Efficient numerical methods for hierarchical dynamic optimization with application to cerebral palsy gait modeling

In the development of complex engineering systems, many engineers from different disciplines collaborate to identify feasible designs that satisfy all system requirements. Analytical target cascading (ATC) is a method for the design optimization of hierarchical, multilevel systems and has been successfully employed in the design of complex engineering systems. In this paper, we propose a novel data‐driven set‐based ATC (SBATC) method for hierarchical design optimization problems using machine learning techniques. The proposed SBATC offers two advantages in engineering processes. First, it decomposes hierarchical design optimization problems into sets of suboptimization problems. The feasible regions of the suboptimization problems are explored and cascaded to lower‐level optimization problems. Using the set‐based approach, couplings between two adjacent levels in the optimization process are not required. Second, an efficient strategy is employed to determine feasible regions based on Bayesian active learning (BAL). In BAL, the Gaussian process (GP) of all cost functions is trained. An acquisition function that combines the probability of feasibility and entropy search is evaluated using posterior distributions of the trained GP. The acquisition function is maximized to generate new sampling points around the feasible regions by balancing the exploitation and exploration of the design space. To verify the effectiveness of the proposed method, numerical examples of hierarchical optimization problems are evaluated.

In this paper, an interactive trade-off sequential optimization method for multiobjective decision problem is integrated with the hierarchical optimization technique,and a hierarchical multiobjective optimization problem for large-scale time-delay system is studied. Decomposition of problem in decision and objective spaces is directly realized through incorporating the necessary conditions for the preferred solution with /spl epsiv/-constraint problem, and a hierarchical multiobjective optimization algorithm for large-scale time-delay system is developed.It can be considered as an extension of the time-delay algorithm of Tamura to multiobjective case. Finally,the algorithm proposed is applied to ten years educational planning with three criteria and five -year time-delay. It is shown that the advantages of less computation time and fast convergence can be gained by using the proposed algorithm.

Analytical Target Cascading method has been widely developed to solve hierarchical design optimization problems. In the Analytical Target Cascading method, a weighted-sum formulation has been commonly used to coordinate the inconsistency between design points and assigned targets in each level while minimizing the cost function. However, the choice of the weighting coefficients is very problem dependent and improper selections of the weights will lead to incorrect solutions. To avoid the problems associated with the weights, single objective functions in the hierarchical design optimization are formulated by a new Bounded Target Cascading method. Instead of point targets assigned for design variables in the Analytical Target Cascading method, bounded targets are introduced in the new method. The target bounds are obtained from the optimal solutions in each level while the response bounds are updated back to the system level. If the common variables exist, they are coordinated based on their sensitivities with respect to design variables. Finally, comparisons of the results from the proposed method and the weighted-sum Analytical Target Cascading are presented and discussed.

A novel approach to find optimal real power generation pattern will be introduced in this study. This has been achieved by maximizing the power system voltage stability margin and minimizing generation cost. The loadability limit index is used to assess static voltage stability security margin. Based on proposed algorithm, a Toolbox has been developed to recognize the loadability limit by using Lagrangian method. The mentioned problem has been modeled as a non-linear and multi objective optimization problem along with Hybrid Genetic Algorithm Particle Swarm Optimization (HGAPSO) method to reach the optimal generation pattern. The method has been performed as a hierarchical optimization problem. In the first stage, HGAPSO method has been used to reach the securest pattern of real power. Also, mentioned voltage stability toolbox has been executed to evaluate the voltage stability margin respectively each real power generation pattern during HGAPSO search. These simulations are performed on IEEE 30 bus test system. To validation of proposed method, obtained results have been compared with Particle Swarm Optimization (PSO) results.

A large-scale machine system often has a general hierarchical structure. For hierarchical structures, optimization is difficult because many local optima exist in the hierarchical optimization problem, however genetic algorithms that have a hierarchical genotype can be applied to treat such problems directly. Relations between the structural components are analyzed and this information used to divide the hierarchical structure. Dividing large-scale problems into sub-problems that can be solved using parallel processed GAs increases the efficiency of the optimization search. The optimization of the large-scale system then becomes possible due to information sharing concerning Pareto optimum solutions for the sub-problems.

