Quadratically Constrained Quadratic Programming Research Articles

Resource Allocation and Computation Offloading for Wireless Powered Mobile Edge Computing.

In this paper, we investigate a resource allocation and computation offloading problem in a heterogeneous mobile edge computing (MEC) system. In the considered system, a wireless power transfer (WPT) base station (BS) with an MEC sever is able to deliver wireless energy to the mobile devices (MDs), and the MDs can utilize the harvested energy for local computing or task offloading to the WPT BS or a Macro BS (MBS) with a stronger computing server. In particular, we consider that the WPT BS can utilize full- or half-duplex wireless energy transmission mode to empower the MDs. The aim of this work focuses on optimizing the offloading decision, full/half-duplex energy harvesting mode and energy harvesting (EH) time allocation with the objective of minimizing the energy consumption of the MDs. As the formulate problem has a non-convex mixed integer programming structure, we use the quadratically constrained quadratic program (QCQP) and semi-definite relaxation (SDR) methods to solve it. The simulation results demonstrate the effectiveness of the proposed scheme.

Fuel-Optimal Guidance for End-to-End Human-Mars Entry, Powered-Descent, and Landing Mission

This article investigates the fuel-optimal guidance problem of the end-to-end human-Mars entry, powered-descent, and landing (EDL) mission. It applies a unified modeling scheme and develops a computationally efficient new optimization algorithm to solve the multiphase optimal guidance problem. The end-to-end EDL guidance problem is first modeled as a multiphase optimal control problem with different dynamics and constraints at each phase. Via polynomial approximation and discretization techniques, this multiphase optimal control problem is then reformulated as a polynomial programming problem. By introducing intermediate variables and quadratic equality constraints, a polynomial program is equivalently converted into a nonconvex quadratically constrained quadratic program (QCQP). Then, a novel customized alternating direction method of multipliers (ADMM) is proposed to efficiently solve the large-scale QCQP with convergence proof to a local optimum under certain conditions on the algorithmic parameters. The fuel savings under the end-to-end human-Mars EDL guidance are verified by comparing to the fuel consumption using the separate phase guidance approach. Furthermore, the computational efficiency of the customized ADMM algorithm is validated by comparing to the state-of-the-art nonlinear programming method. The robustness of the customized ADMM algorithm is verified via extensive simulation cases with random initial conditions.

Quaternion matrix decomposition and its theoretical implications

This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in Ai W et al (Math Progr 128(1):253–283, 2011), Huang Y, Zhang S (Math Oper Res 32(3):758–768, 2007), Sturm JF, Zhang S (Math Oper Res 28(2):246–267 2003). The enhanced property can be used to drive some improved results in joint numerical range, \({\mathcal {S}}\)-Procedure and quadratically constrained quadratic programming (QCQP) in the quaternion domain, demonstrating the capability of our new decomposition technique.

Assessment of a two-step approach for global optimization of mixed-integer polynomial programs using quadratic reformulation

This paper revisits the approach of transforming a mixed-integer polynomial program (MIPOP) into a mixed-integer quadratically-constrained program (MIQCP), in the light of recent progress in global solvers for this latter class of models. We automate this transformation in a new reformulation engine called CANON, alongside preprocessing strategies including local search and bounds tightening. We conduct comparative tests on a collection of 137 MIPOPs gathered from test libraries such as MINLPLib. The solver GUROBI gives the best performance on the reformulated MIQCPs and outperforms the generic global solvers BARON and SCIP. The MIQCP reformulation also improves the performance of SCIP compared to direct MIPOP solution, whereas the performance of BARON is comparable on the original MIPOPs and reformulated MIQCPs. Overall, these results establish the effectiveness of quadratic reformulation for MIPOP global optimization and support its integration into global solvers.

Solving Combined Optimal Transmission Switching and Optimal Power Flow sequentially as convexificated Quadratically Constrained Quadratic Program

A novel approach to efficiently solve a combined optimal transmission switching and optimal power flow problem is derived and evaluated. The approach is based on a sequentially solved quadratically constrained quadratic program with a new way of convexification. This convexification requires adapted modeling and uses two procedural steps: First, nonlinear equality constraints are eliminated by quadratically approximating their inverse system of functions and inserting it into the objective function and into the inequality constraints. The resulting objective function and inequality constraints are approximated quadratically. Second, the remaining nonconvex parts of the objective function and the inequality constraints are identified by eigenvalue analysis of their Hessian matrices and eliminated by using piecewise linear approximations. Issues of accuracy and convergence are examined and countermeasures are presented. Additionally, the avoidance of islanding and the sequentiality of parallel circuits are considered. Case studies show fast und stable convergence of the approach.

