Second-order Cone Programming Solution Research Articles

Conic relaxation approaches for equal deployment problems

An important problem in the breeding of livestock, crops, and forest trees is the optimum selection of genotypes that maximize genetic gain. The key constraint in the optimal selection is a convex quadratic constraint that ensures genetic diversity, so that optimal selection can be cast as a second-order cone programming (SOCP) problem. Yamashita et al. (2015) exploit the structural sparsity of the quadratic constraints and reduce the computation time drastically while attaining the same optimal solution.This paper is concerned with a special case of equal deployment (ED), in which we solve the optimal selection problem with the constraint that contribution of genotypes must either be a fixed size or zero. This involves a form of combinatorial optimization, and the ED problem can be described as a mixed-integer SOCP problem.In this paper, we discuss conic relaxation approaches for the ED problem based on LP (linear programming), SOCP, and SDP (semidefinite programming). We then propose a steepest-ascent method that combines the solution obtained from the conic relaxation with a concept from discrete convex optimization, in order to acquire an approximate solution for the ED problem in a practical time. From numerical tests, we observed that among the LP, SOCP, and SDP relaxation problems, SOCP gave a suitable solution from the viewpoint of the optimality and computation time. The steepest-ascent method starting from the SOCP solution provides high-quality solutions much faster than other existing methods that have been widely used for optimal selection.

Joint Beamforming and Power Splitting for MISO Interference Channel With SWIPT: An SOCP Relaxation and Decentralized Algorithm

This paper considers a power splitting-based MISO interference channel for simultaneous wireless information and power transfer (SWIPT), where each single antenna receiver splits the received signal into two streams of different power for decoding information and harvesting energy separately. We aim to minimize the total transmission power by joint beamforming and power splitting (JBPS) under both the signal-to-interference-plus-noise ratio (SINR) constraints and energy harvesting (EH) constraints. The JBPS problem is nonconvex and has not yet been well addressed in the literature. Moreover, decentralized algorithm design for JBPS based on local channel state information (CSI) and limited information exchange remains open. In this paper, we first propose a novel relaxation method named second-order cone programming (SOCP) relaxation to address the JBPS problem. We formulate the relaxed problem as an SOCP and present two sufficient conditions under which the SOCP relaxation is tight. For the case when the SOCP solution is not necessarily optimal to the JBPS problem, a closed-form feasible-solution-recovery method is provided. Then, we develop a distributed algorithm for the JBPS problem based on primal-decomposition (PD) method. The PD-based distributed algorithm consists of a master problem and a set of subproblems. The former is solved by using subgradient method while the latter are solved using coordinate descent method. Finally, numerical results validates the efficiency of the proposed algorithms.

A Joint Scheme of Antenna Selection and Power Allocation for Localization in MIMO Radar Sensor Networks

We consider a joint scheme of antenna subset selection and optimal power allocation for localization in multiple-input-multiple-output radar sensor networks. Sensor management is accomplished by solving a constrained optimization problem that is formulated to minimize the error in estimating target position, while conserving transmitter number and power budget. We propose a suboptimal scheme for approximately solving this problem. The scheme separates the optimization into two steps, in which each step is transformed into a second-order cone programming (SOCP) by convex relaxation. Simulation results show that the proposed suboptimal two-step SOCP solution provides better localization performance for other strategies and is close to the optimal solution with a simulated annealing technique.

A Conic Quadratic Programming Approach to Physical Layer Multicasting for Large-Scale Antenna Arrays

We investigate the problem of downlink physical layer multicasting that aims at minimizing the transmit power with a massive antenna array installed at the transmitter site. We take a solution based on semidefinite relaxation (SDR) as our benchmark. It is shown that instead of working on the semidefinite program (SDP) naturally produced by the SDR, the dual counterpart of the same problem may provide a more efficient numerical implementation. Later, by using a successive convex approximation strategy, we arrive at a provably convergent iterative second-order cone programming (SOCP) solution. Our thorough numerical investigations report that the newly proposed SOCP solution offers improved power efficiency and a massively reduced computational complexity. Therefore, the SOCP solution is seen as a suitable candidate for obtaining beamformers that minimize transmit power, especially, when a very large number of antennas is used at the transmitter.

A new one-step smoothing Newton method for second-order cone programming

In this paper, we present a new one-step smoothing Newton method for solving the second-order cone programming (SOCP). Based on a new smoothing function of the well-known Fischer-Burmeister function, the SOCP is approximated by a family of parameterized smooth equations. Our algorithm solves only one system of linear equations and performs only one Armijo-type line search at each iteration. It can start from an arbitrary initial point and does not require the iterative points to be in the sets of strictly feasible solutions. Without requiring strict complementarity at the SOCP solution, the proposed algorithm is shown to be globally and locally quadratically convergent under suitable assumptions. Numerical experiments demonstrate the feasibility and efficiency of our algorithm.

MAP-Based Underdetermined Blind Source Separation of Convolutive Mixtures by Hierarchical Clustering and -Norm Minimization

We address the problem of underdetermined BSS. While most previous approaches are designed for instantaneous mixtures, we propose a time-frequency-domain algorithm for convolutive mixtures. We adopt a two-step method based on a general maximum a posteriori (MAP) approach. In the first step, we estimate the mixing matrix based on hierarchical clustering, assuming that the source signals are suciently sparse. The algorithm works directly on the complex-valued data in the time-frequency domain and shows better convergence than algorithms based on self-organizing maps. The assumption of Laplacian priors for the source signals in the second step leads to an algorithm for estimating the source signals. It involves the l1-norm minimization of complex numbers because of the use of the time-frequency-domain approach. We compare a combinatorial approach initially designed for real numbers with a second-order cone programming (SOCP) approach designed for complex numbers. We found that although the former approach is not theoretically justified for complex numbers, its results are comparable to, or even better than, the SOCP solution. The advantage is a lower computational cost for problems with low input/output dimensions.