Quadratically Constrained Quadratic Programming Research Articles

An Efficient Solution to Non-Minimal Case Essential Matrix Estimation.

Finding relative pose between two calibrated images is a fundamental task in computer vision. Given five point correspondences, the classical five-point methods can be used to calculate the essential matrix efficiently. For the case of N ( ) inlier point correspondences, which is called N-point problem, existing methods are either inefficient or prone to local minima. In this paper, we propose a certifiably globally optimal and efficient solver for the N-point problem. First we formulate the problem as a quadratically constrained quadratic program (QCQP). Then a certifiably globally optimal solution to this problem is obtained by semidefinite relaxation. This allows us to obtain certifiably globally optimal solutions to the original non-convex QCQPs in polynomial time. The theoretical guarantees of the semidefinite relaxation are also provided, including tightness and local stability. To deal with outliers, we propose a robust N-point method using M-estimators. Though global optimality cannot be guaranteed for the overall robust framework, the proposed robust N-point method can achieve good performance when the outlier ratio is not high. Extensive experiments on synthetic and real-world datasets demonstrated that our N-point method is 2∼3 orders of magnitude faster than state-of-the-art methods. Moreover, our robust N-point method outperforms state-of-the-art methods in terms of robustness and accuracy.

The Convex Hull of a Quadratic Constraint over a Polytope

A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. ...

Alternating Optimization Based Hybrid Precoding Strategies for Millimeter Wave MIMO Systems

In millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems, the hybrid beamforming architecture has been put forward to reduce the high hardware cost and power consumption, which are resulted from the tremendous requirements of dedicated radio frequency (RF) chains. In this paper, we propose several strategies to design analog and digital precoders for a point-to-point (P2P) hybrid MIMO system. Aiming at minimizing the Euclidean distance between the optimal digital precoder and hybrid precoder, we decouple this matrix factorization problem into a nonconvex quadratically constrained quadratic programming (QCQP) problem and an unit-modulus least-squares (ULS) problem, which can be solved by the presented three alternating optimization algorithms. Simulation and analysis results indicate that the proposed semidefinite relaxation based alternating optimization (SDR-AO) algorithm can approach near-optimal spectral efficiency performance compared with previous algorithms in the literature, but shows extremely high computational complexity. The alternating direction method of multipliers based alternating optimization (ADMM-AO) algorithm is preferred in the case that the number of transmit antennas is much larger than that of receive antennas or the amount of data streams is small. Moreover, when equal number of RF chains and data streams are employed, the analytical constant modulus factorization based alternating optimization (ACMF-AO) algorithm is a better choice. Finally, the proposed algorithms can also be well applied in finite resolution phase shifters (PSs) of the analog component and are extended to wideband mmWave systems.

Cooperative Delay-Constrained Cognitive Radio Networks: Delay-Throughput Trade-Off With Relaying Full-Duplex Capability

In this paper, the problem of maximizing the secondary user (SU) throughput under a primary user (PU) quality of service (QoS) delay requirement is studied. Moreover, the impact of having a full-duplex capability at the SU on the network performance is investigated compared to the case of a SU with half-duplex capability. We consider a cooperative cognitive radio (CR) network in which the receiving nodes have multi-packet reception (MPR) capabilities. In our proposed model, the SU not only benefits from the idle time slots (i.e. when PU is idle), but also chooses between sharing the channel or cooperating with the PU in a probabilistic manner. We formulate our optimization problem to maximize the SU throughput under a PU QoS, defined by a delay constraint; the optimization is performed over the transmission modes selection probabilities of the SU. The resultant optimization problem is found to be a non-convex quadratic constrained quadratic programming (QCQP) optimization problem, which is, generally, an NP-hard problem. We devise an efficient approach to solve it and to characterize the network stability region under a delay constraint set on the PU. Numerical results, surprisingly, reveal that the network performance when full-duplex capability exists at the SU is not always better compared to that of a half-duplex SU. In fact, we demonstrate that a full-duplex capability at the SU can, in some cases, adversely influence the network stability performance, especially if the direct channel conditions between the SU and the destinations are worse than that between the PU and the destinations. In addition, we formulate a multi-objective programming (MOP) optimization problem to investigate the trade-off between the PU delay and the SU throughput. Our MOP approach allows for assigning relative weights for our two conflicting performance metrics, i.e., PU delay and SU throughput. Numerical results also demonstrate that our cooperation policy outperforms conventional cooperative and non-cooperative policies presented in previous works.