Integrated sizing and scheduling of an off-grid integrated energy system for an isolated renewable energy hydrogen refueling station

When designing an isolated renewable energy hydrogen refueling station (HRS), it is important to consider the electrical, heating and cooling demands of the supporting building as it is crucial for maintaining the stable HRS operation. Therefore, this paper proposed a new structure of off-grid integrated energy system (OIES) for an isolated renewable energy HRS, which can meet the hydrogen load of the market and the three kinds of loads of the supporting building. The OIES in question consists of wind generators, PV panels, batteries, electrolyzers, heat storage tanks, hydrogen tanks, and absorption chillers. A mixed integer quadratic constrained programming (MIQCP) model of the OIES is built based on the minimization of the total life cycle cost (TLCC). It is built to obtain the optimal solution for sizing and the equipment units’ scheduling strategy. To accord with the actual situation, the electrical, heating, and cooling loads of the supporting building in the simulation are obtained by the EnergyPlus software. Furthermore, the ‘community’ hydrogen demand is obtained by the HOMER software. The results indicate that the proposed structure meets a wide array of energy demands and is more sensible solution when compared to other configurations with one component less. In addition, sensitivity analyses of the influence of economic data, meteorological data, and hydrogen load on TLCC are performed. These results show that the capital and replacement cost of the PV panels are the major economic factors influencing the TLCC. They also confirm that the PV is the system’s main component due to a high correlation between the hourly hydrogen demand per year and the hourly solar radiation per year. Lastly, the effects of meteorological data and hydrogen load are also quantitatively analyzed.

Learning to Solve the AC-OPF Using Sensitivity-Informed Deep Neural Networks

To shift the computational burden from real-time to offline in delay-critical power systems applications, recent works entertain the idea of using a deep neural network (DNN) to predict the solutions of the AC optimal power flow (AC-OPF) once presented load demands. As network topologies may change, training this DNN in a sample-efficient manner becomes a necessity. To improve data efficiency, this work utilizes the fact OPF data are not simple training labels, but constitute the solutions of a parametric optimization problem. We thus advocate training a sensitivity-informed DNN (SI-DNN) to match not only the OPF optimizers, but also their partial derivatives with respect to the OPF parameters (loads). It is shown that the required Jacobian matrices do exist under mild conditions, and can be readily computed from the related primal/dual solutions. The proposed SI-DNN is compatible with a broad range of OPF solvers, including a non-convex quadratically constrained quadratic program (QCQP), its semidefinite program (SDP) relaxation, and MATPOWER; while SI-DNN can be seamlessly integrated in other learning-to-OPF schemes. Numerical tests on three benchmark power systems corroborate the advanced generalization and constraint satisfaction capabilities for the OPF solutions predicted by an SI-DNN over a conventionally trained DNN, especially in low-data setups.

Fast trajectory planning and control of a lab-scale 3D gantry crane for a moving target in an environment with obstacles

In this work, the real-time optimal trajectory planning, together with a cascaded tracking controller, is presented for a three-dimensional (3D) gantry crane in an environment with static obstacles and a dynamically moving target considering dynamic constraints and control input limits. State-of-the-art trajectory optimization-based approaches require long computation times and cannot quickly respond to changes in the target state. The focus of this paper lies on a novel trajectory planning algorithm, which consists of two steps. First, an offline trajectory planner is implemented to compute a time-optimal, collision-free, and dynamically feasible trajectory database that connects all possible initial states of the gantry crane from a predefined starting subspace to the target states in a target subspace. Second, based on linear constrained quadratic programming, the online trajectory replanner makes use of this trajectory database to generate an optimal trajectory in real time that accounts for all changes in the target state. Additionally, a trajectory tracking controller is developed to take into account the dynamic constraints of the gantry crane and to compensate for possible model inaccuracies, disturbances, and other non-modeled effects. Both simulation and experimental results are presented to demonstrate the performance of the proposed trajectory (re)planning algorithm and the control concept.

