This work considers the numerical optimization of constrained batch and semi-batch processes, for which direct as well as indirect methods exist. Direct methods are often the methods of choice, but they exhibit certain limitations related to the compromise between feasibility and computational burden. Indirect methods, such as Pontryagin's Minimum Principle (PMP), reformulate the optimization problem. The main solution technique is the shooting method, which however often leads to convergence problems and instabilities caused by the integration of the co-state equations forward in time.This study presents an alternative indirect solution technique. Instead of integrating the states and co-states simultaneously forward in time, the proposed algorithm parameterizes the inputs, and integrates the state equations forward in time and the co-state equations backward in time, thereby leading to a gradient-based optimization approach. Constraints are handled by indirect adjoining to the Hamiltonian function, which allows meeting the active constraints explicitly at every iteration step. The performance of the solution strategy is compared to direct methods through three different case studies. The results show that the proposed PMP-based quasi-Newton strategy is effective in dealing with complicated constraints and is quite competitive computationally.

Nonlinear model predictive control (NMPC) is an important tool to perform real-time optimization for batch and semi-batch processes. Direct methods are often the methods of choice to solve the corresponding optimal control problems, in particular for large-scale problems. However, the matrix factorizations associated with large prediction horizons can be computationally demanding. In contrast, indirect methods can be competitive for smallerscale problems. Furthermore, the interplay between states and co-states in the context of Pontryagin's Minimum Principle might turn out to be computationally quite efficient. This work proposes to use an indirect solution technique within shrinking-horizon in the context of NMPC. In particular, the technique deals with path constraints via indirect adjoining, which allows meeting active path constraints explicitly at each iteration. Uncertainties are handled by the introduction of time-varying backoff terms for the path 2 constraints. The resulting NMPC algorithm is applied to a two-phase semi-batch reactor for the hydroformylation of 1-dodecene in the presence of uncertainty, and its performance is compared to that of NMPC that uses a direct simultaneous optimization method. The results show that the proposed algorithm (i) can enforce feasible operation for different uncertainty realizations both within batch or from batch to batch, and (ii) it is significantly faster than direct simultaneous NMPC, especially at the beginning of the batch. In addition, a modification of the PMP-based NMPC scheme is proposed that enforces active constraints via tracking.

Dynamic optimization plays an important role toward improving the operation of chemical systems, such as batch and semibatch processes. The preferred strategy to solve constrained nonlinear dynamic optimization problems is to use a so-called direct approach. Nevertheless, based on the problem at hand and the solution algorithm used, direct approaches may lead to large computational times. Indirect approaches based on Pontryagin's Minimum Principle (PMP) represent an efficient alternative for the optimization of batch and semibatch processes. This paper details the combination of an indirect solution scheme together with a parsimonious input parametrization. The idea is to parametrize the sensitivity-seeking inputs in a parsimonious way so as to decrease the computational load of constrained nonlinear dynamic optimization problems. In addition, this article discusses structural differences between direct and indirect approaches. The proposed method is tested on both a batch binary distillation column with terminal purity constraints and a two-phase semibatch hydroformylation reactor with a complex path constraint. The performance of the proposed indirect parsimonious solution scheme is compared with those of a fully parametrized PMP-based method and a direct simultaneous method. It is observed that the combination of the indirect approach with parsimonious input parametrization can lead to significant reduction in computational time.

Artificial Neural Networks (ANNs) have been used in a wide range of applications for complex datasets with their flexible mathematical architecture. The flexibility is favored by the introduction of a higher number of connections and variables, in general. However, over-parameterization of the ANN equations and the existence of redundant input variables usually result in poor test performance. This paper proposes a superstructure-based mixed-integer nonlinear programming method for optimal structural design including neuron number selection, pruning, and input selection for multilayer perceptron (MLP) ANNs. In addition, this method uses statistical measures such as the parameter covariance matrix in order to increase the test performance while permitting reduced training performance. The suggested approach was implemented on two public hyperspectral datasets (with 10% and 50% sampling ratios), namely Indian Pines and Pavia University, for the classification problem. The test results revealed promising performances compared to the standard fully connected neural networks in terms of the estimated overall and individual class accuracies. With the application of the proposed superstructural optimization, fully connected networks were pruned by over 60% in terms of the total number of connections, resulting in an increase of 4% for the 10% sampling ratio and a 1% decrease for the 50% sampling ratio. Moreover, over 20% of the spectral bands in the Indian Pines data and 30% in the Pavia University data were found statistically insignificant, and they were thus removed from the MLP networks. As a result, the proposed method was found effective in optimizing the architectural design with high generalization capabilities, particularly for fewer numbers of samples. The analysis of the eliminated spectral bands revealed that the proposed algorithm mostly removed the bands adjacent to the pre-eliminated noisy bands and highly correlated bands carrying similar information.