Consider the following quadratic optimization problem with non-convex inhomogeneous quadratic constraints:
where are symmetric positive semidefinite matrices and . Note that if then the problem (1) is easily solved, so assume Generally, this problem is NP-hard    . It has a lot of applications in telecommunications, robust control, portfolio and so on. So it’s helpful to solve it.
By means of the work of Lovász and Schrijver  , Shor  , and others that certain NP-hard combinatorial optimization problems can be approximated by semidefinite programming (SDP) problems, for which efficient solution methods exist   . So many mathematics workers with this motive put forward a lot of algorithms to solving quadratic optimization problem based on the semidefinite programming (SDP) relaxation. Recently, there were several results on solving different forms of quadratic problems. The results can be summarized as follows.
• If all are symmetric positive semidefinite matrices with positive definite sum and A is an arbitrary symmetric matrix. A. Nemirov- ki  produce a feasible solution such that, with constant probability,
• If Luo et al.  showed that the ratio between the original optimal value and the SDP relaxation optimal value is bounded by , have
• If all are symmetric matrices, and two or more of them are indefinite. S. He et al.  compute a feasible solution such that,
• Of special interest is the case of ellipsoid constraints
where denotes the Euclidean norm, so . Nemirovski  show that if all and are positive semidefinite. Then a feasible solution can be generated from (SDP) satisfying
• In particular, if (1) has a ball constraints, Ye and Zhang(Corollary 2.6 in  ) showed that a feasible x satisfying
can be found in polynomial time.
• Ye.  extended the above result to allow the ellipsoids not to have a common center but assuming . Ye showed that a feasible solution can be randomly generated such that
However, the existing algorithms are just for the problem of discrete problems or continuous problems, which is mostly based on homogeneous or inhomogeneous convex constraint problems. For this kind of quadratic optimization problem with non-convex inhomogeneous quadratic constraints, cannot find a very effective algorithm. This paper will propose a new effective algorithm to solve this problem.
This paper is organized as follows. In Section 2, we present a semidefinite pro- gramming (SDP) relaxation of (1). In Section 3, we propose a new effective algorithm to get the feasible solution of quadratic optimization problem (1) with non-convex inhomogeneous quadratic constraints. At last, some conclusions and the future works are given in Section 4.
Notations. Throughout this paper, we denote by and the n-dimen- sional real vector space and positive semidefinite symmetric matrices space. denotes that A is semidefinite. represents the trace of a matrix. The inner product of two matrices A and B is denoted by
stands for the probability.
2. Semidefinite Programming (SDP) Relaxation
In this section, we present a semidefinite programming (SDP) relaxation of (1). To avoid trivial cases, we first make the following assumption.
Assumption Let , has a solution,
Assume t is a constant, and satisfy . (1) is equivalent to:
Let be the global optimal solution of the above problem, the objective value is . Assume , it’s block structure like this:
By letting and dropping the rank one constraint, the semidefinite programming relaxation of (9) can be drawn up as follows.
An optimal solution of SDP relaxation (SDP) can be computed efficiently using, say, interior-point mathods; see  and references therein.
In this section, we bring an effective algorithm for solving (1). The algorithm is divided into two parts. The main idea as follows: the first stage produces a solution which satisfies the first constraint of problem (9). Making a small change to the solution which obtained in the first stage, we can get the solution of (9) in the second stage. We will set the randomization algorithm as follows.
3.1. The First Stage
The first stage of the algorithm uses the randomization algorithm, which is proposed by Luo et al.  . At first, we need obtain an optimal solution of (SDP), then construct a feasible solution for the first constraint of problem (9) using the following randomization procedure:
First, it can be easily verified that satisfy the first constraint of problem (9).
(12) is equivalent to:
Lemma 1 For generated in step 2, we have that
Proof. By the step 2, we first have
where is a vector with the element being 1 and all the other elements being 0. By denoting we obtain that
By using the total probability formula for the last term in (16), we have
By Lemma 3.1 and Lemma 3.2 in  ,
Since is feasible for (SDP), it follows that for all . Since so
According to Lemma 1 in  , we have
Thus, it follows from (17), (18), (19) and (20) that:
The proof is completed by setting
Note that by Lemma 1 and (13), it can be concluded that
So there is a , for any satisfies:
3.2. The Second Stage
In this part, we make a change to the solution which constructed in the first stage in order to satisfy the problem (9). In this stage, we will by ways of the algorithm in  .
The procedure as follows:
Let , so can be seen as a quadratic function for . The symmetry axis of is:
Because the can’t make for all set up. We introduce a parameter , and construct a new solution . It’s the feasible solution of (1).
When , the symmetry axis of satisfies:
So for all can make set up. It’s helpful for us to solve the problem, because we only need to find satisfying in the situation of
When , because are symmetric, . To simplify the writing, we introduce the following notations:
According to the norm inequality:
For using (28), (29). The last term in (27) can be simplified to be
Whenever it can be easily checked that
From (23), we can get
and we also have
Using (31) and (32), the last term in (33) can be simplified as
We will give the analysis of (34) as follows.
we can simplified
Since , we know that is an increasing function about .
However, is a function depend on and the number of the constraints. According to simple calculations, we find, when satisfies , is an increasing function about . In this situation, we also have . When , becomes smaller with the increase of .
we can write (34) as a piecewise function about and
So it can be easily verified that
is a feasible solution of the original problem.
For the quadratic optimization problem with non-convex inhomogeneous quadratic constraints, it’s NP-hard. We can’t find an effective algorithm solving it. In this paper, we put forward an effective algorithm. According to it, many problems in life can be solved. Through the algorithm, we can get the feasible solution of (1). Transforming the original problem into (SDP) is a very important step in solving the problem. So we give the semidefinite programming (SDP) relaxation of (1) in Section 2, then propose an effective algorithm which given in Section 3 to construct the feasible solution of (1).
In the future, I will do the following work: discusses the quality of the feasible solution about (1), and gives some numerical experiments to verify it, we will consider the problem with inhomogeneous objective function. To this problem, we want to find an algorithm solve it by ways of the effective algorithm which put forward in this paper.
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