A Comprehensive Approach for an Approximative Integration of Nonlinear-Bivariate Functions in Mixed-Integer Linear Programming Models

As decentralized energy supply units, microgrids can make a decisive contribution to achieving climate targets. In this context, it is particularly important to determine the optimal size of the energy components contained in the microgrids and their optimal operating schedule. Hence, mathematical optimization methods are often used in association with such tasks. In particular, mixed-integer linear programming (MILP) has proven to be a useful tool. Due to the versatility of the different energetic components (e.g., storages, solar modules) and their special technical characteristics, linear relationships can often only inadequately describe the real processes. In order to take advantage of linear solution techniques but at the same time better represent these real-world processes, accurate and efficient approximation techniques need to be applied in system modeling. In particular, nonlinear-bivariate functions represent a major challenge, which is why this paper derives and implements a method that addresses this issue. The advantage of this method is that any bivariate mixed-integer nonlinear programming (MINLP) formulation can be transformed into a MILP formulation using this comprehensive method. For a performance comparison, a mixed-integer quadratic constrained programming (MIQCP) model—as an MINLP special case—is applied and transformed into a MILP, and the solution of the transformed problem is compared with the one of the MIQCP. Since there are good off-the-shelf solvers for MIQCP problems available, the comparison is conservative. The results for an exemplary microgrid sizing task show that the method delivers a strong performance, both in terms of approximation error (0.08%) and computation time. The method and its implementation can serve as a general user-tool but also as a basis for further methodological developments and research.

Modeling side slopes in vertical alignment resource road construction using convex optimization

AbstractA new convex quadratically‐constrained quadratic programming (QCQP) model is proposed for modeling side‐slopes volumes in the minimization of earthwork operations to compute the vertical alignment of a resource road while satisfying design and safety constraints. The new QCQP model is convex but nonlinear; it is compared to a state‐of‐the‐art mixed integer linear programming (MILP) model. The QCQP model can be viewed as the limit of this MILP model. Numerical results show that for roads with less than 100 stations, the QCQP model has similar computation time to the MILP model. However, the QCQP model significantly outperforms the MILP model for other roads, in some cases finding a global optimum in minutes while the MILP model fails to find any solution in hours. The technique is directly applicable for resource roads and has potential for other types of road.

Intelligent Omni Surface-Assisted Secure MIMO Communication Networks With Artificial Noise

Intelligent omni surface (IOS) technology that can provide both reflection and refraction for signals impinging on its surface, was recently proposed to overcome the coverage area limitation of reconfigurable intelligent surfaces (RISs). This letter investigates the efficacy of IOS in the context of physical layer security (PLS) over wireless channels. In particular, we consider a scenario where the IOS is utilized to enhance the secrecy performance of a legitimate receiver in the presence of a multi-antenna eavesdropper. In addition, artificial noise (AN) aided beamforming is implemented to provide additional security robustness. The resultant optimization problem is non-convex and difficult to solve. Accordingly, the block coordinate descent (BCD) optimization approach is adopted, and the Lagrangian dual method is applied to reduce the complexity of the AN-aided beamforming design. Furthermore, the reflecting and refracting phase shifts are optimized via the quadratically constrained quadratic programming (QCQP) method. Simulation results validate the efficacy of the proposed algorithm and confirm the superiority of IOS over traditional RIS.

Adaptive Waveform Design for Communication-Enabled Automotive Radars

Large-scale deployment of connected vehicles with cooperative sensing technologies increases the demand on the vehicular communication spectrum in 5.9 GHz allocated for the exchange of safety messages. To support the high data rates needed by such applications, the millimeter-wave (mmWave) automotive radar spectrum at 76–81 GHz can be utilized for wideband communication as well. For this purpose, various joint automotive radar-communication (JARC) systems have been proposed in the literature to perform both functions using the same wideband waveform. However, the wideband joint waveforms encounter frequency-selectivity in both radar and communication channels due to multi-path propagation. In this paper, we address the optimal joint waveform design problem to exploit the frequency-selectivity for wideband JARC operations via orthogonal frequency-division multiplexing (OFDM) wherein subcarrier coefficients are designed for optimal power allocation and phase coding. We show that the problem is a non-convex quadratically constrained quadratic programming (QCQP) problem which is known to be NP-hard. Existing approaches to solve QCQP include semidefinite relaxation (SDR) which incurs high time complexity. Instead, we propose approximation methods to solve QCQP more efficiently by leveraging structured matrices and using convex approximations. Finally, we demonstrate the efficacy of the proposed approaches through numerical simulations.