Optimal program control in the class of quadratic splines for linear systems

This article describes an algorithm for solving the optimal control problem in the case when the considered process is described by a linear system of ordinary differential equations. The initial and final states of the system are fixed and straight two-sided constraints for the control functions are defined. The purpose of optimization is to minimize the quadratic functional of control variables. The control is selected in the class of quadratic splines. There is some evolution of the method when control is selected in the class of piecewise constant functions. Conveniently, due to the addition/removal of constraints in knots, the control function can be piecewise continuous, continuous, or continuously differentiable. The solution algorithm consists in reducing the control problem to a convex mixed-integer quadratically-constrained programming problem, which could be solved by using well-known optimization methods that utilize special software.

Robust FDA-STAP approach based on multiple response constraints

Frequency diverse array (FDA) has the ability to suppress the range-dependent interference and resolve the range ambiguity by using its additional degree-of-freedom (DOFs) in range domain. However, the spatial-temporal spectrum of fast-moving target is not ideally focused in spatial-temporal plane, which causes a large mismatch between the real and presumed target locations and significantly degrades the performance of space-time adaptive processing (STAP). To solve this problem, in this paper, a robust FDA-STAP approach based on multiple magnitude response constraints is proposed to improve the detection performance of fast-moving target. Specifically, several small uncertainty sets are employed to constraint the magnitude response, and then the worst-case based optimization problem is formed. To reduce the influence of the spatial-temporal-spectrum-defocusing of fast-moving target, a non-focused constraint on the STAP weight vector is devised. Finally, the proposed method is formulated as a non-convex quadratically constrained quadratic programming (QCQP) problem, which is efficiently resolved by using multiple constraints relaxation technique. Simulation results demonstrate that the proposed method outperforms other state-of-the-arts methods including well-controlled mainbeam and superior target detection performance.

Bilevel programming approach to demand response management with day-ahead tariff

This paper introduces a bilevel programming approach to electricity tariff optimization for the purpose of demand response management (DRM) in smart grids. In the multi-follower Stackelberg game model, the leader is the profit-maximizing electricity retailer, who must set a time-of-use variable energy tariff in the grid. Followers correspond to groups of prosumers (simultaneous producers and consumers of the electricity. They response to the observed tariff, schedule controllable loads and determine the charging/discharging policy of their batteries to minimize the cost of electricity and to maximize the utility at the same time. A bilevel programming formulation of the problem is defined, and its fundamental properties are proven. The primal-dual reformulation is proposed in this paper to convert the bilevel optimization problem into a single-level quadratically constrained quadratic program (QCQP), and a successive linear programming (SLP) algorithm is applied to solve it. It is demonstrated in computational experiments that the proposed approach outperforms typical earlier methods based on the Karush–Kuhn–Tucker (KKT) reformulation regarding both solution quality and computational efficiency on practically relevant problem sizes. Besides, it also offers more flexible modeling capabilities.

An Efficient Large Resource-User Scale SCMA Codebook Design Method

Sparse code multiple access (SCMA) is an attractive non-orthogonal multiple access (NOMA) technology. An effective SCMA codebook design metrics is maximizing minimum Euclidean distance between superimposed codewords. However, the number of superimposed codewords will significantly expand with the growing number of resources and users, making the max–min problem impractical. To this end, an efficient suboptimal SCMA codebook design method is proposed for large source-user scale. We first project multi-dimensional superimposed codewords to resources to reduce the number of superimposed codewords in the constellation. Then we formulate the max–min optimization as a nonconvex quadratically constrained quadratic programming (QCQP) problem and solve it by semidefinite relaxation (SDR). Simulation result shows that the method, which can dramatically reduce computational complexity, is able to solve large-scale codebook design problem while incurring little loss in performance compared with the existing codebook design method.

