Non-convex Quadratically Constrained Quadratic Programming Research Articles

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.

Precoded Optical Spatial Modulation for Indoor Visible Light Communications

This paper proposes a precoded optical space-domain index modulation scheme for indoor visible light communications, which is based on the optimization of the minimum Euclidean distance of optical spatial modulation (OSM) with real-valued modulation constellations. We find that the precoding matrix design can be formulated as a non-convex quadratically constrained quadratic program (QCQP), whose solution is generally intractable. To tackle this problem, we first consider the case of two optical transmit antennas ( N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> = 2) in the precoded OSM and derive a closed-form solution for arbitrary M-order pulse amplitude modulation (PAM). Based on the derived solutions and the error vector reduction method, we then propose a low-complexity iterative (LCI) algorithm to identify the precoding matrix for the setup N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> > 2. To strike a flexible complexity-BER (bit error rate) tradeoff, we propose a successive convex approximation (SCA)-assisted matrix-based optimization method to transform the non-convex QCQP problem into a series of linear convex subproblems, which can be solved by low-complexity solvers. Simulation results show that these proposed algorithms are capable of substantially improving the system error performance compared with conventional OSM systems. Besides, a symbol-based SCA algorithm is introduced and it is shown to outperform the matrix-based SCA and the suboptimal LCI algorithm in terms of the BER.

Penalized semidefinite programming for quadratically-constrained quadratic optimization

In this paper, we give a new penalized semidefinite programming approach for non-convex quadratically-constrained quadratic programs (QCQPs). We incorporate penalty terms into the objective of convex relaxations in order to retrieve feasible and near-optimal solutions for non-convex QCQPs. We introduce a generalized linear independence constraint qualification (GLICQ) criterion and prove that any GLICQ regular point that is sufficiently close to the feasible set can be used to construct an appropriate penalty term and recover a feasible solution. Inspired by these results, we develop a heuristic sequential procedure that preserves feasibility and aims to improve the objective value at each iteration. Numerical experiments on large-scale system identification problems as well as benchmark instances from the library of quadratic programming demonstrate the ability of the proposed penalized semidefinite programs in finding near-optimal solutions for non-convex 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 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.

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.

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.

Data-Driven Reserve Allocation With Frequency Security Constraint Considering Inverter Air Conditioners

Inverter air conditioners (IACs) with considerable total capacity and fast response speed are ideal demand response resources, which are of significant potential to provide reserve capacity for the power system frequency regulation. However, due to the complexity and implicitness of the frequency response models, it is difficult to formulate the optimization problem considering frequency dynamics to allocate reserve capacity precisely. In this paper, a data-driven method is proposed for reserve allocation with the frequency security constraint considering IACs. Firstly, the equivalent frequency response model of aggregated IACs is developed considering electrical-thermal characteristics and then incorporated into the frequency regulation framework of power systems along with conventional generators. Then, simulations are implemented to generate massive reserve samples with deterministic frequency security labels. Later, a support vector machine (SVM) based frequency security classifier is trained to convert the implicit frequency security constraint into polynomials and reshape the reserve allocation problem into a solvable general quadratically constrained quadratic program (QCQP). Finally, a heuristic Suggest-and-Improve (SI) method is adopted to deal with the nonconvex QCQP of interest. It is demonstrated by numerical studies that the proposed data-driven method enables power systems to operate closer to the frequency security boundaries and thus achieve lower costs.

Unified Power Allocation for Receive Spatial Modulation Based on Approximate Optimization

In this paper, a novel unified power allocation (PA) framework is proposed for receive (pre-coding aided) spatial modulation (RSM). We find that the PA matrix design can be formulated as a non-convex quadratically constrained quadratic program (QCQP) problem, whose solution is generally intractable. To tackle this problem, we propose a pair of solvers having different trade-offs in terms of bit-error-rate and complexity. Specifically, we first propose a successive convex approximation (SCA) method, to convert the non-convex QCQP problem under consideration into a series of linear convex subproblems, where the latter can be easily solved by the classic polynomial-time-based optimization method, i.e., the interior point method. To further reduce the computational complexity, we propose an augmented Lagrangian multiplier (ALM) method, which transforms the challenging non-convex constrained PA optimization problem into its unconstrained counterpart, which can be efficiently solved by an iterative manner. Our simulation results show that both the proposed SCA and ALM methods are capable of substantially improving the system error performance compared with the conventional RSM system without PA as well as conventional PA-aided RSM schemes.