Necessary and Sufficient Conditions for Rank-One-Generated Cones

A closed convex conic subset [Formula: see text] of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of [Formula: see text] is closely related to the exactness of semidefinite program (SDP) relaxations of nonconvex quadratically constrained quadratic programs (QCQPs) related to [Formula: see text]. We consider the case where [Formula: see text] is obtained as the intersection of the PSD cone with finitely many homogeneous linear matrix inequalities and conic constraints and identify sufficient conditions that guarantee that [Formula: see text] is ROG. Our general framework allows us to recover a number of well-known results from the literature. In the case of two linear matrix inequalities, we also establish the necessity of our sufficient conditions. This extends one of the few settings from the literature—the case of one linear matrix inequality and the S-lemma—where an explicit characterization for the ROG property exists. Finally, we show how our ROG results on cones can be translated into inhomogeneous SDP exactness results and convex hull descriptions in the original space of a QCQP. We close with a few applications of these results; specifically, we recover the well-known perspective reformulation of a simple mixed-binary set via the ROG toolkit. Funding: This work was supported by the Office of Naval Research [Grant N00014-19-1-2321] with the National Science Foundation (NSF) Division of Civil, Mechanical and Manufacturing Innovation [Grant CMMI-1454548] and Frank A. and Helen E. Risch Faculty Development Chair. Part of this work was done while the second author was visiting the Simons Institute for the Theory of Computing, which was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through the NSF Division of Computing and Communication Foundations [Grant CCF-1740425].

Optimal Power Quality Compensation of Energy Storage System in Distribution Networks Based on Unified Multi-Phase OPF Model

Energy Storage System (ESS) is a promising solution to suppress the peak-valley difference of residential distribution networks (RDN) with high penetration of distributed photovoltaic generations. Meanwhile, it can also provide certain power quality compensations to RDN due to the flexible adjusting ability of its converter interface. To make full use of this feature, this paper investigates the optimal power quality compensation of the ESS. This is achieved by formulating and solving an optimal power flow (OPF) problem, the objectives of which are to minimize the power loss, harmonic distortion, and voltage unbalance of the network. Noticeably, the established model includes the neutral line and inter-phase loads; thus, it is unified and can be easily extended to networks with different structures. However, the coupling between different phases and the inclusion of harmonic variables significantly increase the complexity of the OPF problem. To address this issue, the optimization model is further transformed into a convex quadratic constrained quadratic program (QCQP) by replacing power flow equations with current injection equations. As a result, the problem can be solved by the primal-dual interior-point method (PDIPM) with guaranteed convergence and computational efficiency. An iterative solution method is further proposed to improve the accuracy of the solution. Finally, the proposed method is verified using a three-phase three-wire 25-node distribution network and a three-phase four-wire 162-node distribution network in North America.

Robust adaptive beamforming based on covariance matrix reconstruction with RCB principle

In this paper, a novel robust adaptive beamforming (RAB) technique based on the covariance matrix reconstruction is proposed to solve the array model mismatch. The proposed technique provides an alternative method to reconstruct the interference-plus-noise covariance matrix (IPNCM) and estimate the steering vector (SV) of the desired signal. In particular, an unknown error exists in the SV of the considered array. The proposed RAB algorithm adopts the robust Capon beamformer (RCB) principle to roughly estimate the SVs of the desired and interference signals. Based on those preliminary estimated SVs, an improved Capon power spectrum is constructed. Then the interference covariance matrix is reconstructed by utilizing the modified Capon power spectrum integrated over the union of several disjoint angular sectors. Then, the reconstructed covariance matrix is further refined by exploring the low rank property. Meanwhile, the SV of the desired signal is estimated by solving a modified quadratically constrained quadratic programming (QCQP) problem. The simulation results show that the RCB principle provides an accurate IPNCM reconstruction, and the proposed RAB algorithm outperforms the existing RAB techniques over a wide range of input signal-to-noise ratio (SNR) region under various mismatch conditions.

Adaptive dual model predictive control for linear systems with parametric uncertainties

Adaptive dual model predictive control (DMPC) for linear systems with constant parametric uncertainties is investigated in this paper. In particular, chance constraints on output and input are considered. The online set-membership identification and recursive least squares are utilized for shrinking the uncertain parameter set and getting the estimation point of the parameters. The dual effect represented by the parameters' prediction error is considered in the receding horizon optimization problem for easing the impact of the uncertainties and improving the performance simultaneously. Chance constraints on output are tackled by converting into convex alternatives. The output tracking problem is transformed into a quadratically constrained quadratic-programming (QCQP) problem, which is computationally tractable. A numerical example is provided to illustrate the effectiveness of the proposed method.