Signal Shaping for Generalized Spatial Modulation and Generalized Quadrature Spatial Modulation

This paper investigates generic signal shaping methods for multiple-data-stream generalized spatial modulation (GenSM) and generalized quadrature spatial modulation (GenQSM) based on the maximizing the minimum Euclidean distance (MMED) criterion. Three cases with different channel state information at the transmitter (CSIT) are considered, including no CSIT, statistical CSIT and perfect CSIT. A unified optimization problem is formulated to find the optimal transmit vector set under size, power and sparsity constraints. We propose an optimization-based signal shaping (OBSS) approach by solving the formulated problem directly and a codebook-based signal shaping (CBSS) approach by finding sub-optimal solutions in discrete space. In the OBSS approach, we reformulate the original problem to optimize the signal constellations used for each transmit antenna combination (TAC). Both the size and entry of all signal constellations are optimized. Specifically, we suggest the use of a recursive design for size optimization. The entry optimization is formulated as a non-convex large-scale quadratically constrained quadratic programming (QCQP) problem and can be solved by existing optimization techniques with rather high complexity. To reduce the complexity, we propose the CBSS approach using a codebook generated by quadrature amplitude modulation (QAM) symbols and a low-complexity selection algorithm to choose the optimal transmit vector set. Simulation results show that the OBSS approach exhibits the optimal performance in comparison with existing benchmarks. However, the OBSS approach is impractical for large-size signal shaping and adaptive signal shaping with instantaneous CSIT due to the demand of high computational complexity. As a low-complexity approach, CBSS shows comparable performance and can be easily implemented in large-size systems.

Optimal finite-dimensional spectral densities for the identification of continuous-time MIMO systems

This paper presents a method for designing inputs to identify linear continuous-time multiple-input multiple-output (MIMO) systems. The goal here is to design a T-optimal band-limited spectrum satisfying certain input/output power constraints. The input power spectral density matrix is parametrized as the product $$\phi_u(\text{j}\omega)=\frac{1}{2}H(\text{j}\omega)H^\text{H}(\text{j}\omega)$$ , where H(jω) is a matrix polynomial. This parametrization transforms the T-optimal cost function and the constraints into a quadratically constrained quadratic program (QCQP). The QCQP turns out to be a non-convex semidefinite program with a rank one constraint. A convex relaxation of the problem is first solved. A rank one solution is constructed from the solution to the relaxed problem. This relaxation admits no gap between its solution and the original non-convex QCQP problem. The constructed rank one solution leads to a spectrum that is optimal. The proposed input design methodology is experimentally validated on a cantilever beam bonded with piezoelectric plates for sensing and actuation. Subspace identification algorithm is used to estimate the system from the input-output data.

Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming

In this paper, we present new convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP) problems. While recent research has focused on strengthening convex relaxations of QCQP using the reformulation-linearization technique (RLT), the state-of-the-art methods lose their effectiveness when dealing with (multiple) nonconvex quadratic constraints in QCQP, except for direct lifting and linearization. In this research, we decompose and relax each nonconvex constraint to two second order cone (SOC) constraints and then linearize the products of the SOC constraints and linear constraints to construct some new effective valid constraints. Moreover, we extend the reach of the RLT-like techniques for almost all different types of constraint-pairs (including valid inequalities by linearizing the product of a pair of SOC constraints, and the Hadamard product or the Kronecker product of two respective valid linear matrix inequalities), examine dominance relationships among different valid inequalities, and explore almost all possibilities of gaining benefits from generating valid constraints. We also successfully demonstrate that applying RLT-like techniques to additional redundant linear constraints could reduce the relaxation gap significantly. We demonstrate the efficiency of our results with numerical experiments.

