We consider nine elementary problems in optimization. We simply explore the conditions for optimality as known from the duality theory for convex optimization. This yields a quite straightforward solution method for each of these problems. The main contribution of this paper is that we show that even in the harder cases the solution needs only O(n) time.

Tropospheric ozone is a secondary pollutant which can affect human health and plant growth. In this paper, we investigated transferred convolutional neural network long short-term memory (TL-CNN-LSTM) model to predict ozone concentration. Hourly CNN-LSTM model is used to extract features and predict ozone for next hour, which is superior to commonly used models in previous studies. In the daily ozone prediction model, prediction over a large time-scale requires more data, however, only limited data are available, which causes the CNN-LSTM model to fail to accurately predict. Network-based transfer learning methods based on hourly models can obtain information from smaller temporal resolution. It can reduce prediction errors and shorten run time for model training. However, for extreme cases where the amount of data is severely insufficient, transfer learning based on smaller time scale cannot improve model prediction accuracy.

Graph-based semi-supervised learning (G-SSL) methods play an increasingly important role in machine learning systems. Recently, latent low-rank representation (LatLRR) graph has gained great success in subspace clustering. However, LatLRR only considers the global structure, while the local geometric information, which is often important to many real applications, is ignored. In this paper, we propose a locality regularized LatLRR model (LR-LatLRR) for semi-supervised subspace clustering problems. This model incorporates two regularization terms into LatLRR by taking the local structure of data into account. Then, we develop an efficient splitting algorithm for solving LR-LatLRR. In addition, we also prove the global convergence of the proposed algorithm. Furthermore, we extend the LR-LatLRR model to a case of including the non-negative constraint. Finally, we conduct experiments on a synthetic data and several real data sets for the semi-supervised clustering problems. Experimental results show that our method can obtain high classification accuracy and outperforms several state-of-the-art G-SSL methods.

Subspace segmentation is a fundamental issue in computer vision and machine learning, which segments a collection of high-dimensional data points into their respective low-dimensional subspaces. In this paper, we first propose a model for segmenting the data points from incomplete and noisy observations. Then, we develop an inexact splitting method for solving the resulted model. Moreover, we prove the global convergence of the proposed method. Finally, the inexact splitting method is implemented on the clustering problems in synthetic and benchmark data, respectively. Numerical results demonstrate that the proposed method is computationally efficient, robust as well as more accurate compared with the state-of-the-art algorithms.