Pre-filters design for weighted sum rate maximization in multiuser time reversal downlink systems

Second-order cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order (Lorentz) cones. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems, as can many other problems that do not fall into these three categories. These latter problems model applications from a broad range of fields from engineering, control and finance to robust optimization and combinatorial optimization. On the other hand semidefinite programming (SDP)—that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices—includes SOCP as a special case. Therefore, SOCP falls between linear (LP) and quadratic (QP) programming and SDP. Like LP, QP and SDP problems, SOCP problems can be solved in polynomial time by interior point methods. The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and QP problems but less than that required to solve SDP’s of similar size and structure. Because the set of feasible solutions for an SOCP problem is not polyhedral as it is for LP and QP problems, it is not readily apparent how to develop a simplex or simplex-like method for SOCP. While SOCP problems can be solved as SDP problems, doing so is not advisable both on numerical grounds and computational complexity concerns. For instance, many of the problems presented in the survey paper of Vandenberghe and Boyd [VB96] as examples of SDPs can in fact be formulated as SOCPs and should be solved as such. In §2, 3 below we give SOCP formulations for four of these examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadratic functions ∗RUTCOR, Rutgers University, e-mail:alizadeh@rutcor.rutgers.edu. Research supported in part by the U.S. National Science Foundation grant CCR-9901991 †IEOR, Columbia University, e-mail: gold@ieor.columbia.edu. Research supported in part by the Department of Energy grant DE-FG02-92ER25126, National Science Foundation grants DMS-94-14438, CDA-97-26385 and DMS-01-04282.

A note on second order cone programming approach to two-stage network data envelopment analysis

The ever-increasing demand for more reliable and flexible energy supply for customers has pushed up the distributed generation technology to the center stage in the development of distribution networks. If managed and coordinated efficiently, the distributed generators (DGs) can provide distinct benefits to the suppliers as well as energy consumers. In this paper, a distributed optimization method is proposed to coordinate the multiple DGs in the distribution network (DN) as well as to achieve the optimal operation among them. To optimize the power generation DGs in the DN is actually equivalent to solve the AC optimal power flow (OPF) problem. By defining new variables, the OPF problem is first transformed to a convex second-order cone program (SOCP) problem. By applying alternating direction method of multipliers (ADMM), the SOCP problem is decomposed to the bus level, and it is then solved in a distributed manner. The proposed control enables the autonomous operation of DGs in the distribution network while still achieves the optimality. Simulation studies further demonstrate the effectiveness of the proposed approach.

The core of the support vector machine (SVM) problem is a quadratic programming problem with a linear constraint and bounded variables. This problem can be transformed into the second order cone programming (SOCP) problems. An interior-point-method (IPM) can be designed for the SOCP problems in terms of storage requirements as well as computational complexity if the kernel matrix has low-rank. If the kernel matrix is not a low-rank matrix, it can be approximated by a low-rank positive semi-definite matrix, which in turn will be fed into the optimizer. In this paper we present two SOCP formulations for each SVM classification and regression problem. There are several search direction methods for implementing SOCPs. Our main goal is to find a better search direction for implementing the SOCP formulations of the SVM problems. Two popular search direction methods: HKM and AHO are tested analytically for the SVM problems, and efficiently implemented. The computational costs of each iteration of the HKM and AHO search direction methods are shown to be the same for the SVM problems. Thus, the training time depends on the number of IPM iterations. Our experimental results show that the HKM method converges faster than the AHO method. We also compare our results with the method proposed in Fine and Scheinberg (2001) that also exploits the low-rank of the kernel matrix, the state-of-the-art SVM optimization softwares SVMTorch and SVMlight. The proposed methods are also compared with Joachims ‘Linear SVM’ method on linear kernel.

