Quadratically Constrained Quadratic Programming Problem Research Articles

Energy-Efficient Joint Collaborative and Passive Beamforming for Intelligent-Reflecting-Surface-Assisted Wireless Sensor Networks

Intelligent reflecting surface (IRS) provides a promising technology that can improve the energy efficiency of wireless communications, by building the controllable wireless propagation environment with the utilization of massive passive reflecting elements. Motivated by this, we apply the IRS to assist the data collection based on collaborative beamforming in Wireless Sensor Networks (WSNs). With the objective to maximize the network lifetime, we present an energy efficient joint collaborative and passive beamforming design for an IRS-assisted WSN, which jointly optimizes the beamforming vector for all nodes and phase shifts for IRS’s reflecting elements. To efficiently resolve the joint optimization problem, we develop a penalty dual decomposition (PDD) based algorithm combined with augmented Lagrangian method, penalty-based method and block coordinate descent algorithm, which decomposes the joint optimization problem into four simplified subproblems. Especially in solving IRS’s phase shifts optimization subproblem, we formulate this subproblem as a quadratically constrained quadratic programming (QCQP) problem and further propose a low computational complexity approximate iteration algorithm to resolve this QCQP problem, which iteratively linearizes this QCQP problem to a sequence of approximate programming with linear equality constraints, whose closed-form solution is presented. Theoretical analysis indicates the proposed algorithm can converge to a KKT solution. Simulation results indicate that the IRS-assisted design and proposed algorithm improve the energy efficiency and prolong the network lifetime as compared to alternative algorithms.

Multi-time scale stochastic production simulation under VHVDC long-term contract trading electricity quantity constraint

A two-layer multi-time scale stochastic production simulation framework is constructed to account for the long-term contract electricity quantity of ultra-high voltage direct current (UHVDC) transmission. On the upper layer, based on the characteristics of load demand and renewable energy output extracted from the historical operating data, monthly and daily production simulation models are carried out considering the seasonal characteristics of hydropower during a high-water period and low-water period to optimize the distribution of contract electric quantity sending through UHVDC transmission in the target year or month. According to the DC transmission electric quantity optimized by the daily production simulation in the upper layer, together with the forecast scenario, the lower layer of the framework provides the optimization of day-ahead scheduling and intra-day rolling dispatch in the implementation process. The day-ahead dispatch optimization makes full use of the adjustment capability of transmission and optimizes the DC transmission electric quantity correction. Its compensation is based on the result of the daily production simulation, then the correction will be returned to the upper layer to restart the optimization of the remaining UHVDC contract electric quantity of the subsequent period and its distribution plan. Combined with the day-ahead DC transmission plan, the intra-day rolling optimization is carried out to adjust the output of the unit using more accurate forecasting scenarios. The distributionally robust optimization model is used in the lower layer to convert an uncertain problem into a deterministic quadratically constrained quadratic programming (QCQP) problem according to the form of an uncertain distribution set. Then the QCQP problem is further converted into a linear programming (LP) problem by using the reformulation linearization technique (RLT). A test system with the energy composition and distribution referring to a real provincial power grid in northwest China is established for verification. The results show that the proposed method can effectively improve the economics of system operation and the accommodation of renewable energy based on ensuring security.

A hybrid method for solving non-convex min–max quadratic fractional problems under quadratic constraints

In this paper, we study a non-convex min–max fractional problem of quadratic functions subject to convex and non-convex quadratic constraints. First, by using the Dinkelbach-type method, we transform the fractional problem into a univariate nonlinear equation. To evaluate this equation, we need to solve a non-convex quadratically constrained quadratic programming (QCQP) problem. To solve this problem, we propose a new method. In the proposed method, first, by using relaxation and convexification of non-convex constraints of non-convex QCQP problem, an upper bound and a lower bound of the optimal value is obtained. By using these bounds, we construct a parametric QCQP problem with two constraints. Then, by solution of the new problem, the parameters of this problem are updated for the next iteration. We show that the sequence of solutions of new problems is convergent to a global optimal solution of the non-convex QCQP problem. Numerical results are given to show the applicability of the proposed method.

