Algorithmic Design For Separable Convex Optimization And Its Application In Portfolio Selection

The separable convex optimization problem is an essential model in operational research and decision making,it has important applications in the fields of management science,finance and machine learning.In the financial field,portfolio selection is a sig-nificant research direction,which provides investors more scientific investment advice.One important type of investment portfolio problems is the robust portfolio selection,which mainly considers the uncertainty of the parameters(return,variance)estimation in the model,and how to guarantee the optimal portfolio under the worst case.Another problem is the short-term sparse portfolio optimization,which is based on some empir-ical financial rules,significantly increases the potential return of a smaller proportion of assets in the portfolio to maximize the cumulative wealth of the portfolio.Both types of problems can be transferred into a separable convex optimization problem.In addition,there are many separable convex optimization problems in machine learning,such as Lasso and sparse inverse covariance selection,which can be used for various data predictions and have many important applications.The above model requires a high speed for the algorithm,and needs the algorithm to obtain the optimal solution of the model fast or even in near real time.Classical first-order optimization algorithms,such as the Alternating Direction Method of multi-pliers(ADMM),which converges slowly near the solution point,often can not meet the requirements,so our thesis proposes an over-relaxed alternating direction method of multipliers(ADMM),which can greatly improve the solution speed by increasing the step size in each iteration.Also,we prove the global convergence of the over-relaxed ADMM algorithm and demonstrate the convergence rate of o(1/?).Besides,we implement our proposed algorithm to solve Lasso,sparse inverse co-variance selection problem and two optimization models in portfolio selection,and compare its performance with the relaxed customized PPA and the classical ADMM.The numerical results show that our algorithm has faster speed and higher efficiency.