In this paper, we propose an algorithm based on Fletcher’s Sl1QP method and the trust region technique for solving Nonlinear Second-Order Cone Programming (NSOCP) problems. The Sl1QP method was originally developed for nonlinear optimization problems with inequality constraints. It converts a constrained optimization problem into an unconstrained problem by using the l1 exact penalty function, and then finds an optimum by solving approximate quadratic programming subproblems successively. In order to apply the Sl1QP method to the NSOCP problem, we introduce an exact penalty function with respect to second-order cone constraints and reformulate the NSOCP problem as an unconstrained optimization problem. However, since each subproblem generated by the Sl1QP method is not differentiable, we reformulate it as a second-order cone programming problem whose objective function is quadratic and constraint functions are affine. We analyze the convergence property of the proposed algorithm, and show that the generated sequence converge to a stationary point of the NSOCP problem under mild assumptions. We also confirm the efficiency of the algorithm by means of numerical experiments.

9 - Double-timescale distributed voltage control for unbalanced distribution networks based on MPC and ADMM

An iterative second-order cone programming (SOCP) approach is proposed in this paper. The original nonconvex design problem is first relaxed into an SOCP problem, which can provide a lower bound on the optimal value of the original problem. For reducing the gap between the original and the convex problem, an iterative procedure is developed. The initial point of the iterative procedure can be chosen as the solution obtained from the relaxed SOCP problem. Unlike other iterative approaches, the convergence of the proposed iterative procedure is definitely guaranteed. Design examples demonstrate the effectiveness of the proposed method.

Second-order cone programming problems are a tractable subclass of convex optimization problems that can be solved using polynomial algorithms. In the last decade, stochastic second-order cone programming problems have been studied, and efficient algorithms for solving them have been developed. The mixed-integer version of these problems is a new class of interest to the optimization community and practitioners, in which certain variables are required to be integers. In this paper, we describe five applications that lead to stochastic mixed-integer second-order cone programming problems. Additionally, we present solution algorithms for solving stochastic mixed-integer second-order cone programming using cuts and relaxations by combining existing algorithms for stochastic second-order cone programming with extensions of mixed-integer second-order cone programming. The applications, which are the focus of this paper, include facility location, portfolio optimization, uncapacitated inventory, battery swapping stations, and berth allocation planning. Considering the fact that mixed-integer programs are usually known to be NP-hard, bringing applications to the surface can detect tractable special cases and inspire for further algorithmic improvements in the future.

A memristor crossbar, which is constructed with memristor devices, has the unique ability to change and memorize the state of each of its memristor elements. It also has other highly desirable features such as high density, low power operation and excellent scalability. Hence the memristor crossbar technology can potentially be utilized for developing low-complexity and high-scalability solution frameworks for solving a large class of convex optimization problems, which involve extensive matrix operations and have critical applications in multiple disciplines. This paper, as the first attempt towards this direction, proposes a novel memristor crossbar-based framework for solving two important convex optimization problems, i.e., second-order cone programming (SOCP) and homogeneous quadratically constrained quadratic programming (QCQP) problems. In this paper, the alternating direction method of multipliers (ADMM) is adopted. It splits the SOCP and homogeneous QCQP problems into sub-problems that involve the solution of linear systems, which could be effectively solved using the memristor crossbar in O(1) time complexity. The proposed algorithm is an iterative procedure that iterates a constant number of times. Therefore, algorithms to solve SOCP and homogeneous QCQP problems have pseudo-O(N) complexity, which is a significant reduction compared to the state-of-the-art software solvers (O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3.5</sup> )-O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> )).

In this paper, we present a new fast optimization method to solve large margin estimation (LME) of continuous density hidden Markov models (CDHMMs) for speech recognition based on second order cone programming (SOCP). SOCP is a class of nonlinear convex optimization problems which can be solved very efficiently. In this work, we have formulated the LME of CDHMMs as an SOCP problem and proposed two improved tighter SOCP relaxation methods for LME. The new LME/SOCP methods have been evaluated in a connected digit string speech recognition task using the standard TIDIGITS database. Experimental results demonstrate efficiency and effectiveness of the proposed LME/SOCP methods in speech recognition.