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.

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or Moré’s algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the (usually black-box) iterations for computing the eigenpair. Numerical experiments illustrate the efficiency and accuracy of our algorithm. We also analyze the QCQP solution extensively, including difficult cases, and show that the canonical form of a matrix pair gives a complete classification of the QCQP in terms of boundedness and attainability, and explain how to obtain a global solution whenever it exists.

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.

Design perfect reconstruction cosine-modulated filter banks via quadratically constrained quadratic programming and least squares optimization

In this paper, the design of perfect reconstruction (PR) cosine-modulated filter banks (CMFBs) is implemented via quadratically constrained quadratic programming (QCQP) and least squares (LS) optimization. To this end, a PR CMFB design problem is formulated as a nonconvex QCQP after re-arranging the coefficients of the prototype filter. Then a deep insight is offered into the algebraic relationship between the PR conditions and near-perfect reconstruction (NPR) ones for CMFB designs. Here we theoretically show that the NPR conditions are just the summations of the PR conditions. Firmly in the light of this relationship, a two-stage method is proposed for PR CMFB design. We firstly solve an NPR CMFB problem to obtain its optimal solution as a reference point, then model the PR CMFB design problem as a series of small-sized LS problems near the reference point. And we solve the LS problems in parallel with cheap iteration. Our analysis and numerical results show that the proposed method bears superior performance on effectiveness and efficiency, especially in the case of designing PR CMFBs with large number of channels.

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.

Bound Tightening for the Alternating Current Optimal Power Flow Problem

We consider the Alternating Current Optimal Power Flow (ACOPF) problem, formulated as a nonconvex Quadratically-Constrained Quadratic Program (QCQP) with complex variables. ACOPF may be solved to global optimality with a semidefinite programming (SDP) relaxation in cases where its QCQP formulation attains zero duality gap. However, when there is positive duality gap, no optimal solution to the SDP relaxation is feasible for ACOPF. One way to find a global optimum is to partition the problem using a spatial branch-and-bound method. Tightening upper and lower variable bounds can improve solution times in spatial branching by potentially reducing the number of partitions needed. We propose special-purpose closed-form bound tightening methods to tighten limits on nodal powers, line flows, phase angle differences, and voltage magnitudes. Computational experiments are conducted using a spatial branch-and-cut solver. We construct variants of IEEE test cases with high duality gaps to demonstrate the effectiveness of the bound tightening procedures.

Plane Localization for MIMO Radar

We consider the problem of localizing a ground plane with respect to a multiple-input multiple-output (MIMO) radar within a three-dimensional environment. An example application is a low-visibility landing radar as described by Longstaff (“MIMO radar developments at Teledyne Australia,” Proc. Int. Conf. Radar, 2013). We propose an extension to the common tangent algorithm (COTA), which has been developed by the echo location community for localizing planes with single input multiple output (SIMO) audio systems. The method adopted for this paper is to generalize COTA for the multiple transmitters of the MIMO radar case. Generalizing the COTA equations generates a Non-Convex Quadratically Constrained Quadratic Program (NC-QCQP). We show that by first replacing the quadratic constraints by linear approximations we can solve the NC-QCQP problem by iteratively solving a series of quadratic programming problems. This method demonstrates both global convergence and convergence to the Cramer-Rao Lower Bound (CRLB). We then extend this approach to additionally linearize the quadratic objective function, which in turn reduces the iterate to a simple pseudoinverse. This method retains the global convergence properties of the previous method, but does not quite converge to the CRLB. We also investigate two methods which operate directly on the unconstrained cost function, the first a Newton–Raphson based method and the second a singular value decomposition (SVD)-based approach to solving the optimization problem. The SVD method exhibits approximately quadratic convergence and reaches the CRLB. However, it is only locally convergent. The Newton–Raphson method also achieves the CRLB. We use numerical simulations to demonstrate the validity of the techniques presented and show a comparison to the CRLB for the plane localization problem. Finally, we discuss a preprocessing technique that can be used to improve the estimation accuracy. We conclude by demonstrating the proposed algorithms on data collected from the low visibility landing aid radar (Longstaff , 2013) developed at Teledyne Defence Australia.

