## Transforming a QP Problem to an SOCP Problem

A Quadratic Programming problem (QP) in the form of where , can be transformed to a Second-Order Cone Programming (SOCP) problem in the form of Consider , and As is non-negative, minimizing is equivalent to minimizing , and hence is equivalent to minimizing . If we have  and , then the objective function in QP  can be written as . We can thus minimize . Thus, the QP problem can now be written as As , by definition of QP, is symmetric, a symmetric can be found such that . If the QP is assumed to be a convex QP, is positive semidefinite, applying Cholesky factorization gives (or ). In this case, (or ). Next, as is always non-negative, the equality constraint can be written as Finally, each row in the inequality constraint can be written as where is the i-th row of , and is the i-th element of . Therefore, a QP problem can be transformed to an equivalent SOCP problem in the following way. We need to introduce a few variables first. The sub-vector with the first elements in the solution of the transformed SOCP problem is the solution of the original QP problem. SuanShu has implementations to solve both SOCP and QP problems. SOCP interior point solver QP active set...

## Solving the "Corner Solution Problem" of Portfolio Optimization

Many portfolio optimization methods (e.g., Markowitz/Modern Portfolio Theory in 1952) face the well-known predicament called the “corner portfolio problem”. When short selling is allowed, they usually give efficient allocation weighting that is highly concentrated in only a few assets in the portfolio. This means that the portfolio is not as diversified as we would like, which makes the optimized portfolio less practically useful. In [Corvalan, 2005], the author suggests to look for instead an “almost efficient” but “more diversified” portfolio within the close neighborhood of the Mean-Variance (MV) optimal solution. The paper shows that there are many eligible portfolios around the MV optimal solution on the efficient frontier. Specificially, given the MV optimal solution, those “more diversified” portfolios can be computed by relaxing the requirements for the portfolio return and risk in an additional optimization problem: where , is the Markowitz MV optimal weights, are the relaxation tolerance parameters, and is a diversification measure for the portfolio (for example, , ). In other words, the new optimization problem looks for a portfolio with the maximal diversification around the optimal solution. Corvalan’s approach can be extended to create an approximate, sufficiently optimal and well diversified portfolio from the optimal portfolio. The approximate portfolio keeps the constraints from the original optimization problem. References: SuanShu Javadoc Alejandro Corvalan (2005). Well Diversified Efficient Portfolios. Documentos de trabajo del Banco Central, no....

## Mean-Variance Portfolio Optimization When Means And Covariances Are Unknown

[Lai, Xing and Chen, 2010], in the paper “Mean-Variance Portfolio Optimization When Means And Covariances Are Unknown”, proposed a ground breaking method to do portfolio optimization. In what follows we summarize their idea and use it to implement a periodic rebalancing strategy based on the AlgoQuant framework. Harry Markowitz won the Nobel prize for his work in mean-variance (MV) portfolio optimization in 1950s. The theory is widely regarded as fundamental in financial economics. It says, given a target return of a portfolio of m assets, the optimal (in terms of information ratio) weighting is given by where is the expected future returns, and is the expected covariance matrix of future returns. This problem is readily solved by quadratic programming. Nonetheless, the assumption that and are known in advance is very dubious. This has been referred to as the “Markowitz optimization engima”. The attempts made so far are to better forecast these estimators, namely and , as accurately as possible. The maximum likelihood estimate (MLE) from the training sample is an example. It turns out, however, that MLE performs poorly because the estimators are quite different from the realized values [Michaud, 1989]. Since then, three approaches have been proposed to address the difficulty. The first approach uses a multi-factor model to reduce the dimensionality in estimating [Fan, Fan and Lv, 2008]. The second approach uses Bayes or other shrinkage estimates of [Ledoit and Wolf, 2004]. Both approaches attempt to use improved estimates of for the plug-in efficient frontier. They have also been modified to provide better estimates of , for example, in the quasi-Bayesian approach of [Black and Litterman, 1990]. The third approach uses bootstrapping...