Nonconvex Quadratic Programming Problems Research Articles

Multi-View Randomized Kernel Classification via Nonconvex Optimization

Multi kernel learning (MKL) is a representative supervised multi-view learning method widely applied in multi-modal and multi-view applications. MKL aims to classify data by integrating complementary information from predefined kernels. Although existing MKL methods achieve promising performance, they fail to consider the tradeoff between diversity and classification accuracy of kernels, preventing further improvement of classification performance. In this paper, we tackle this problem by generating a number of high-quality base learning kernels and selecting a kernel subset with maximum pairwise diversity and minimum generalization errors. We first formulate this idea as a nonconvex quadratic integer programming problem. Then we transform this nonconvex problem into a convex optimization problem and prove it is equivalent to a semidefinite relaxation problem, which a semidefinite-based branch-and-bound algorithm can quickly solve. Experimental results on the real-world datasets demonstrate the superiority of the proposed method. The results also show that our method works for the support vector machine (SVM) classifier and other state-of-the-art kernel classifiers.

Robust Adaptive Beamformer Based on Constant Modulus Penalty Criteria

In this study, a robust adaptive beamformer based on constant modulus (CM) criteria is developed to improve the robustness of the array beamforming, which is a reconstructing minimal optimization for solving the mismatch problem of weight vector caused by steering vector mismatch. In the global positioning system (GPS) L1 band, firstly, a GPS array signal is modelled by designing a dual-polarized antenna array. Secondly, the distortion problem of beamforming is formulated in the traditional minimum variance distortionless response (MVDR) beamformer. For repairing the weight vector mismatch problem, a novel beamformer based on the CM envelope response is proposed to reconstruct MVDR beamforming in the array processing. Besides, min-max penalty criteria are used to enable the beamformer to allocate more degrees of freedom (DOFs) when penalizing the MVDR beamformer responses. Finally, an auxiliary two-element real variable is designed to plan the proposed beamformer. But it is still a nonconvex quadratic programming problem, so an alternating direction method of multipliers (ADMM) is employed to transform the objective function into several subproblems. Illustrative numerical simulation results are provided for validating the effectiveness of the proposed beamformer by comparing it with other existing approaches.

Operation Optimization Adapting to Active Distribution Networks Under Three-Phase Unbalanced Conditions: Parametric Linear Relaxation Programming

Asymmetric phase distributed generation, which is widely used in distribution systems, aggravates the severity of the intrinsic three-phase unbalanced condition of distribution systems. To overcome these problems, operation optimization formulas of the three-phase unbalanced active distribution network (ADN) are established considering soft open point (SOP) and belong to a nonconvex quadratic programming (NQP) problem. However, in conventional methods, either approximations that are not valid for an unbalanced distribution system are used or convex relaxation methods are applied to primitive nonlinear optimization models. However, these methods do not guarantee a feasible optimization solution. Therefore, determining the solution to the NQP problem is challenging. In this article, we propose a parametric linear relaxation algorithm to obtain the optimal feasible solution by solving the operation optimization problem of the three-phase unbalanced ADN with the SOP. The parametric linear relaxation technique is constructed based on the properties of quadratic functions and the mean value theorem, and the NQP is converted to a series of parametric linear relaxation programming problems. The results revealed that the proposed algorithm can achieve the optimal and feasible solution and considerably reduce the computation time compared with an equivalent NLP optimization model for the same distribution system.