Signal Shaping for Non-Uniform Beamspace Modulated mmWave Hybrid MIMO Communications

This paper investigates adaptive signal shaping methods for millimeter wave (mmWave) multiple-input multiple-output (MIMO) communications based on the maximizing the minimum Euclidean distance (MMED) criterion. In this work, we utilize the indices of analog precoders to carry information and optimize the symbol vector sets used for each analog precoder activation state. Specifically, we firstly propose a joint optimization based signal shaping (JOSS) approach, in which the symbol vector sets used for all analog precoder activation states are jointly optimized by solving a series of quadratically constrained quadratic programming (QCQP) problems. JOSS exhibits good performance, however, with a high computational complexity. To reduce the computational complexity, we then propose a full precoding based signal shaping (FPSS) method and a diagonal precoding based signal shaping (DPSS) method, where the full or diagonal digital precoders for all analog precoder activation states are optimized by solving two small-scale QCQP problems. Simulation results show that the proposed signal shaping methods can provide considerable performance gain in reliability in comparison with existing mmWave transmission solutions.

Optimal Design of Antenna Arrays Focused on Multiple Targets

This article presents an optimal design method for antenna arrays focused on multiple targets. The optimal design process is characterized by a quadratically constrained quadratic programming (QCQP). The complicated QCQP problem is solved by introducing a weighted diagonal matrix. The radiated power from the transmitting antenna array can be distributed among the multiple targets according to predetermined pattern by adjusting the weighted diagonal matrix. Once the positions of the targets and the predetermined field focusing pattern are all set, the test receiving antennas may be introduced to replace the targets, and the power transmission efficiency between the transmitting antenna array and the test receiving antennas is then maximized, which guarantees the maximum possible power delivery to each target in a focused manner. A unique optimal solution for the distribution of excitations can be found explicitly by analytically solving the QCQP. Several systems are designed for focusing the power to multiple targets in closed and open regions to demonstrate the focusing performance of the transmitting array excited with the optimized distributions of excitations obtained from QCQP.

Robust adaptive beamforming against large steering vector mismatch based on minimum dispersion

Many efforts have been recently devoted to adaptive beamformers based on the minimum dispersion (MD) criterion for non-Gaussian signals, which results in improved signal-to-noise ratio performance. However, large steering vector mismatch and inappropriate choice of uncertainty set will cause performance degradation to the existing MD-based methods. To address these issues, we develop a robust MD-based beamformer against large steering vector mismatch, which utilizes multiple small uncertainty sets to cover the whole uncertainty region. The ℓp−norm (p≥1) minimization problem turns to be nonlinear and nonconvex because of these multi-uncertainty-set constraints. To solve the problem, a projected gradient algorithm is utilized to transform the original problem into a nonconvex quadratically constrained quadratic programming (QCQP) problem. The semidefinite programming relaxation technique is then employed in each iteration to deal with the nonconvex QCQP problem. Numerical results demonstrate that the proposed beamformer offers a significant performance improvement in case of large steering vector mismatch for non-Gaussian signals compared with the conventional schemes.

An Iterative Rank Penalty Method for Nonconvex Quadratically Constrained Quadratic Programs

This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using an iterative method. A QCQP problem can be handled as a linear matrix programming problem wit...

First-Order Methods for Fast Feasibility Pursuit of Non-convex QCQPs

Quadratically constrained quadratic programming (QCQP) is NP-hard in its general non-convex form, yet it frequently arises in various engineering applications. Several polynomial-time approximation algorithms exist for non-convex QCQP problems (QCQPs), but their success hinges upon the ability to find at least one feasible point—which is also hard for a general problem instance. In this paper, we present a heuristic framework for computing feasible points of general non-convex QCQPs using simple first-order methods. Our approach features low computational and memory requirements, which makes it well suited for application to large-scale problems. While a priori it may appear that these benefits come at the expense of technical sophistication, rendering our approach too simple to even merit consideration for a non-convex and NP-hard problem, we provide compelling empirical evidence to the contrary. Experiments on synthetic as well as real-world instances of non-convex QCQPs reveal the surprising effectiveness of first-order methods compared to more established and sophisticated alternatives.

