungsgemeinschaft     

Selecting preferred solutions in the minimax approach to dynamic programming problems under flexible constraints

Dynamic Programming is a powerful approach to the optimization of sequential or multistage decision processes, e.g., in planning or in system control. In this paper, we consider both theoretical and algorithmic issues in sequential decision processes under flexible constraints. Such processes must attain a given goal within some tolerance. Tolerances or preferences also apply to the values the decision variables may take or on the action chosen at each step. Such problems boil down to maximin optimization. Unfortunately, this approach suffers from the so-called “drowning effect” (lack of discrimination) and the optimality principle of dynamic programming is not always verified. In this context, we introduce a general framework for refined minimax optimization procedures in order to compare and select preferred alternatives. This framework encompasses already introduced methods such as LexiMin and DiscriMin, but it allows their extension to the comparison of vectors of unequal lengths. We show that these refined comparisons restore compatibility with the optimality principle, and that classical algorithms can be adapted to compute such preferred solutions, by exploiting existing results on idempotent semirings.