Convexity Study Of Nonconvex Quadratic Constrained Optimization Problems

Quadratic function is a kind of simpler function in nonlinear functions.Many functions can be approximated by it.Therefore,the study of quadratic optimization is helpful for the study of general nonlinear problems.At the same time,the quadratic constrained optimization problem has a wide range of practical applications in many fields.Therefore,it is very meaningful to explore the problem of quadratic constrained optimization.This paper mainly studies the non-convex quadratic optimization problem with quadratic constraints,and mainly studies one of the special problems:the CDT problem.The main research contents are as follows:(1)We study a class of CDT problem with two quadratic constraints,one of which is the unit ball constraint and the other is the ellipsoid constraint.Select the appropriate hyperplane through the optimal line segment,without dividing the feasible region.In the case of the second-order cone recombination technique and the SDP relaxation method,the necessary and sufficient conditions for the existence of the dual gap in the second-order cone reshaping problem of the CDT problem are obtained,and the theoretical proof is given which is paved to reduce or even eliminate the dual gap of the CDT problem.(2)A class of classic CDT problem that the dual gap can be completely eliminated is found.The theoretical proof is given,and it is proved that all the problems are satisfied in the two-dimensional case,and a counterexample of three-dimensional is given.This is prepared for the subsequent research.