Multi-Beam Design for Near-Field Extremely Large-Scale RIS-Aided Wireless Communications

As the energy-saving array composed of passive elements, reconfigurable intelligent surface (RIS) will evolve to the extremely large-scale RIS (XL-RIS) to overcome serious path loss. This change leads to the near-field propagation becoming dominant. There are some works to explore the near-field beam design via beam training. Unfortunately, due to the constant modulus constraint for XL-RIS, most of works in the near-field scenario focus on single-beam design. For massive connectivity requirement scenario, these works will face a serious loss of beam gains, resulting in a decrease in transmission rate. To solve this problem, we propose a block coordinate descent-based scheme with majorization-minimization (MM) algorithm for multi-beam design. The proposed scheme handles constant modulus constraint from two aspects. Firstly, under this constraint, the multi-beam design is an intractable non-convex quadratic programming problem. We utilize MM algorithm to solve this problem as several iterative sub-problems which are easily to be sloved. Secondly, the solution space for multi-beam optimization is confined to a limited space due to this constraint, so we introduce the phases for beam gains as an extra optimizable variable to enrich the degree of freedom for optimization. Simulation results show that the proposed scheme could achieve a superior rate 50% higher than the existing schemes.

Global optimization for non-convex programs via convex proximal point method

<p style='text-indent:20px;'>In this study, a convex proximal point algorithm (CPPA) is considered for solving constrained non-convex problems, and new theoretical results are proposed. It is proved that every cluster point of CPPA is a stationary point, and the initial point of CPPA is key to global optimization. Several sufficient conditions for the initial point selection are provided for CPPA to find the global minimum. Motivated by these results, numerical experiments were conducted on non-convex quadratic programming problems with convex quadratic constraints. The performance of CPPAs was compared, with the initial point randomly selected or obtained through the Lagrangian dual problem. The numerical results demonstrate that the quality of the CPPA with the computed Lagrangian dual initial point is much better than that with the random initial point, in terms of the objective function value.</p>

Outcome-space branch-and-bound outer approximation algorithm for a class of non-convex quadratic programming problems

Outcome-space branch-and-bound outer approximation algorithm for a class of non-convex quadratic programming problems

A new SOCP relaxation of nonconvex quadratic programming problems with a few negative eigenvalues

We present a new second order cone programming (SOCP) relaxation of nonconvex quadratic programs with a few negative eigenvalues (NQP-r-NE) by employing the difference of convex (DC) decomposition and simultaneous matrix diagonalization together. The auxiliary variables are bounded from above by one more convex quadratic constraint in the proposed SOCP relaxation than that in the classical SOCP relaxation provided by Kim and Kojima (2001). We prove that the proposed SOCP relaxation is strictly tighter than the classical SOCP relaxation under certain circumstances, especially when there exists one constraint matrix being positive definite in the primal problem. Three types of numerical experiments including the large-scale NQP-r-NE problem with the number of negative eigenvalues r≤20, the optimal spectrum sharing problem in MIMO cognitive radio networks and the two-trust-region subproblem are provided to illustrate that the proposed SOCP relaxation could achieve a better lower bound than that of the classical SOCP relaxation. Moreover, when the number of variables is larger than 250, the proposed SOCP relaxation shows its great superiority both in the bound quality and computing efficiency compared to the classical SOCP relaxation.

Energy efficient air-to-ground communication networks with reconfigurable intelligent surface

Reconfigurable intelligent surface (RIS) assisted air-to-ground wireless powered communication networks (WPCNs) are constructed to promote the energy efficiency of the system, in which unmanned aerial vehicles (UAV) reflects energy towards the energy-constrained hybrid access points (HAP) through RIS. Subsequently, HAP uses the collected energy to communicate with users in blind areas through the RIS-UAV. By optimizing parameters such as charging time ratio, transmit power and phase shifts matrix, the energy efficiency for single and multiuser scenarios are maximized, respectively. The energy efficiency is modeled as a non-convex quadratic programming problem with inequality constraints in the single-user scenario. Then the problem is converted into an equivalent convex problem by using the Taylor expansion and convex approximation method. A joint optimization algorithm of power and phase shifts is proposed to obtain a feasible solution. The energy efficiency is modeled as a fractional programming problem with non-convex inequality constraints in the multi-user scenario, and the feasible solution is obtained by proposing an alternating algorithm. Finally, both two scenarios are compared with the system of traditional amplify-and-forward (AF) relay, respectively. Finally, the simulation results show that the proposed algorithms can provide up with 150% and 160% improvement of the energy efficiency in the single and multi-user scenarios, respectively, compared to the traditional AF relay.