A New Mixed Integer Linear Programming Formulation for Protection Relay Coordination Using Disjunctive Inequalities

Numerical optimization-based solution to directional overcurrent relay (DOCR) coordination problem has been a widely addressed research problem in the recent past. Many linear (LP), nonlinear (NLP), mixed integer nonlinear (MINLP), mixed integer linear (MILP), and quadratically constrained quadratic programming (QCQP)-based formulations have been presented in the past literature. This paper proposes a new MILP-based formulation using disjunctive inequalities. The nonlinear DOCR protection coordination model is formulated as MILP by linearizing the bilinear terms existing in the original formulation. One of the variables in each bilinear term is discretized over its interval into a fixed number of steps. After assigning binary variables to each discrete interval, the resulting bilinear terms with binary variables are written in terms of disjunctive inequalities. The results have shown that the proposed MILP formulation fetch better optimal solutions compared with past MILP and MINLP formulations. The MILP problem is programmed in GAMS package with CPLEX solver and tested on standard 3 bus, 9 bus, 15 bus, and 30 bus systems and results are found to be satisfactory.

L1-NORM FITTING OF ELLIPTIC PARABOLOIDS WITH PRIOR INFORMATION FOR ENHANCED CONIFEROUS TREE LOCALIZATION IN ALS POINT CLOUDS

Abstract. Airborne laser scanning (ALS) is an established tool for deriving various tree characteristics in forests. In some applications, an accurate pointwise estimate of the tree position is required. For dense data acquired by TLS or UAV-mounted scanners, this can be achieved by locating the stem, whose center coordinates are then used for deriving the planimetric tree position. However, in case of standard ALS data this is often not an option due to the low probability of obtaining stem hits in operational scenarios of forest mapping campaigns. This paper presents an alternative, indirect approach where the tree position is approximated as the center of a quadric surface which best represents the tree crown shape. The study targets coniferous trees due to their distinct crown shape which may be approximated by an elliptic paraboloid. It is assumed that individual tree point clusters are given and the task is to find the tree center for each cluster. We first consider the general problem of fitting an elliptic paraboloid with a known axis and an L1 residual norm error criterion, which is more robust to outliers compared to least-squares fitting. We formulate this problem as a quadratically constrained quadratic program (QCQP), and show how prior knowledge on the crown shape and center position can be incorporated. Next, a computationally simpler problem is considered where the paraboloid semiaxis lengths are constrained to be equal, and a corresponding linear program is constructed. Experiments on ALS datasets of forest plots from Bavaria, Germany and Oregon, USA reveal that a reduction in median tree position error of up to 20% can be attained compared to both least-squares fitting and other baseline techniques, resulting in an absolute error of ca. 22 cm on both datasets.

A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint

In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the sparsity of the involved matrices and solves the problem via solving a sequence of positive definite system of linear equations after identifying suitable generalized eigenvalues. Specifically, we analyze how to recognize hard case (case 2) in a preprocessing step, fixing an error in Sect. 2.2.2 of Pong and Wolkowicz (Comput Optim Appl 58(2):273–322, 2014) which studies the same problem with the two-sided constraint. Some numerical experiments are given to show the effectiveness of the proposed method and to compare it with some recent algorithms in the literature.

Certifiably Globally Optimal Extrinsic Calibration From Per-Sensor Egomotion

We present a certifiably globally optimal algorithm for determining the extrinsic calibration between two sensors that are capable of producing independent egomotion estimates. This problem has been previously solved using a variety of techniques, including local optimization approaches that have no formal global optimality guarantees. We use a quadratic objective function to formulate calibration as a quadratically constrained quadratic program (QCQP). By leveraging recent advances in the optimization of QCQPs, we are able to use existing semidefinite program (SDP) solvers to obtain a certifiably global optimum via the Lagrangian dual problem. Our problem formulation can be globally optimized by existing general-purpose solvers in less than a second, regardless of the number of measurements available and the noise level. This enables a variety of robotic platforms to rapidly and robustly compute and certify a globally optimal set of calibration parameters without a prior estimate or operator intervention. We compare the performance of our approach with a local solver on extensive simulations and multiple real datasets. Finally, we present necessary observability conditions that connect our approach to recent theoretical results and analytically support the empirical performance of our system.

