In this work we consider a continuous-time mean-variance portfolio selection problem that is formulated as a hi-criteria optimization problem. The objective is to maximize the expected return and minimize the variance of the terminal wealth. By putting weights on the two criteria one obtains a single objective stochastic control problem which is however not in the standard form. We show that this non-standard problem can be "embedded" into a class of auxiliary stochastic linear-quadratic (LQ) problems. By solving the latter based on the recent development on stochastic LQ problems with indefinite control weighting matrices, we derive the efficient frontier in a closed form for the original mean-variance problem. 1 INTRODUCTION Portfolio selection is to seek a best allocation of wealth among a basket of securities. The mean-variance approach by Markowitz [9] provides a fundamental basis for portfolio construction in a single period. The most important contribution of this model is that it quantifies the risk by using the variance which enables investors to seek highest return after specifying their acceptable risk level. This approach becomes the foundation of modern finance theory and inspires literally hundreds of extensions and applications. In particular, in the case where the variance matrix is positive definite and short-selling is allowed, an analytic solution was obtained by Merton [10]. Perold [12] developed a more
We explore in this paper certain rich geometric properties hidden behind quadratic 0-1 programming. Especially, we derive new lower bounding methods and variable fixation techniques for quadratic 0-1 optimization problems by investigating geometric features of the ellipse contour of a (perturbed) convex quadratic function. These findings further lead to some new optimality conditions for quadratic 0-1 programming. Integrating these novel solution schemes into a proposed solution algorithm of a branch-and-bound type, we obtain promising preliminary computational results.
Abstract. We present in this paper new results on the duality gap between the binary quadratic optimization problem and its Lagrangian dual or semidefinite programming relaxation. We first derive a necessary and sufficient condition for the zero duality gap and discuss its relationship with the polynomial solvability of the primal problem. We then characterize the zeroness of the duality gap by the distance, δ, between {−1, 1} n and certain affine subspace C and show that the duality gap can be reduced by an amount proportional to δ 2 . We finally establish the connection between the computation of δ and cell enumeration of hyperplane arrangement in discrete geometry and further identify two polynomially solvable cases of computing δ.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.