Relaxation when the dimension of variables was 79. Hence, designing a convex
Relaxation when the dimension of variables was 79. Thus, designing a Uniconazole medchemexpress convex relaxation that can be efficiently used even for huge-size problem though preserving the strength of your convex relaxation is vital. Within this paper, we design and style an enhanced second order cone programming (SOCP) relaxation for (1) instead. We first reformulate the primal problem into a quadratic programming trouble having a quadratic equality and linear constraints. Furthermore, we present an enhanced SOCP relaxation exploiting the simultaneous matrix diagonalization tool. We evaluate the enhanced SOCP relaxation using the classical SOCP relaxation, and extensive numerical experiments confirm that the enhanced SOCP relaxation shows superiority in both the relaxation effect and computational complexity. In particular, the superiority is magnified when the number of the unfavorable eigenvectors of Q increases. Then we design a branch and bound algorithm based on the enhanced SOCP relaxation to seek out the optimal resolution. Numerical experiments show that though the reduced bound offered by the enhanced SOCP relaxation is worse than that of the CP relaxation, the computational complexity is substantially lower. As a result, the enhanced SOCP relaxation-based branch and bound algorithm spends a great deal much less time to obtain the optimal solution than that in the CP relaxation when the dimension from the variables is additional than 100. The following notations are adopted all through the paper. Provided a actual symmetric matrix X, X 0 suggests X is constructive semidefinite. I denotes an identity matrix. For n n by n real matrices A = ( Aij ) and B = ( Bij ), A B =trace( A T B) = i,j=1 Aij Bij . a represents that a R is rounded down towards the nearest integer. Provided a vector b Rn , diag(b) corresponds to an n n diagonal matrix with its diagonal components equal to b. The paper is organized as follows. In Section 2, we recast the issue into a quadratic programming dilemma using a quadratic equality and linear constraints and then present an enhanced SOCP relaxation. Section three describes a branch and bound algorithm. Section four delivers numerical experiments to confirm that the enhanced SOCP relaxation-based branchand-bound technique is successful to solve the problem. Conclusions are offered in Section 5. 2. A Reformulation of (1) and an Enhanced SOCP Relaxation Some constrained quadratic fractional issues are equivalent to quadratically constrained quadratic programming challenges [13,15]. Following this idea, in this section we initial equivalently reformulate (1) into a quadratically constrained quadratic programming difficulty and after that design an enhanced SOCP relaxation. c qT 1 0T For comfort, let A = – a A , Q = ,P= , then (1) equals to the q Q 0 I following homogeneous quadratic fractional system with linear constraints:Mathematics 2021, 9,three ofmin s.t. If we define y =z , z T Pzz T Qz , T Pz z z 0, z1 = 1, Az = 0.(2)then (2) is recast into: min s.t. y T Qy, y 0, Ay = 0, T y Py = 1. (three)Lemma 1. (2) is equivalent to (3). Proof. If z can be a feasible Methyl aminolevulinate custom synthesis option of (2), then let y = y T Qyz T Qz . z T Pzz . z T PzIt is easy to verify that y isa feasible option of (3) and = Therefore, the optimal worth of (3) is no more than that of (2). Conversely, if y is a feasible answer of (three), then y T Py = 1 implies n . If y = 0, then Ay = 0 and that y = 0. Let y = [y1 ; y2 ] with y1 R and y2 R 2 1 y2 0. Hence, y2 ker A Rn = 0, which contradicts with all the conclusion that y = 0. + y For that reason, y1 0. Let z = y , then it is straightforward to confirm that.