Large-scale machine system optimization problems usually resist solution because large-scale machine systems often have a hierarchical general structure that produces numerous local optirna during hierarchical optimization. Genetic algorithms having hierarchical genotypes can be directly applied to the optimization of such problems, however the appearance of local optima during hierarchical optimization procedures often foils effective searching. To enable the global optimization of such problems, skillful analysis of the component relationships comprising the hierarchical problem allows the hierarchical structure to be divided into more tractable units. Divided optimization problems can then be optimized via multiple crenetic algorithm processes, and global large-scale optimization problems can be represented as hierarchical sets of genetic algorithm processes. The exchange of local searching information between these genetic algorithm processes enables global searching and parallel processing, both of which enhance the efficiency and quality of searching for optimum solutions to large-scale optimization problems.

In this paper, the hierarchical optimization problem of multi-player systems with matched uncertainties is investigated via adaptive dynamic programming. In the hierarchical optimization problem, there exist one leader and multiple followers, the leader chooses a policy in advance based on the actions of the followers, and the followers make optimal responses to the leader's policy. The optimal policies of the leader and the followers form the Stackelberg equilibrium. By designing appropriate value functions for the leader and the followers, the hierarchical optimization problem is formulated as a Stackelberg game and the robust stabilization problem is transformed into an optimal regulation problem. Moreover, the critic-only structure is established to obtain the approximate Stackelberg equilibrium. Theoretical analysis shows that the developed ADP-based robust control guarantees the closed-loop system to be asymptotically stable. Finally, simulation example is adopted to verify the effectiveness of the present scheme.

This paper studies a hierarchical optimization problem of scheduling n jobs on a serial-batching machine, in which two objective functions are maximum costs. By a hierarchical optimization problem, we mean the problem of optimizing the secondary criterion under the constraint that the primary criterion is optimized. A serial-batching machine is a machine that can handle up to b jobs in a batch and jobs in a batch start and complete respectively at the same time and the processing time of a batch is equal to the sum of the processing times of jobs in the batch. When a new batch starts, a constant setup time s occurs. We confine ourselves to the bounded model, where b < n. We present an O(n4)-time algorithm for this hierarchical optimization problem. For the special case where two objective functions are maximum lateness, we give an O(n3 log n)-time algorithm.

The management of a food supply chain is difficult and complex because of the product's short shelf-life, time-sensitivity, and perishable nature which must be carefully considered to minimize food waste. Temperature-controlled perishable food supply chain provides the highly crucial facilities necessary to maintain the quality and safety of the product. The storage temperature is the most vital factor in maintaining both the quality and shelf-life of a perishable food. Adequate storage temperature control ensures that perishable foods are transported to the end-users in good quality and safe to consume. This paper presents perishable food storage temperature control through mathematical optimal control model where the storage temperature is regarded as the control variable and the deterioration of the perishable food’s quality follows the first-order reaction. The optimal storage temperature for a single perishable food is determined by applying the Pontryagin's maximum principle to solve the optimal control model problem. For multi-temperature commodities supply chain, an unsupervised machine learning (ML) method, called k-means clustering technique is used to determine the temperature clusters for a range of perishables. Based on descriptive analysis, it is observed that the k-means clustering technique is effective in identifying the best suitable storage temperature clusters for quality control of multi-commodity supply chain.

A new product's introduction to the market is greatly influenced by effective and enough advertising. In this paper, we have considered a problem in which the firm partitions the market into various segments to reduce costs associated with advertising and targets to maximize the total profit. By integrating single channel and differentiated advertising to segmented market, we extend Nerlove-goodwill Arrow's dynamic model in which advertising variables are control variables. It is assumed that the whole available budget is imprecise and fuzzy in nature in order to create a realistic model. Using necessity and possibility constraints, the optimal control model with fuzzy parameters is transformed into crisp form, and the Pontryagin Maximum principle is then used to solve the problem. Numerical examples are provided to support the theoretical analysis.

In this paper, we consider furtivity and masking problems in time-dependent three-dimensional electromagnetic obstacle scattering. That is, we propose a criterion based on a merit function to minimize or to mask the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and, with respect to this criterion, we derive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time-dependent acoustic obstacle scattering presented in Ref. 1. We propose mathematical models of the furtivity and masking time-dependent three-dimensional electromagnetic scattering problems that consist in optimal control problems for systems of partial differential equations derived from the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first-order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover, the first-order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series of the type considered in Refs. 2-3. Finally, we assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Impressive speedup factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in the website http://www.econ.unian.it,/recchioni/w10/.

