Nonconvex Quadratic Programming Research Articles

Coordinated allocation of distributed generation, soft open points and energy storage systems in unbalanced distribution networks considering flexibility deficiency risk

Optimizing the location and capacity of flexible resources is critical. The increase in the permeability of distributed generation (DG) causes the net load to drastically fluctuate, which sharply increases flexibility demand for the distribution network. The asymmetric access of distributed generators, asymmetric line parameters, and unbalanced loads aggravate the three-phase voltage unbalance of distribution systems. Energy storage systems (ESSs) and soft open points (SOPs) can flexibly adjust power flow direction, which can address the problem. This study analyzed the characteristics of flexibility resources based on the flexibility theory. Furthermore, the potential loss of flexibility demand is quantitatively assessed by conditional value-at-risk, and the coordination planning model of DG, SOP, and ESS for the unbalanced distribution network is constructed considering flexibility and economy. Because the resulting mathematical formulation belongs to a nonconvex quadratic programming problem, an adaptive linear relaxation programming method is proposed to improve computational efficiency. The adaptive linear relaxation technique is constructed based on the properties of quadratic functions and the mean value theorem, and the non-convex quadratic programming problem is converted to a series of parametric linear relaxation programming problems by the technique. Finally, the effectiveness of the proposed model is verified on IEEE 33-bus and IEEE 123-bus systems.

An efficient branch-and-bound algorithm using an adaptive branching rule with quadratic convex relaxation for globally solving general linear multiplicative programs

This article proposes an efficient global algorithm for solving general linear multiplicative programming problem (GLMP). The new algorithm combines the quadratic convex relaxation problem, adaptive branching rule, region reduction technique and branch-and-bound scheme. Firstly, a transformation technique can transform GLMP into a non-convex quadratic program with quadratic and linear constraints. The non-convexity parts of the equivalent problem are addressed by employing secant lines, so that a quadratic convex relaxation problem is structured. Secondly, we introduce the adaptive branching rule to improve the upper bound of the optimal value. Thirdly, the convergence of the proposed algorithm is analyzed and its worst-case complexity is provided. Finally, numerical experiments demonstrate the efficiency and advantage of the proposed algorithm for obtaining the global ɛ-optimal solutions of test instances.

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.

A slightly lifted convex relaxation for nonconvex quadratic programming with ball constraints

A slightly lifted convex relaxation for nonconvex quadratic programming with ball constraints

Polyhedral properties of RLT relaxations of nonconvex quadratic programs and their implications on exact relaxations

We study linear programming relaxations of nonconvex quadratic programs given by the reformulation–linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible regions of a quadratic program and its RLT relaxation. We establish various connections between recession directions, boundedness, and vertices of the two feasible regions. Using these properties, we present a complete description of the set of instances that admit an exact RLT relaxation. We then give a thorough discussion of how our results can be converted into simple algorithmic procedures to construct instances of quadratic programs with exact, inexact, or unbounded RLT relaxations.

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.

Effective algorithms for separable nonconvex quadratic programming with one quadratic and box constraints

Effective algorithms for separable nonconvex quadratic programming with one quadratic and box constraints

Effective algorithms for optimal portfolio deleveraging problem with cross impact

AbstractWe investigate the optimal portfolio deleveraging (OPD) problem with permanent and temporary price impacts, where the objective is to maximize equity while meeting a prescribed debt/equity requirement. We take the real situation with cross impact among different assets into consideration. The resulting problem is, however, a nonconvex quadratic program with a quadratic constraint and a box constraint, which is known to be NP‐hard. In this paper, we first develop a successive convex optimization (SCO) approach for solving the OPD problem and show that the SCO algorithm converges to a KKT point of its transformed problem. Second, we propose an effective global algorithm for the OPD problem, which integrates the SCO method, simple convex relaxation, and a branch‐and‐bound framework, to identify a global optimal solution to the OPD problem within a prespecified ε‐tolerance. We establish the global convergence of our algorithm and estimate its complexity. We also conduct numerical experiments to demonstrate the effectiveness of our proposed algorithms with both real data and randomly generated medium‐ and large‐scale OPD instances.

Model-Driven Deep Learning ADMM Decoder for Irregular Binary LDPC Codes

In this letter, a model-driven deep learning (DL) decoder for irregular binary low-density parity-check (LDPC) codes is proposed via the alternating direction method of multipliers (ADMM) technique. Our technical contributions three twofold: 1) we formulate the maximum likelihood decoding problem as a non-convex quadratic program with a new penalty term and present an ADMM based decoder; 2) to alleviate hyper-parameter tuning, we employ the deep unfolding strategy to derive a model-driven ADMM-DL decoder; and 3) to save the number of learning parameters, we propose an initialization strategy which can apply to different codeword structures. Specifically, the designing procedure for the DL network structure and DL network parameters training algorithm are presented in detail for efficient implementation. Numerical results demonstrate that the proposed model-driven ADMM-DL decoder for irregular binary LDPC codes is competitive in comparison with the state-of-the-arts.

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

Geometric Properties of Time-Optimal Controls With State Constraints Using Strong Observability

This article considers minimum time optimal control problems with linear dynamics subject to state equality constraints and control inequality constraints. In the absence of state constraints, there are well-known sufficient conditions to guarantee that optimal controls are at the boundary or extreme points of the control set. With strong observability as the key tool, analogous conditions are derived for problems subject to both extrinsic and intrinsic state constraints. Understanding these geometric properties enables exact convex relaxations. The relaxation technique is used to convert a nonconvex quadratic program to a second-order cone program and a mixed integer linear program to a linear program. The relaxations accelerate numerical solution times by factors of 18 000 and 150, respectively. As such, the theorems and relaxations are seen as important tools for real-time optimization-based control.

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.

On a solution method in indefinite quadratic programming under linear constraints

We establish some properties of the Proximal Difference-of-Convex functions decomposition algorithm in indefinite quadratic programming under linear constraints. The first property states that any iterative sequence generated by the algorithm is root linearly convergent to a Karush–Kuhn–Tucker point, provided that the problem has a solution. The second property says that iterative sequences generated by the algorithm converge to a locally unique solution of the problem if the initial points are taken from a suitably chosen neighbourhood of it. Through a series of numerical tests, we analyse the influence of the decomposition parameter on the rate of convergence of the iterative sequences and compare the performance of the Proximal Difference-of-Convex functions decomposition algorithm with that of the Projection Difference-of-Convex functions decomposition algorithm. In addition, the performances of the above algorithms and the Gurobi software in solving some randomly generated nonconvex quadratic programs are compared.

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.

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.

A neurodynamic optimization approach to nonconvex resource allocation problem

A neurodynamic optimization approach characterized by a negative subgradient flow (NSF) is proposed to solve distributed nonconvex resource allocation problem (DNRAP). Under some mild assumptions, it is proved that the state solutions of the proposed approach converge to the critical point set of the considered DNRAP. Moreover, benefiting from the well-known nonsmooth Lojasiewicz inequality, analysis results show that the state solutions converge toward a critical point (not set) of the considered DNRAP. The convergence rate of the state solution, including exponential or finite-time convergence, can be evaluated through the quantitative calculation of Lojasiewicz exponent. Furthermore, we assess the related Lojasiewicz exponent of presented NSF in some special situation, for instance, in nonconvex quadratic programming. Finally, simulation illustrates the well performance of presented neurodynamic optimization approach.

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.