Many problems in statistical estimation, classification, and regression can be cast as optimization problems.Gradient descent, which is one of the simplest and easy to implement multivariate optimization techniques, lies at the heart of many powerful classes of optimization methods. However, its major disadvantage is the slower rate of convergence with respect to the other more sophisticated algorithms. In order to improve the convergence speed of gradient descent, we simultaneously determine near-optimal scalar step size and momentum factor for gradient descent in a deterministic quadratic bowl from the largest and smallest eigenvalues of the Hessian. The resulting algorithm is demonstrated on specific and randomly generated test problems and it converges faster than any previous batch gradient descent method.

The aim of index-tracking approaches in portfolio optimization is to create a mimicking portfolio which tracks a specific market index. However, without regularization, this mimicking behavior of the index-tracking model is susceptible to the volatility in the market index and has negative effects on the tracking portfolio. We recast the index-tracking optimization problem by applying a form of regularization using the convex combination of 1 and squared 2 norm constraints on the portfolio weights. The proposed optimization model enables us to control the tracking performance and the sparsity structure of the portfolio simultaneously. A sample of assets from Borsa Istanbul (BIST) is used to demonstrate the performance of the regularized portfolios with various levels of regularization. Results indicated that the regularized portfolios obtained using this approach had better tracking performances with a desired sparsity structure compared to the standard index-tracking portfolio where no regularization is applied.

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The main problem in the classification problems encountered with gene samples is that the dimension of the data is high although the sample size is small. In such problems, the classifier to be used must be a classifier that allows the processing of high dimensional data and extracts maximum information from a small number of samples at hand. In this context, a classification methodology has been developed, which first transforms the problem of binary or multiple classification into separate pair-wise classification problems. To this end, an online classifier has been adapted to solve pair-wise binary classification problems. The resulting classifier performed better on most of the real problems compared to other popular classifiers.