NISQ Algorithm for Hamiltonian simulation via truncated Taylor series

Simulating the dynamics of many-body quantum systems is believed to be one of the first fields that quantum computers can show a quantum advantage over classical computers. Noisy intermediate-scale quantum (NISQ) algorithms aim at effectively using the currently available quantum hardware. For quantum simulation, various types of NISQ algorithms have been proposed with individual advantages as well as challenges. In this work, we propose a new algorithm, truncated Taylor quantum simulator (TQS), that shares the advantages of existing algorithms and alleviates some of the shortcomings. Our algorithm does not have any classical-quantum feedback loop and bypasses the barren plateau problem by construction. The classical part in our hybrid quantum-classical algorithm corresponds to a quadratically constrained quadratic program (QCQP) with a single quadratic equality constraint, which admits a semidefinite relaxation. The QCQP based classical optimization was recently introduced as the classical step in quantum assisted eigensolver (QAE), a NISQ algorithm for the Hamiltonian ground state problem. Thus, our work provides a conceptual unification between the NISQ algorithms for the Hamiltonian ground state problem and the Hamiltonian simulation. We recover differential equation-based NISQ algorithms for Hamiltonian simulation such as quantum assisted simulator (QAS) and variational quantum simulator (VQS) as particular cases of our algorithm. We test our algorithm on some toy examples on current cloud quantum computers. We also provide a systematic approach to improve the accuracy of our algorithm.

Operational optimization of a 4th generation district heating network with mixed integer quadratically constrained programming

Decreasing CO2-Emissions is a common goal for efficient energy system design and operation. An efficient design for the heating sector are 4th-generation district heating networks with waste heat integration. This paper focuses on operational optimization of a 4th-generation district heating network with regards to temperatures and use of thermal inertia in the network and in thermal energy storages. A case study is presented, where the system is formulated as a Mixed-Integer Quadratically Constrained Program. The quadratic formulations allow to simultaneously optimize energy balances and temperatures, with regards to efficiency correlations and temperature thresholds of components. The case study consists of seven consumers with heat pumps and thermal energy storages, which are supplied by waste heat from a nearby high-performance computer. The case study is solved for eight representative days. The results show, that savings in operational costs up to 18.2% compared to a reference system with constant temperatures are possible. All in all, this paper demonstrates the use and limitations of quadratic formulations in an operational optimization and emphasizes the possibility of temperature optimizations in operation of 4th-generation district heating.

A Collaborative Computational Offloading Strategy for Latency-Sensitive Applications in Fog Networks

Network data traffic has expanded exponentially over the past decade, resulting in massive congestion in heterogeneous networks. Nevertheless, it is nearly impossible to run latency-intensive applications to the end-users local processing unit due to limited system resources. Recently, fog computing has come up with a solution to reduce such data congestion by offloading some or the whole part of the task to the nearby fog nodes (FNs) or the clouds. But this offloading policy becomes more complex when the FN is unable to process the task and further offload it to another neighboring FN or the cloud. In this context, in this study, we have analyzed the offloading strategy in a hierarchical fog-cloud network consisting of several heterogeneous fog devices along with a helping fog and a centralized cloud server. We have considered the most possible practical situation where the FNs are equipped with different CPU frequencies and hence, the power consumption is also different. The total system cost is formulated as a mixed-integer nonlinear problem that aims to reduce the overall delay in the proposed network. To solve the NP-hard problem, we transform it into quadratically constrained quadratic programming (QCQP) formation and further solve it by the separable semidefinite relaxation (SDR) method. Finally, by adopting several benchmark data, we conduct comprehensive simulations to test the efficiency of the proposed offloading profile. The simulation results depict that the proposed strategy outperforms in many aspects when compared to various baseline algorithms.

Error analysis for denoising smooth modulo signals on a graph

In many applications, we are given access to noisy modulo samples of a smooth function with the goal being to robustly unwrap the samples, i.e., to estimate the original samples of the function. In a recent work, Cucuringu and Tyagi [11] proposed denoising the modulo samples by first representing them on the unit complex circle and then solving a smoothness regularized least squares problem – the smoothness measured w.r.t. the Laplacian of a suitable proximity graph G – on the product manifold of unit circles. This problem is a quadratically constrained quadratic program (QCQP) which is nonconvex, hence they proposed solving its sphere-relaxation leading to a trust region subproblem (TRS). In terms of theoretical guarantees, ℓ 2 error bounds were derived for (TRS). These bounds are however weak in general and do not really demonstrate the denoising performed by (TRS). In this work, we analyse the (TRS) as well as an unconstrained relaxation of (QCQP). For both these estimators we provide a refined analysis in the setting of Gaussian noise and derive noise regimes where they provably denoise the modulo observations w.r.t. the ℓ 2 norm. The analysis is performed in a general setting where G is any connected graph.