3-D Inter-Robot Relative Localization via Semidefinite Optimization

In this letter, the 3-D inter-robot relative localization problem is addressed using noise-corrupted odometric and distance measurements. Unlike the existing solutions, we are devoted to providing a relative localization method that has an “overall best performance,” which means that the tradeoffs between the estimation accuracy (EA), the number of measurements (NoMs), and the computation efficiency (CE) are considered. We demonstrate that an existing formulation of the 3-D relative localization problem, the square distances weighted least square (SD-WLS), can be equivalently reformulated as a non-convex quadratic constrained quadratic programming (QCQP) problem. Further, to handle the non-convex nature of the QCQP problem, we adopt the semidefinite programming (SDP) relaxation approach, which drops the rank constraint and recovers the solution of the QCQP via an eigenvalue decomposition strategy. Finally, a refinement step is introduced to solve the problem that the quadratic constraints might not be satisfied due to the SDP relaxation. The simulation and experiment results show that, compared to existing methods, our method has the best overall performance when the three factors, i.e., EA, NoMs, and CE, are important for a relative localization application.

Tight compact extended relaxations for nonconvex quadratic programming problems with box constraints

Cutting planes from the Boolean Quadric Polytope can be used to reduce the optimality gap of the mathcal {NP}-hard nonconvex quadratic program with box constraints (BoxQP). It is known that all cuts of the Chvátal–Gomory closure of the Boolean Quadric Polytope are A-odd cycle inequalities. We obtain a compact extended relaxation of allA-odd cycle inequalities, which allows to optimize over the Chvátal–Gomory closure without repeated calls to separation algorithms and has less inequalities than the formulation provided by Boros et al. (SIAM J Discrete Math 5(2):163–177, 1992) for sparse matrices. In a computational study, we confirm the strength of this relaxation and show that we can provide very strong bounds for the BoxQP, even with a plain linear program. The resulting bounds are significantly stronger than these from Bonami et al. (Math Program Comput 10(3):333–382, 2018), which arise from separating A-odd cycle inequalities heuristically.

Алгоритм дифференциальной эволюции для оптимизации направленности фазированных антенных решеток

The problem of optimization of the directivity of phased antenna arrays for shortwave radio communication is formulated as a nonconvex quadratic programming problem. Taking into account the specifics of the problem under consideration, a hybrid algorithm based on the method of differential evolution, the method of penalty functions and gradient optimization is proposed. Correction of invalid solutions is carried out by means of their scaling. The penalty coefficient is chosen adaptively during the algorithm execution. A computational experiment was carried out, where the proposed algorithm is compared with the commercial solver BARON, based on repeated application of the local search algorithm and branch and bound method, as well as with the ANTIGONE solver and the gradient algorithm. In addition, the effect of accounting for symmetries of the considered problem is studied. Accordingto the results experiment, we can conclude that the proposed hybrid algorithm is competitive, especially on problems of higher dimension.

Robust Energy-Efficient Hybrid Beamforming Design for Massive MIMO LEO Satellite Communication Systems

In low earth orbit (LEO) satellite communication systems, the limited energy supply capacity and the difficulty in obtaining channel state information (CSI) are the practical challenges. Motivated by this, we focus on the design of robust hybrid beamforming to maximize the energy efficiency of LEO satellite communication systems. Assuming that the LEO satellite transmitter adopts the massive multi-input multi-output (MIMO) technology, considering the CSI errors caused by propagation delay and Doppler shift, under the constraints of transmit power and quality of service (QoS), a robust energy-efficient hybrid beamforming scheme is proposed. Since that there are no explicit expressions for the ergodic user rate and the ergodic signal-to-interference-plus-noise ratio, the approximate values with closed-form expressions are adopted. Then, we invoke the semidefinite programming (SDP) algorithm to transform the nonconvex quadratic constrained quadratic programming (QCQP) problem equivalently, and an inner and outer nested iterative algorithm combining quadratic transformation fractional programming (QTFP) and concave convex process (CCCP) is utilized to transfer a nonconvex problem into a convex problem. Meanwhile, we adopt a penalty function algorithm to solve the rank-one constraint in semidefinite programming algorithm. Finally, we invoke the normal form distance minimization (NFDM) algorithm and the alternating optimization (AltOpt) algorithm to jointly solve the digital beamforming matrix and analog beamforming matrix in hybrid beamformer. Numerical results validate that our proposed robust approach significantly outperforms the conventional one.

