Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a noncon...Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation,which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints.In this paper,we provide exact computable representations for some more subclasses of the QCQP problem,in particular,the subclass with one secondorder cone constraint and two special linear constraints.展开更多
Marginal risk represents the risk contribution of an individual asset to the risk of the entire portfolio In this paper, we investigate the portfolio selection problem with direct marginal risk control in a linear con...Marginal risk represents the risk contribution of an individual asset to the risk of the entire portfolio In this paper, we investigate the portfolio selection problem with direct marginal risk control in a linear conic programming framework. ＇The optimization model involved is a nonconvex quadratically constrained quadratic programming （QCQP） problem. We first transform the QCQP problem into a linear conic programming problem, and then approximate the problem by semidefinite programming （SDP） relaxation problems over some subrectangles. In order to improve the lower bounds obtained from the SDP relaxation problems, linear and quadratic polar cuts are introduced for designing a branch-and-cut algorithm, that may yield an e -optimal global solution （with respect to feasibility and optimality） in a finite number of iterations. By exploring the special structure of the SDP relaxation problems, an adaptive branch-and-cut rule is employed to speed up the computation. The proposed algorithm is tested and compared with a known method in the literature for portfolio selection problems with hundreds of assets and tens of marginal risk control constraints.展开更多
基金supported by US Army Research Office Grant(No.W911NF-04-D-0003)by the North Carolina State University Edward P.Fitts Fellowship and by National Natural Science Foundation of China(No.11171177)。
文摘Solving the quadratically constrained quadratic programming(QCQP)problem is in general NP-hard.Only a few subclasses of the QCQP problem are known to be polynomial-time solvable.Recently,the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation,which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints.In this paper,we provide exact computable representations for some more subclasses of the QCQP problem,in particular,the subclass with one secondorder cone constraint and two special linear constraints.
基金supported by the Edward P.Fitts Fellowship at North Carolina State Universitythe National Natural Science Foundation of China Grant Numbers 11171177,11371216 and 11371242the US National Science Foundation Grant No.DMI-0553310
文摘Marginal risk represents the risk contribution of an individual asset to the risk of the entire portfolio In this paper, we investigate the portfolio selection problem with direct marginal risk control in a linear conic programming framework. ＇The optimization model involved is a nonconvex quadratically constrained quadratic programming （QCQP） problem. We first transform the QCQP problem into a linear conic programming problem, and then approximate the problem by semidefinite programming （SDP） relaxation problems over some subrectangles. In order to improve the lower bounds obtained from the SDP relaxation problems, linear and quadratic polar cuts are introduced for designing a branch-and-cut algorithm, that may yield an e -optimal global solution （with respect to feasibility and optimality） in a finite number of iterations. By exploring the special structure of the SDP relaxation problems, an adaptive branch-and-cut rule is employed to speed up the computation. The proposed algorithm is tested and compared with a known method in the literature for portfolio selection problems with hundreds of assets and tens of marginal risk control constraints.