Feasible Point Pursuit and Successive Approximation of Non-Convex QCQPs

Quadratically constrained quadratic programs (QCQPs) have a wide range of applications in signal processing and wireless communications. Non-convex QCQPs are NP-hard in general. Existing approaches relax the non-convexity using semi-definite relaxation (SDR) or linearize the non-convex part and solve the resulting convex problem. However, these techniques are seldom successful in even obtaining a feasible solution when the QCQP matrices are indefinite. In this paper, a new feasible point pursuit successive convex approximation (FPP-SCA) algorithm is proposed for non-convex QCQPs. FPP-SCA linearizes the non-convex parts of the problem as conventional SCA does, but adds slack variables to sustain feasibility, and a penalty to ensure slacks are sparingly used. When FPP-SCA is successful in identifying a feasible point of the non-convex QCQP, convergence to a Karush-Kuhn-Tucker (KKT) point is thereafter ensured. Simulations show the effectiveness of our proposed algorithm in obtaining feasible and near-optimal solutions, significantly outperforming existing approaches.

Interval Power Flow Analysis Using Linear Relaxation and Optimality-Based Bounds Tightening (OBBT) Methods

With increasingly large scale of intermittent and non-dispatchable resources being integrated into power systems, the power flow problem presents greater uncertainty. In order to obtain the upper and lower bounds of power flow solutions including voltage magnitudes, voltage angles and line flows, Cartesian coordinates-based power flow is utilized in this paper. A quadratically constrained quadratic programming (QCQP) model is then established to formulate the interval power flow problem. This non-convex QCQP model is relaxed to linear programming problem by introducing convex and concave enclosures of the original feasible region. To improve the solutions bounds while still encompassing the true interval solution, optimality-based bounds tightening (OBBT) method is employed to find a better outer hull of the feasible region. Numerical results on IEEE 9-bus, 30-bus, 57-bus, and 118-bus test systems validate the effectiveness of the proposed method.

A Novel Optimization Method for Nonconvex Quadratically Constrained Quadratic Programs

This paper presents a novel optimization method for effectively solving nonconvex quadratically constrained quadratic programs (NQCQP) problem. By applying a novel parametric linearizing approach, the initial NQCQP problem and its subproblems can be transformed into a sequence of parametric linear programs relaxation problems. To enhance the computational efficiency of the presented algorithm, a cutting down approach is combined in the branch and bound algorithm. By computing a series of parametric linear programs problems, the presented algorithm converges to the global optimum point of the NQCQP problem. At last, numerical experiments demonstrate the performance and computational superiority of the presented algorithm.

Semidefinite Programming-Based Method for Implementing Linear Fitting to Interval-Valued Data

Building a linear fitting model for a given interval-valued data set is challenging since the minimization of the residue function leads to a huge combinatorial problem. To overcome such a difficulty, this article proposes a new semidefinite programming-based method for implementing linear fitting to interval-valued data. First, the fitting model is cast to a problem of quadratically constrained quadratic programming (QCQP), and then two formulae are derived to develop the lower bound on the optimal value of the nonconvex QCQP by semidefinite relaxation and Lagrangian relaxation. In many cases, this method can solve the fitting problem by giving the exact optimal solution. Even though the lower bound is not the optimal value, it is still a good approximation of the global optimal solution. Experimental studies on different fitting problems of different scales demonstrate the good performance and stability of our method. Furthermore, the proposed method performs very well in solving relatively large-scale interval-fitting problems.