A branch-and-cut algorithm using polar cuts for solving nonconvex quadratic programming problems

In this paper, we propose a branch-and-cut algorithm for solving a nonconvex quadratically constrained quadratic programming (QCQP) problem with a nonempty bounded feasible domain. The problem is first transformed into a linear conic programming problem, and then approximated by semidefinite programming problems over different intervals. In order to improve the lower bounds, polar cuts, generated from corresponding cut-generation problems, are embedded in a branch-and-cut algorithm to yield a globally --optimal solution (with respect to feasibility and optimality, respectively) in a finite number of iterations. To enhance the computational speed, an adaptive branch-and-cut rule is adopted. Numerical experiments indicate that the number of explored nodes required for solving QCQP problems can be significantly reduced by employing the proposed polar cuts.

Fast Approximation Algorithms for a Class of Non-convex QCQP Problems Using First-Order Methods

A number of important problems in engineering can be formulated as non-convex quadratically constrained quadratic programming (QCQP). The general QCQP problem is NP-Hard. In this paper, we consider a class of non-convex QCQP problems that are expressible as the maximization of the point-wise minimum of homogeneous convex quadratics over a “simple” convex set. Existing approximation strategies for such problems are generally incapable of achieving favorable performance-complexity tradeoffs. They are either characterized by good performance but high complexity and lack of scalability, or low complexity but relatively inferior performance. This paper focuses on bridging this gap by developing high performance, low complexity successive non-smooth convex approximation algorithms for problems in this class. Exploiting the structure inherent in each subproblem, specialized first-order methods are used to efficiently compute solutions. Multicast beamforming is considered as an application example to showcase the effectiveness of the proposed algorithms, which achieve a very favorable performance-complexity tradeoff relative to the existing state of the art.

ADMM-Based Fast Algorithm for Multi-Group Multicast Beamforming in Large-Scale Wireless Systems

Multi-group multicast beamforming in wireless systems with large antenna arrays and massive audience is investigated in this paper. Multicast beamforming design is a well-known non-convex quadratically constrained quadratic programming (QCQP) problem. A conventional method to tackle this problem is to approximate it as a semi-definite programming problem via semi-definite relaxation, whose performance, however, deteriorates considerably as the number of per-group users goes large. A recent attempt is to apply convex-concave procedure (CCP) to find a stationary solution by treating it as a difference of convex programming problem, whose complexity, however, increases dramatically as the problem size increases. In this paper, we propose a low-complexity high-performance algorithm for multi-group multicast beamforming design in large-scale wireless systems by leveraging the alternating direction method of multipliers (ADMM) together with CCP. In specific, the original non-convex QCQP problem is first approximated as a sequence of convex subproblems via CCP. Each convex subproblem is then reformulated as a novel ADMM form. Our ADMM reformulation enables that each updating step is performed by solving multiple small-size subproblems with closed-form solutions in parallel. Numerical results show that our fast algorithm maintains the same favorable performance as state-of-the-art algorithms but reduces the complexity by orders of magnitude.

Hidden Convexity in QCQP with Toeplitz-Hermitian Quadratics

Quadratically Constrained Quadratic Programming (QCQP) has a broad spectrum of applications in engineering. The general QCQP problem is NP–Hard. This article considers QCQP with Toeplitz-Hermitian quadratics, and shows that it possesses hidden convexity: it can always be solved in polynomial-time via Semidefinite Relaxation followed by spectral factorization. Furthermore, if the matrices are circulant, then the QCQP can be equivalently reformulated as a linear program, which can be solved very efficiently. An application to parametric power spectrum sensing from binary measurements is included to illustrate the results.

