Exact Computable Representation of Some Second-Order Cone Constrained Quadratic Programming Problems

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.

A BRANCH-AND-CUT APPROACH TO PORTFOLIO SELECTION WITH MARGINAL RISK CONTROL IN A LINEAR CONIC PROGRAMMING FRAMEWORK

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.

Real-time microgrid economic dispatch based on model predictive control strategy

To deal with uncertainties of renewable energy,demand and price signals in real-time microgrid operation,this paper proposes a model predictive control strategy for microgrid economic dispatch, where hourly schedule is constantly optimized according to the current system state and latest forecast information. Moreover, implicit network topology of the microgrid and corresponding power flow constraints are considered, which leads to a mixed integer nonlinear optimal power flow problem. Given the non-convexity feature of the original problem, the technique of conic programming is applied to efficiently crack the nut. Simulation results from a reconstructed IEEE-33 bus system and comparisons with the routine day-ahead microgrid schedule sufficiently substantiate the effectiveness of the proposed MPC strategy and the conic programming method.

Quadratic Optimization over a Second-Order Cone with Linear Equality Constraints

This paper studies the nonhomogeneous quadratic programming problem over a second-order cone with linear equality constraints.When the feasible region is bounded,we show that an optimal solution of the problem can be found in polynomial time.When the feasible region is unbounded,a semidefinite programming(SDP)reformulation is constructed to find the optimal objective value of the original problem in polynomial time.In addition,we provide two sufficient conditions,under which,if the optimal objective value is finite,we show the optimal solution of SDP reformulation can be decomposed into the original space to generate an optimal solution of the original problem in polynomial time.Otherwise,a recession direction can be identified in polynomial time.Numerical examples are included to illustrate the effectiveness of the proposed approach.