A New Global Optimization Scheme for Quadratic Programs with Low-Rank Nonconvexity

We consider the classical convex constrained nonconvex quadratic programming problem where the Hessian matrix of the objective to be minimized has r negative eigenvalues, denoted by (QPr). Based on a biconvex programming reformulation in a slightly higher dimension, we propose a novel branch-and-bound algorithm to solve (QP1) and show that it returns an [Formula: see text]-approximate solution of (QP1) in at most [Formula: see text] iterations. We further extend the new algorithm to solve the general (QPr) with r > 1. Computational comparison shows the efficiency of our proposed global optimization method for small r. Finally, we extend the explicit relaxation approach for (QP1) to (QPr) with r > 1. Summary of Contribution: Nonconvex quadratic program (QP) is a classical optimization problem in operations research. This paper aims at globally solving the QP where the Hessian matrix of the objective to be minimized has r negative eigenvalues. It is known to be nondeterministic polynomial-time hard even when r = 1. This paper presents a novel algorithm to globally solve the QP for r = 1 and then extends to general r. Numerical results demonstrate the superiority of the proposed algorithm in comparison with state-of-the-art algorithms/software for small r.

Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation

The mean-variance (MV) portfolio is typically formulated as a quadratic programming (QP) problem that linearly combines the conflicting objectives of minimizing the risk and maximizing the expected return through a risk aversion profile parameter. In this formulation, the two objectives are expressed in different units, an issue that could definitely hamper obtaining a more competitive set of portfolio weights. For example, a modification in the scale in which returns are expressed (by one or percent) in the MV portfolio, implies a modification in the solution of the problem. Motivated by this issue, a novel mean squared variance (MSV) portfolio is proposed in this paper. The associated optimization problem of the proposed strategy is very similar to the Markowitz optimization, with the exception of the portfolio mean, which is presented in squared form in our formulation. The resulting portfolio model is a non-convex QP problem, which has been reformulated as a mixed-integer linear programming (MILP) problem. The reformulation of the initial non-convex QP problem into an MILP allows for future researchers and practitioners to obtain the global solution of the problem via the use of current state-of-the-art MILP solvers. Additionally, a novel purely data-driven method for determining the optimal value of the hyper-parameter that is associated with the MV and MSV approaches is also proposed in this paper. The MSV portfolio has been empirically tested on eight portfolio time series problems with three different estimation windows (composing a total of 24 datasets), showing very competitive performance in most of the problems.

Задачи устойчивости круговых арок с неудерживающими связями

The work deals with the problems of stability of arches and arch systems supported by inextensible threads that cannot withstand compressive forces. These questions relate to contact problems of elastic structures with an unknown region of active interaction of their elements, and the study is reduced to the search and analysis of bifurcation points ofsolutions to variational problems in the presence of inequality constraints on the sought functions. The paper considers the problem of stability of an arch supported by threads, one end of which is attached to some point of the arch, the other to the projection of this point on the chord connecting its ends. In the numerical solution, the problem is reducedto finding a load parameter in which the nonconvex quadratic programming problem with constraints on variables in the form of inequalities has a non-unique solution. Numerical calculations have shown that even with a small number of reinforcing threads, the critical load approximately doubles.

