The automated architecture search methodology for neural networks is known as Neural Architecture Search (NAS). In recent times, Convolutional Neural Networks (CNNs) designed through NAS methodologies have achieved very high performance in several fields, for instance image classification and natural language processing. Our work is in the same domain of NAS, where we traverse the search space of neural network architectures with the help of an evolutionary algorithm which has been augmented with a novel approach of piecemeal-training. In contrast to the previously published NAS techniques, wherein the training with given data is considered an isolated task to estimate the performance of neural networks, our work demonstrates that a neural network architecture and the related weights can be jointly learned by combining concepts of the traditional training process and evolutionary architecture search in a single algorithm. The consolidation has been realised by breaking down the conventional training technique into smaller slices and collating them together with an integrated evolutionary architecture search algorithm. The constraints on architecture search space are placed by limiting its various parameters within a specified range of values, consequently regulating the neural networktextquoterights size and memory requirements. We validate this concept on two vastly different datasets, namely, the CIFAR-10 dataset in the domain of image classification, and PAMAP2 dataset in the Human Activity Recognition (HAR) domain. Starting from randomly initialized and untrained CNNs, the algorithm discovers models with competent architectures, which after complete training, reach an accuracy of of 92.5% for CIFAR-10 and 94.36% PAMAP2. We further extend the algorithm to include an additional conflicting search objective: the number of parameters of the neural network. Our multi-objective algorithm produces a Pareto optimal set of neural networks, by optimizing the search for both the accuracy and the parameter count, thus emphasizing the versatility of our approach.