In this paper, we consider one optimal control problem with a multipoint quality functional described by a system of nonlinear hyperbolic equations with Goursat boundary conditions. Using a modified version of the increment method, various necessary first-order optimality conditions such as the Pontryagin maximum principle and the linearized maximum condition are proved.

Part I of the thesis investigates novel urban traffic state estimation methods utilizing probe vehicle data. Chapter 2 proposes a method to integrate the collective effect of dispersed probe data with traffic kinematic wave theory and data mining techniques to model the spatial and temporal dynamics of queue formation and dissipation in arterials. The queue estimation method captures interdependencies in queue evolutions of successive intersections, and moreover, the method is applicable in oversaturated conditions and includes a queue spillover statistical inference procedure. Chapter 3 develops a travel time reliability model to estimate arterial route travel times distribution (TTD) considering spatial and temporal correlations between traffic states in consecutive links. The model uses link-level travel time data and a heuristic grid clustering method to estimate the state structure and transition probabilities of a Markov chain. By applying the Markov chain procedure, the correlation between states of successive links is integrated and the route-level TTD is estimated. The methods in Part I are tested with various probe vehicle penetration rates on case studies with field measurements and simulated data. The methods are straightforward in implementation and have demonstrated promising performance and accuracy through numerous experiments. Part II studies network-level modeling and control of large-scale urban networks. Chapter 4 is the pioneer that introduces the urban perimeter control for two-region urban cities as an elegant control strategy to decrease delays in urban networks. Perimeter controllers operate on the border between the two regions, and manipulate the percentages of transfer flows between the two regions, such that the number of trips reaching their destinations is maximized. The optimal perimeter control problem is solved by the model predictive control (MPC) scheme, where the prediction model and the plant (reality) are formulated by macroscopic fundamental diagrams (MFD). Chapter 5 extends the perimeter control strategy and MFD modeling to mixed urban-freeway networks to provide a holistic approach for large-scale integrated corridor management (ICM). The network consists of two urban regions, each one with a well-defined MFD, and a freeway, modeled by the asymmetric cell transmission model, that is an alternative commuting route which has one on-ramp and one off-ramp within each urban region. The optimal traffic control problem is solved by the MPC approach to minimize total delay in the entire network considering several control policies with different levels of urban-freeway control coordination. Chapter 6 integrates traffic heterogeneity dynamics in large-scale urban modeling and control to develop a hierarchical control strategy for heterogeneously congested cities. Two aggregated models, region- and subregion-based MFDs, are introduced to study the effect of link density heterogeneity on the scatter and hysteresis of MFD. A hierarchical perimeter flow control problem is proposed to minimize the network delay and to homogenize the distribution of congestion. The first level of the hierarchical control problem is solved by the MPC approach, where the prediction model is the aggregated parsimonious region-based MFD and the plant is the subregion-based MFD, which is a more detailed model. At the lower level, a feedback controller tries to maximize the network outflow, by increasing regional homogeneity.

For optimal control problems, described by the Gursat-Darboux stochastic system, a number of first-order necessary optimality conditions are formulated and proved, which are the stochastic analogue - the Pontryagin maximum principle, the linearized maximum principle and the Euler equation.

This paper deals with the reconstruction of the desired demand in an optimal control problem, stated over a tree-shaped transportation network which is governed by a linear hyperbolic conservation law. As desired demands typically undergo fluctuations due to seasonality or unexpected events making short-term adjustments necessary, such an approach can exemplary be used for forecasting from past data. We suggest to model this problem as a so-called inverse optimal control problem, i.e., a hierarchical optimization problem whose inner problem is the optimal control problem and whose outer problem is the reconstruction problem. In order to guarantee the existence of solutions in the function space framework, the hyperbolic conservation law is interpreted in weak sense allowing for control functions in Lebesgue spaces. For the computational treatment of the model, we transfer the hierarchical problem into a nonsmooth single-level one by plugging the uniquely determined solution of the inner optimal control problem into the outer reconstruction problem before applying techniques from nonsmooth optimization. Some numerical experiments are presented to visualize various features of the model including different types of noise in the demand and strategies of how to observe the network in order to obtain good reconstructions of the desired demand.