Recover feasible solutions for SOCP relaxation of optimal power flow problems in mesh networks

The AC optimal power flow (OPF) problem is essential for the schedule and operation of power systems. Convex relaxation methods have been studied and used extensively to obtain an optimal solution to the OPF problem. When the exactness of convex relaxations is not guaranteed, it is important to recover a feasible solution for the convex relaxation methods. This paper presents an alternative convex optimisation (ACP) approach that can efficiently recover a feasible solution from the result of second-order cone programming (SOCP) relaxed OPF in mesh networks. The OPF problem is first formulated as a difference-of-convex programming (DCP) problem, then efficiently solved by a penalty convex concave procedure (CCP). By using the solution of a tightened SOCP OPF as an initial point, the proposed algorithm is able to find a global or near-global optimal solution to the AC OPF problem. Numerical tests show that the proposed method outperforms those semi-definite programming (SDP) and quadratically constrained quadratic programming (QCQP)-based algorithms.

Optimization Design of M-Channel Oversampled Graph Filter Banks via Successive Convex Approximation

This letter presents an efficient algorithm to design an M-channel oversampled graph filter bank, whose overall performance is governed by the spectral characteristic of the filters and the reconstruction error. The design of the filters is formulated into a constrained optimization problem that minimizes the stopband energy of the filters subject to the reconstruction error constraint. The problem is a non-convex quadratically constrained quadratic program (QCQP), which is typically NP-hard. In order to overcome this challenge, we apply the successive convex approximation to successively solve it. At each iteration, the non-convex QCQP is approximately transformed into the convex one by linearizing the constraint. Numerical examples and comparison are included to show that the proposed approach can lead to the oversampled graph filter banks with satisfactory overall performance, particularly good spectral selectivity.

Architecting a fully fuzzy information model for multi-level quadratically constrained quadratic programming problem

Fully fuzzy quadratic programming became emerge naturally in numerous real-world applications. Therefore, an effective model based on the bound and decomposition method and the separable programming method is proposed in this paper for solving Fully Fuzzy Multi-Level Quadratically Constrained Quadratic Programming (FFMLQCQP) problem, where the objective function and the constraints are quadratic, also all the coefficients and variables of both objective functions and constraints are described fuzzily as fuzzy numbers. The bound and decomposition method is recommended to decompose the given (FFMLQCQP) problem into series of crisp Quadratically Constrained Quadratic Programming (QCQP) problems with bounded variable constraints for each level. Each (QCQP) problem is then solved independently by utilizing the separable programming method, which replaces the quadratic separable functions with linear functions. At last, the fuzzy optimal solution to the given (FFMLQCQP) problem is obtained. The effectiveness of the proposed model is illustrated through an illustrative numerical example.

Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

We study a class of quadratically constrained quadratic programs (QCQPs), called {\em diagonal QCQPs\/}, which contain no off-diagonal terms $x_j x_k$ for $j \ne k$, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a general QCQP into diagonal form, we establish new, polynomial-time-checkable sufficient conditions for the semidefinite relaxations of general QCQPs to be exact. Finally, these ideas are extended to show that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables. To the best of our knowledge, this is the first result establishing the exactness of the semidefinite relaxation for random general QCQPs.

A hybrid technique for joint PAPR reduction and sidelobe suppression in NC-OFDM based cognitive radio systems

Non-Contiguous Orthogonal Frequency Division Multiplexing (NC-OFDM) based Cognitive Radio (CR) system automatically identifies available frequency gaps in spectrum and utilize them to meet the requirement of higher data rates in 5G technology. It allocates some unauthorized users to access those frequency gaps without causing interference to the adjacent users. Besides the advantages of NC-OFDM based CR system, it suffers from two main drawbacks which are high Peak-to-Average Power Ratio (PAPR) and large spectrum sidelobe power. In order to overcome these drawbacks, we propose a hybrid technique that jointly reduces PAPR and sidelobe power. This technique involves a two level or modified clipping scheme for PAPR reduction. For sidelobe power suppression, signal cancellation symbols are generated by calculating Error Vector Magnitude (EVM) that are added to the original data bits. Here, the problem of sidelobe suppression can be formulated as a quadratically constrained quadratic program (QCQP), and the optimal solution can be obtained by the standard interior-point method. By combining two techniques we can solve these two issues in NC-OFDM based CR system simultaneously. From the simulation results, it is inferred that our proposed method outperforms interms of both PAPR, sidelobe power suppression over other existing methods and also improves the BER performance of the system.