Randomized Algorithms for Optimal Solutions of Double-Sided QCQP With Applications in Signal Processing

Quadratically constrained quadratic programming (QCQP) with double-sided constraints has plenty of applications in signal processing as have been addressed in recent years. QCQP problems are hard to solve, in general, and they are typically approached by solving a semidefinite programming (SDP) relaxation followed by a postprocessing procedure. Existing postprocessing schemes include Gaussian randomization to generate an approximate solution, rank reduction procedure (the so-called purification), and some specific rank-one matrix decomposition techniques to yield a globally optimal solution. In this paper, we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double-sided QCQP (i.e., instances with a small number of constraints). We illustrate their applicability in robust receive beamforming, radar optimal code design, and broadcast beamforming for multiuser communications. As a byproduct, we derive an alternative (shorter) proof for the Sturm-Zhang rank-one matrix decomposition theorem.

A branch-and-cut approach to portfolio selection with marginal risk control in a linear conic programming framework

Marginal risk represents the risk contribution of an individual asset to the risk of the entire portfolio. In this paper, we investigate the portfolio selection problem with direct marginal risk control in a linear conic programming framework. The optimization model involved is a nonconvex quadratically constrained quadratic programming (QCQP) problem. We first transform the QCQP problem into a linear conic programming problem, and then approximate the problem by semidefinite programming (SDP) relaxation problems over some subrectangles. In order to improve the lower bounds obtained from the SDP relaxation problems, linear and quadratic polar cuts are introduced for designing a branch-and-cut algorithm, that may yield an ∈-optimal global solution (with respect to feasibility and optimality) in a finite number of iterations. By exploring the special structure of the SDP relaxation problems, an adaptive branch-and-cut rule is employed to speed up the computation. The proposed algorithm is tested and compared with a known method in the literature for portfolio selection problems with hundreds of assets and tens of marginal risk control constraints.

Exact Computable Representation of Some Second-Order Cone Constrained Quadratic Programming Problems

Solving the quadratically constrained quadratic programming (QCQP) problem is in general NP-hard. Only a few subclasses of the QCQP problem are known to be polynomial-time solvable. Recently, the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation, which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints. In this paper, we provide exact computable representations for some more subclasses of the QCQP problem, in particular, the subclass with one second-order cone constraint and two special linear constraints.

Cooperative Transmission for Relay Networks Based on Second-Order Statistics of Channel State Information

Cooperative beamforming in relay networks is considered, in which a source transmits to its destination with the help of a set of cooperating nodes. The source first transmits locally. The cooperating nodes that receive the source signal retransmit a weighted version of it in an amplify-and-forward (AF) fashion. Assuming knowledge of the second-order statistics of the channel state information, beamforming weights are determined so that the signal-to-noise ratio (SNR) at the destination is maximized subject to two different power constraints, i.e., a total (source and relay) power constraint, and individual relay power constraints. For the former constraint, the original problem is transformed into a problem of one variable, which can be solved via Newton's method. For the latter constraint, the original problem is transformed into a homogeneous quadratically constrained quadratic programming (QCQP) problem. In this case, it is shown that when the number of relays does not exceed three the global solution can always be constructed via semidefinite programming (SDP) relaxation and the matrix rank-one decomposition technique. For the cases in which the SDP relaxation does not generate a rank one solution, two methods are proposed to solve the problem: the first one is based on the coordinate descent method, and the second one transforms the QCQP problem into an infinity norm maximization problem in which a smooth finite norm approximation can lead to the solution using the augmented Lagrangian method.

Nonlinear dimensionality reduction with relative distance comparison

This paper proposes a new algorithm for nonlinear dimensionality reduction. Our basic idea is to explore and exploit the local geometry of the manifold with relative distance comparisons. All such comparisons derived from local neighborhoods are enumerated to constrain the manifold to be learned. The task is formulated as a problem of quadratically constrained quadratic programming (QCQP). However, such a QCQP problem is not convex. We relax it to be a problem of semi-definite programming (SDP), from which a globally optimal embedding is obtained. Experimental results illustrate the validity of our algorithm.