Convexification Technique and Portfolio Optimization
1 The Bucharest University of Economic Studies
6, Piaţa Romană 374 Bucharest 1, Romania
2 “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics
and Applied Mathematics of the Romanian Academy
13, Calea 13 Septembrie, 050711 Bucharest 5, Romania
Abstract: In this paper, a general transformation method which converts a nonconvex optimization problem to an equivalent problem with better properties is proposed. Under certain assumptions, the local convexity of the Lagrangian function of the equivalent problem is guaranteed and thus the class of optimization models to which dual methods can be applied is extended. Practical classes of problems where the proposed method can be applied are given. They include the class of portfolio selection models. Numerical examples illustrate the main results.
Keywords: Nonconvex optimization; local convexification; Lagrangian function, portfolio optimization, efficient frontier.
CITE THIS PAPER AS:
Cristinca FULGA, Convexification Technique and Portfolio Optimization, Studies in Informatics and Control, ISSN 1220-1766, vol. 22 (4), pp. 285-290, 2013.
Convexity is one of the most frequently used hypotheses in optimization theory and certainly the most beneficial property in the practice of optimization because the convexity of an optimization model ensures global validity to properties otherwise only locally true. But in real life problems, the hypothesis of convexity is not very often fulfilled. The primal-dual method has been one of the most efficient solution algorithms in solving constrained optimization problems, see for example Lasdon , Luenberger . The success of the primal-dual method depends on the local convexity of the Lagrangian function at the optimal solution of the problem. Several convexification schemes have been proposed in the literature to extend the primal- dual method for certain nonconvex problems, see for example Bertsekas , Li  and , Xu .
In this paper we propose a convexification technique for a general differentiable nonlinear programming problem with inequality constraints. We begin by giving some examples which illustrate the usefulness of the convexification techniques in practical situations of portfolio optimization.
Examples of nonconvex models for portfolio optimization.
The modern portfolio theory trades off the risk and expected return of a portfolio, Markowitz . The classical Mean-Variance (MV) , where the notations used are: n the number of securities available, the random vector of the asset returns or simply r (we use bold symbols for vectors), the return of the portfolio the feasible set, estimates of each asset expected return and the variance , correlation coefficient of any two assets , , the expected return and the variance of the portfolio and where . The efficient frontier set is defined as the set of efficient portfolios. An efficient portfolio is a portfolio whose expected return cannot increase unless its risk represented here by its variance increases as well. But determining the set of efficient portfolios is just the first step of the portfolio selection procedure. The second step is the actual selection of one optimal portfolio out of the entire efficient frontier corresponding to the specific investor’s preference structure. In order to determine an optimal portfolio corresponding to a given degree of risk aversion within the MV approach, we consider the MV utility function ( and ): The optimal portfolio corresponding to the investor’s degree of risk aversion is the solution of the problem, see Markowitz :
The covariance matrix is positive semi-definite, represents the variance of the portfolio x. But in practice we work with real data and it is not rare the case when the real data have missing values. In this case, one can estimate a covariance matrix by considering the matrix that will be called NAN – covariance matrix and denoted here by which is constructed in a simple and natural way starting from the rate of return matrix by ignoring the terms containing unknown return values (NAN is the Matlab convention for Not a Number). Unfortunately, the positive semi-definiteness of the resulting matrix is may not be assured. The immediate consequence is the lack of convexity of the objective function of the optimization problem even in this case when the optimal decision is determined based on solely two numbers: the mean and the variance of the portfolio.
The MV approach is consistent with the von Neumann-Morgenstern expected utility theory only if (i) the returns of the assets are normally distributed and consequently it is legitimate to ignore higher moments beginning with third order and (ii) the utility function characterizing the investor’s attitude towards risk is quadratic. But there is a plethora of empirical studies showing that portfolio returns are generally not normally distributed.
Consequently, in some recent studies, the concept of Mean-Variance trade-off has been extended to include the skewness of return in portfolio selection, see Konno and Suzuki , Leung et al. , Liu et al. , Joro and Na , Briec et al. , or the kurtosis, see Maringer and Parpas , Lai et al. , Jondeau
and Rockinger  among others and also applied in various practical problems, see Rădulescu et al. [17-19]. Many empirical studies show that investors prefer positive skewness, because it implies a low probability of obtaining a large negative return. It was also observed that an increased diversification leads frequently to skewness loss. In order to integrate the skewness, we use the Mean-Variance-Skewness (MVS) utility: where S(R) is the portfolio skewness, a>0, b>0 and . The optimal portfolio corresponding to the investor’s specific degree of risk aversion and degree of absolute prudence , see Eeckhoudt , is the solution of the maximization problem, see Briec et al. :
We note that this model is not convex also. It is not necessary for our present purposes to pursue the generalization of this result by taking into account the kurtosis or other higher moments, or to discuss further the possibility of nonexistence of some of these moments. But in general, we can construct an utility function defined over the first m moments of the probability distribution of the underlying random variable, provided all of the first m moments exist and are finite. We remark the general lack of convexity of these models (nonconcavity for max problems). Next we propose a convexification technique for a general differentiable nonlinear programming problem with inequality constraints which includes models like (1) and (2).
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