In this paper, a downlink wireless communication channel is considered. The base station (BS) has common data for all users, unicast data for a set of intended users, and transmits the superposition of these messages. This setting neither falls into the non-orthogonal multiple access (NOMA) nor into the multi-group multicasting literatures. In NOMA systems, the BS has unicast data for all users, and multiple users share the same resources. In multi-group multicasting, there are non-overlapping groups, each demanding a different multicast message. This paper studies precoder design to achieve maximum weighted sum rate (WSR). It is first shown that the precoders designed for WSR maximization and weighted minimum mean square error (WMMSE) minimization are equivalent. Second, an iterative low complexity algorithm (named WMMSE), based on WMMSE transmit precoders and receivers, is proposed. Another low-complexity precoder, the phase aligned zero forcing (PAZF) precoder, is also introduced. The results show that both algorithms converge fast. The WMMSE algorithm outperforms both PAZF and the zero-forcing (ZF) precoder for all signal-to-noise ratio ranges. It offers better interference management and high coherent combining gains for common data while PAZF finds the optimal phase rotation on the ZF precoder, and increases coherent combining gains.

The extended trust region subproblem (ETRS) of minimizing a quadratic objective over the unit ball with additional linear constraints has attracted a lot of attention in the last few years due to its theoretical significance and wide spectra of applications. Several sufficient conditions to guarantee the exactness of its semidefinite programming (SDP) relaxation or second order cone programming (SOCP) relaxation have been recently developed in the literature. In this paper, we consider a generalization of the extended trust region subproblem (GETRS), in which the unit ball constraint in ETRS is replaced by a general, possibly nonconvex, quadratic constraint. We demonstrate that the SDP relaxation can further be reformulated as an SOCP problem under a simultaneous diagonalization condition of the quadratic form. We then explore several sufficient conditions under which the SOCP relaxation of GETRS is exact under Slater condition.

Amplify-and-forward (AF) is one of the most popular and simple approaches to transmit information over a cooperative multi-input multi-output (MIMO) relay channel. In this paper, we propose three novel power allocation methods for the downlink of cooperative multi-user MIMO AF system, which are designed to maximize the weighted sum-rate (WSR) of the cooperative system, by considering two different levels of channel state information (CSI). We design efficient precoding algorithms at the relay by assuming that either only the receive CSI or both receive and transmit CSI is available at the relay. Results show the performance improvement that our schemes can achieve in terms of sum-rate and WSR metrics.

The new networking paradigm of opportunistic spectrum sharing is a promising technology for mitigating the scarcity of spectrum, which has resulted from the exponential increase in the number of wireless devices and ubiquitous services. In light of the new concept of Authorized/Licensed Shared Access (ASA/LSA), in this work, we consider the coexistence and spectrum sharing between a collocated multiple-input-multiple-output (MIMO) radar and a full-duplex (FD) multi-user MIMO cellular system consisting of a FD base station (BS) serving multiple downlink and uplink users simultaneously, and accordingly focus on maximizing the detection probability of the radar. The main objective of this paper is to develop an optimization technique for jointly optimizing the beamforming weights at the BS and transmit power for uplink users that can maximize the detection probability of radar while guaranteeing the quality-of-service requirements of each user and power budget for the uplink users and the BS. The joint beamforming design is a non-convex problem which, we convert into a second-order cone programming (SOCP) problem and propose an iterative algorithm for finding the optimal solution. Numerical results demonstrate the feasibility of the spectral coexistence and show a scalable trade-off in performance of both systems.

We present a method for tracking deformable surfaces in 3D using a stereo rig. Different from traditional recursive tracking approaches that provide a strong prior on the pose for each new frame, the proposed method tracks deformable surfaces by detecting them in individual frames. In our method, the model of the surface is represented by a triangulated mesh. The constraints for model to image keypoint correspondences, together with the constraints that preserve the lengths of mesh edges, are formulated as Second Order Cone Programming (SOCP) constraints, leading this tracking-by-detection method to be an SOCP problem that can be effectively solved. Experiments on a piece of deformed paper demonstrate the capability of the proposed tracking-by-detection method.