On fractional quadratic optimization problem with two quadratic constraints

In this paper, we study the problem of minimizing the ratio of two quadratic functions subject to two quadratic constraints in the complex space. Using the classical Dinkelbach method, we transform the problem into a parametric nonlinear equation. We show that an optimal parameter can be found by employing the S-procedure and semidefinite relaxation technique. A key element to solve the original problem is to use the rank-one decomposition procedure. Finally, within the new algorithm, semidefinite relaxation is compared with the bisection method for finding the root on several examples. For further comparison, the solution of fmincon command of MATLAB also is reported.

1. Introduction. In this paper, we consider the following fractional optimization problem: are complex vectors, c i , e i ∈ R are real constants, ∀i = 1, 2. The superscript H also denotes the conjugate transpose. Furthermore, we require that there exists α > 0 such that f 2 (x) > α for all x ∈ G, namely problem (1) is well defined. It is worth to note that in general problem (1) is not convex. Two commonly used approaches to transform (1) to a non-fractional problem, are generalized Charnes-Cooper transformation [9] and Dinkelbach method [11,16], where the classical complex optimization duality theory 1, the received SNR of SD is and the power of the interference and the noise in PD can be written as Equations (2) and (3) are equivalent to: where Also, the power of the retransmit signal is | w i | 2 ≤ P for i = 1, 2, · · · , N and the total power of the relays is w 2 ≤ P T , where P T is the maximum total transmitted power of relays. Indeed, SNR has been defined as the ratio of the signal power S p to the noise power N p where the signal power and the noise power of SD is obtained as: Then, the problem is formulated as the following optimization problem: s.t w 2 ≤ P T , where I th is the maximum interference power at PD. Since the set of quadratically constrained quadratic optimization problems is NP-hard, thus the quadratically constrained ratio of two quadratic functions as a special case of it also belongs to the class of NP-hard problems. Cai, et al. [7] first changed the complex fractional problem into a non-fractional one. Then, they used SDO relaxation and obtained the exact optimal solution. The authors in [22] studied quadratic fractional optimization problem with a quadratic constraint. They have applied Dinkelbach method and proposed a bisection method and a generalized Newton algorithm to solve the parametric problem. In [12], the problem of minimizing the ratio of two complex indefinite quadratic functions subject to a strictly convex quadratic constraint is studied. First, After reformulating the fractional problem as a univariate equation, to find the root of the univariate equation, the generalized Newton method is utilized that requires solving a nonconvex quadratic optimization problem at each iteration. To solve these nonconvex quadratic problems, the authors have presented an efficient algorithm by a diagonalization scheme that requires solving a univariate minimization problem at each iteration. It is worth to note also that Nguyen, et al. [20] considered the problem in which the ratio of two indefinite quadratic functions in real space is minimized subject to a two-sided quadratic constraint. Using the relationship between fractional and parametric optimization problems, they proposed the stronger version of the extended S-Lemma to achieve the optimal parameter.
In this paper, we use the classical Dinkelbach method and transform the main problem into a non-fractional one in Section 2. Using the S-Lemma, we show that the optimal parameter can be computed by solving an SDO relaxation problem. Then by solving another SDO relaxation problem and using the rank-one decomposition procedure of [15], the optimal solution of the original problem is extracted. We present the new algorithm and the bisection method in Section 3. Finally in Section 4, we give some numerical results for two sets of examples.
Throughout the paper, H n and H n + denote the space of n × n complex Hermitian matrices and complex Hermitian positive semidefinite matrices, respectively. The notation C 0( ) means C is a positive semidefinite (definite) matrix. For two complex matrices E and D, their inner product is defined as where tr (·) denotes the trace of a matrix, T denotes the transpose of a matrix and Re Y and Im Y stand for the real and imaginary parts of Y ∈ H n .

