Mixed-Integer Semidefinite Relaxation of Joint Admission Control and Beamforming: An SOC-Based Outer Approximation Approach with Provable Guarantees

We consider the joint admission control and multicast downlink beamforming (JABF) problem, which is a fundamental problem in signal processing and admits a natural mixed-integer quadratically constrained quadratic program (MIQCQP) formulation. One popular approach to tackling such MIQCQP formulation is to develop convex relaxations of both the binary and continuous variables. However, most existing convex relaxations impose rather weak relationships between the binary and continuous variables and thus do not yield high-performance solutions. To overcome this weakness, we propose to keep the binary constraints intact and apply the semidefinite relaxation (SDR) technique to continuous variables. Although the resulting relaxation takes the form of a mixed-integer semidefinite program (MISDP) and is theoretically intractable in general, by exploiting the fact that such MISDP arises as a mixed-integer SDR of an MIQCQP and harnessing recent computational advances in solving large-scale mixed-integer second-order cone programming (MISOCP) problems, we develop a novel, practically efficient algorithm that provably converges to an optimal solution to the MISDP in a finite number of steps. The key idea of our algorithm is to construct successively tighter second-order cone (SOC) outer approximations of the constraints in the MISDP and solve a sequence of MISOCPs to obtain an optimal solution to the MISDP. Our work also provides, to the best of our knowledge, the first general framework for solving MISDPs that arise as mixed-integer SDRs of MIQCQPs. Next, we show that by applying a Gaussian randomization procedure to the optimal solution to the MISDP, we obtain a feasible solution to the JABF problem whose approximation accuracy is on the order of M, the number of users in the network. This improves upon the approximation accuracy guarantee of an existing convex relaxation method. Lastly, we present numerical results to demonstrate the viability of our proposed approach.