Local Linear Convergence of the Alternating Direction Method of Multipliers for Nonconvex Separable Optimization Problems

In this paper, we consider the convergence rate of the alternating direction method of multipliers for solving the nonconvex separable optimization problems. Based on the error bound condition, we prove that the sequence generated by the alternating direction method of multipliers converges locally to a critical point of the nonconvex optimization problem in a linear convergence rate, and the corresponding sequence of the augmented Lagrangian function value converges in a linear convergence rate. We illustrate the analysis by applying the alternating direction method of multipliers to solving the nonconvex quadratic programming problems with simplex constraint, and comparing it with some state-of-the-art algorithms, the proximal gradient algorithm, the proximal gradient algorithm with extrapolation, and the fast iterative shrinkage–thresholding algorithm.

Optimal correction of the absolute value equations

In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares is applied for stabilizing the solution of the fractional problem. The regularized problem is reduced to a unimodal single variable minimization problem and to solve it a bisection algorithm is proposed. The main difficulty of the algorithm is a complicated constraint optimization problem, for which two novel methods are suggested. We also present optimality conditions and bounds for the norm of the optimal solutions. Numerical experiments are given to demonstrate the effectiveness of suggested methods.

Trade-off of Tactical Technical Indicators Based on Improved Branch and Bound Method

In view of the relative lack of research on the method of determining the Tactical Technical Indicators requirements through the typical indexes trade-off model in the domestic combat technology index demonstration, based on the non-convex quadratic programming problem widely used in engineering design and economies of scale, a new linear relaxation is proposed with the help of the characteristics of the quadratic function The method uses the new ultra-rectangular reduction and pruning techniques according to the differential median theorem, and uses the branch and bound method to solve, which provides a quantified and efficient method for determining the technical parameters of the weapon equipment design.

Exploiting aggregate sparsity in second-order cone relaxations for quadratic constrained quadratic programming problems

Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for nonconvex quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. [Exploiting sparsity in semidefinite programming via matrix completion I: General framework, SIAM J. Optim. 11(3) (2001), pp. 647–674] and second-order cone programming (SOCP) relaxation have been popular. In this paper, we exploit the aggregate sparsity of SOCP relaxation of nonconvex QCQPs. Specifically, we prove that exploiting the aggregate sparsity reduces the number of second-order cones in the SOCP relaxation, and that we can simplify the matrix completion procedure by Fukuda et al. in both primal and dual of the SOCP relaxation problem without losing the max-determinant property. For numerical experiments, nonconvex QCQPs from the lattice graph and pooling problem are tested as their SOCP relaxations provide the same optimal value as the SDP relaxations. We demonstrate that exploiting the aggregate sparsity improves the computational efficiency of the SOCP relaxation for the same objective value as the SDP relaxation, thus much larger problems can be handled by the proposed SOCP relaxation than the SDP relaxation.

Wireless-Powered Over-the-Air Computation in Intelligent Reflecting Surface-Aided IoT Networks

Fast wireless data aggregation and efficient battery recharging are two critical design challenges of Internet-of-Things (IoT) networks. Over-the-air computation (AirComp) and energy beamforming (EB) turn out to be two promising techniques that can address these two challenges, necessitating the design of wireless-powered AirComp. However, due to severe channel propagation, the energy harvested by IoT devices may not be sufficient to support AirComp. In this article, we propose to leverage the intelligent reflecting surface (IRS) that is capable of dynamically reconfiguring the propagation environment to drastically enhance the efficiency of both downlink EB and uplink AirComp in IoT networks. Due to the coupled problems of downlink EB and uplink AirComp, we further propose the joint design of energy and aggregation beamformers at the access point, downlink/uplink phase-shift matrices at the IRS, and transmit power at the IoT devices, to minimize the mean-squared error (MSE), which quantifies the AirComp distortion. However, the formulated problem is a highly intractable nonconvex quadratic programming problem. To solve this problem, we first obtain the closed-form expressions of the energy beamformer and the device transmit power, and then develop an alternating optimization framework based on difference-of-convex programming to design the aggregation beamformers and IRS phase-shift matrices. Simulation results demonstrate the performance gains of the proposed algorithm over the baseline methods and show that deploying an IRS can significantly reduce the MSE of AirComp.