Preliminaries.
Let G ⊂ C n be a given set. We denote C + (G) as all Hermitian matrices which are co-positive over G, i.e.
Obviously, C + (G) is a closed convex cone in H n . We define FC + (G), the cone of complex quadratic functions which are non-negative on G as follows: Moreover, for a quadratic function g(w) = w H Bw − 2Re(d H w) + e, we denote its matrix representation by For our further development, the following theorem which is an extended version of the S-procedure [15] plays an important role. One may see [17,18,21] for some recent developments of S-procedure.
2.1. Parametric approach. The following proposition is a key element for our algorithmic devolvement.
The following two statements are equivalent: Thus we focus on the parametric optimization problem (7) instead of the main fractional optimization problem (1). In the following theorem we outline some properties of the univariate function F(λ). 3. Main results. First, we give the following lemmas generalizing some primary results in fractional optimization. Then we apply the S-procedure and SDO relaxation to compute λ * . Moreover we show strong duality holds for (7).
[attainment] ( [23]) Suppose that problem (1) is well defined. Then, λ * is attained at x ∈ G if and only if λ * is a root of F(λ) and x * is an optimal solution to (7).
In order to be able to apply Theorem 2.1, we make the following assumption.  Proof. In contrary, suppose Assumption A does not hold. Then Proof. We have = sup λ∈R λ : where (10) follows from Theorem 2.1.
Note that λ * can be efficiently computed by solving problem (8), using CVX [13]. Now to find the optimal solution of (1) using Proposition 2.1, it is sufficient to solve (7). The classical SDO relaxation of (7) is The dual of (11) is given by Note that if A 1 −λA 2 = 0 and Lemma 3.3 holds then Assumption A is satisfied while the strict feasibility condition for dual problem (12) may be violated. Consequently, solving problem (11) is hard in these cases, so we consider the following assumption. Assumption B. There exist nonnegative η 1 , η 2 such that Theorem 3.5. Suppose that problem (1) has a strictly feasible solution x 0 and Assumption B holds. Then both problems (11) and (12) also satisfy the strict feasibility condition. Hence, both problems attain their optimal values and the duality gap is zero.
Proof. Let X 0 be as follows: where Q = diag(q 1 , · · · , q n ) with all q j > 0 and sufficiently small. Obviously by the Schur complement theorem, X is positive definite. Moreover, Then, X 0 is a strictly feasible solution for the problem (1). For the dual problem we have

Now by Schur complement theorem
Z 0 ⇐⇒ (A + y 1 B 1 + y 2 B 2 ) − 1 c + e 1 y 1 + e 2 y 2 + y 3 then by choosing y 3 sufficiently large and y 1 = η 1 , y 2 = η 2 , Z is positive definite which implies the strict feasibility condition of (11). From Theorem 3.5, X * and (y 1 * , y 2 * , y 3 * , Z * ) are optimal solutions of (11) and (12), respectively if and only if the following conditions are satisfied Next theorem shows that there exists a rank-one decomposition of X * that enables us to obtain an optimal solution for (7). Theorem 3.6. ( [15]) Suppose that X ∈ H n + is a complex Hermitian positive semidefinite matrix of rank r, and M 1 , M 2 ∈ H n be two given Hermitian matrices. Then, there is a rank-one decomposition of X, In addition, with regard to (g) is the optimal solution of (11). Furthermore, letx j = 1 x * j , then j is the optimal solution of (7).
11. F 7 =randn(n); F 8 =randn(n); B 2 = (F7+i * F8)+(F7+i * F8) 2 ; 12. f 7 =randn(n,1); f 8 =randn(n,1); The numerical results of Example 1 are provided in Tables 1 and 2. As indicated in Table 1, by considering zero as the starting point for 'fmincon' command in MATLAB, although it stops at the shortest possible time in comparison with other algorithms, but failed to reach the optimal solution. In Table 2, by choosing a random starting point, the 'fmincon' command cannot solve the problems and the bisection algorithm is not affordable because it extremely slow, thus, it is not applicable for dimensions 200 to 400 problems. While our algorithm solves all the problems within an acceptable period of time. Tables 3 and 4 also show the results of Example 2 which has similar analysis as Example 1. Table 1. Numerical Results for Example 1 5. Conclusions. In this paper, we proposed a new method to solve a quadratic fractional optimization problem with two quadratic constraints in complex space. The method is based on S-procedure, parametrization approach of Dinkelbach and the rank-one decomposition. Computational results show that the method solves all test problems up to global optimality. 6. Acknowledgements. The authors would like to thank all reviewers for their useful comments and suggestions. The last two authors are partially supported by the Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan.