Quadratically Constrained Quadratic Programming Research Articles

Efficient Computation of Minimal Wind-Power Deviations That Induce Temporal Line Overloading

The paper develops an optimization method for assessing transmission network vulnerability to small changes in generation (as caused, for example, by wind forecast inaccuracy). The method computes the smallest deviation (in a weighted 2-norm sense) from the nominal generation pattern that would drive a particular line to a specified temperature, over a given time horizon. The 2-norm weighting matrix provides a means of capturing spatial and temporal coupling between generation sites and time intervals. The temperature constraint is second-order in voltage angle differences. The problem is therefore a quadratically-constrained quadratic program (QCQP). Solving the QCQP for each line in the network yields a set of candidate generation deviation patterns which may then be sorted to determine the lines that are most vulnerable to overloading. The paper develops a computationally efficient algorithm for solving this QCQP. An example explores line-overload vulnerability due to changes in wind patterns. Numerical results emphasize the framework’s ability to incorporate evolving ambient and system conditions, as well as computational scaling properties.

A penalty decomposition algorithm with greedy improvement for mean‐reverting portfolios with sparsity and volatility constraints

AbstractMean‐reverting portfolios with few assets, but sufficient variance, are of great interest for investors in financial markets. Such portfolios are straightforwardly profitable because they include a small number of assets whose prices not only oscillate predictably around a long‐term mean but also possess enough volatility. Roughly speaking, sparsity minimizes trading costs, volatility provides arbitrage opportunities, and mean‐reversion property equips investors with ideal investment strategies. Finding such appealing portfolios can be formulated as a nonconvex quadratic optimization problem with an additional sparsity constraint. To the best of our knowledge, there is no method in the literature that directly solves this problem with its solution achieving favorable theoretical properties. In this paper, we develop an efficient two‐stage algorithm for solving this problem which, in the worst case, provably obtains a stationary point and further, demonstrates numerical efficiency. In the first stage of our algorithm, we apply a tailored penalty decomposition method for finding a stationary point of this nonconvex problem. For a fixed penalty parameter, the block coordinate descent method is utilized to find a stationary point of the associated penalty subproblem. In the second stage, we improve the result from the first stage via a greedy scheme that solves restricted nonconvex quadratically constrained quadratic programs (QCQPs). We show that the optimal value of such a QCQP can be obtained by solving their semidefinite relaxations. Numerical experiments on S&P 500 are conducted to demonstrate the efficiency of the proposed algorithm.

Double Intelligent Reflecting Surface-Assisted Multi-User MIMO Mmwave Systems With Hybrid Precoding

This work investigates the effect of double intelligent reflecting surface (IRS) in improving the spectrum efficient of multi-user multiple-input multiple-output (MIMO) network operating in the millimeter wave (mmWave) band. Specifically, we aim to solve a weighted sum rate maximization problem by jointly optimizing the digital precoding at the transmitter and the analog phase shifters at the IRS, subject to the minimum achievable rate constraint. To facilitate the design of an efficient solution, we first reformulate the original problem into a tractable one by exploiting the majorization-minimization (MM) method. Then, a block coordinate descent (BCD) method is proposed to obtain a suboptimal solution, where the precoding matrices and the phase shifters are alternately optimized. Specifically, the digital precoding matrix design problem is solved by the quadratically constrained quadratic programming (QCQP), while the analog phase shift optimization is solved by the Riemannian manifold optimization (RMO). The convergence and computational complexity are analyzed. Finally, simulation results are provided to verify the performance of the proposed design, as well as the effectiveness of double-IRS in improving the spectral efficiency.

New Methodology for Robust-Optimal Design with Acceptable Risks: An Application to Blended/Wing/Body Aircraft

In this paper, a new design methodology guaranteeing optimal performance and constraint satisfaction with acceptable risks is presented. This methodology is particularly tailored to conceptual design, wherein variability stemming from modifications that will inevitably occur in downstream design phases must be taken into account. Based on second-order Taylor expansions, a new method allowing the transformation of this robust-optimal design problem with acceptable risks into an equivalent deterministic quadratically constrained quadratic programming (QCQP) problem is provided and then applied to an example problem for validation. Hence, all the available QCQP optimizers can be used to find a robust-optimal solution. The proposed methodology is applied to the blended-wing/body (BWB) at the conceptual design phase, where robust drag minimization is sought under a pitching moment constraint. Numerical results show the impacts on both the BWB optimal planform parameters and guaranteed drag as a function of the acceptable risk levels taken. The study shows that two of the six planform shape variables are significantly affected by the risk level, namely, the kink parameter and the outer wing span. Variations of almost 20% are obtained on these two variables for a risk level of 0.1% as compared to a standard deterministic optimization.

Lifting convex inequalities for bipartite bilinear programs

The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality that is valid for a restriction obtained by fixing variables to their bounds, when the lifting is accomplished using affine functions of the fixed variables. In this setting, sequential lifting involves solving a non-convex nonlinear optimization problem each time a variable is lifted, just as in Mixed Integer Linear Programming. To reduce the computational burden associated with this procedure, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bipartite bilinear sets. In particular, this framework permits the derivation of closed-form valid inequalities. We then study a separable bipartite bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We argue that this bilinear cover inequality is strong by showing that it yields a constant-factor approximation of the convex hull of the original set. We study its lifting function and construct a two-slope subadditive upper bound. Using this subadditive approximation, we lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bipartite bilinear sets with box constraints.

A Relaxed and Bound Algorithm Based on Auxiliary Variables for Quadratically Constrained Quadratic Programming Problem

Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible.

New Algorithm to Solve Mixed Integer Quadratically Constrained Quadratic Programming Problems Using Piecewise Linear Approximation

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

QCQP With Extra Constant Modulus Constraints: Theory and Application to SINR Constrained Mmwave Hybrid Beamforming

The constant modulus constraint is widely used in analog beamforming, hybrid beamforming, intelligent reflecting surface design, and radar waveform design. The quadratically constrained quadratic programming (QCQP) problem is also widely used in signal processing. However, the QCQP with extra constant modulus constraints was not systematically studied in mathematic programming and signal processing. For example, the multiple quality of service (QoS) constrained analog beamforming is rare, while the QoS constrained digital beamforming methods are abundant. We propose to tackle the QCQP with extra constant modulus constraints problem by solving a series of subproblems with linear programming (LP) under extra constant modulus constraints. Under mild condition, the strong duality between the LP with extra constant modulus constraints and its dual problem is established. Then, by using the optimal solutions from the subproblems, the QCQP with extra constant modulus constraints problem is solved with a monotonically converged algorithm, and all converged solutions are Karush-Kuhn-Tucker (KKT) points. As an application of the proposed method, the signal-to-interference-plus-noise-ratio (SINR) constrained hybrid beamforming is solved. Simulation results show that the transmit power of the proposed method is similar to that of the semidefinite relaxation (SDR) method, while the computational time of proposed method is much smaller than that of the SDR method. The proposed method always provides feasible solution, while the SDR method does not always provide feasible solution.

Learning Discrete-Time Uncertain Nonlinear Systems With Probabilistic Safety and Stability Constraints

This paper presents a discrete-time dynamical system model learning method from demonstration while providing probabilistic guarantees on the safety and stability of the learned model. The controlled dynamic model of a discrete-time system with a zero-mean Gaussian process noise is approximated using an Extreme Learning Machine (ELM) whose parameters are learned subject to chance constraints derived using a discrete-time control barrier function and discrete-time control Lyapunov function in the presence of the ELM reconstruction error. To estimate the ELM parameters a quadratically constrained quadratic program (QCQP) is developed subject to the constraints that are only required to be evaluated at sampled points. Simulations validate that the system model learned using the proposed method can reproduce the demonstrations inside a prescribed safe set while converging to the desired goal location starting from various different initial conditions inside the safe set. Furthermore, it is shown that the learned model can adapt to changes in goal location during reproductions without violating the stability and safety constraints.

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.

Adaptive Beamforming With Continuous/Discrete Phase Shifters via Convex Relaxation

Adaptive antenna array employing phase shifters only can reduce the hardware cost and power consumption, but the related optimization is well understood to be difficult to solve. In this paper, we examine the use of both continuous phase shifters (CPS) and discrete phase shifters (DPS) in antenna array for adaptive interference suppression. First, we formulate the phase-only tuning problem with CPS via quadratically constrained quadratic program (QCQP), which can be approximated through semidefinite relaxation (SDR) and solved by convex-optimization solvers efficiently. We also demonstrate that quantizing such results directly may not necessarily give acceptable performance. Then, we propose to use binary quadratic program (BQP) for the optimization of the adaptive beamformer utilizing DPS with an arbitrary number of control bits to specify the phase-shift levels. The resultant BQP can be relaxed and solved efficiently too. Moreover, due to the similarity between the CPS and DPS formulation, we can mix them in a single beamforming scheme to provide additional compromises between the output signal-to-interference and noise ratio (SINR) performance and implementation cost. Simulations show that in the considered interference-limited environment, our proposed beamformers with CPS, DPS, and hybrid phase shifters (HPS) give desirable results as expected.

Efficient one asset replacement scheme for an optimized portfolio

<abstract><p>The traditional mean-variance portfolio optimization models in practice have suffered from complexity and heavy computation loads in the process of selecting the best assets for constructing a portfolio. If not, they are considerably departed from the theoretically optimized values. In this work, we develop the optimized portfolio investment strategy in which only one asset substitution occurs when re-balancing a portfolio. To do this, we briefly look into a quadratically constrained quadratic programming (QCQP), which has been well-studied for the non-negative solution. Based on the quadratic programming, an efficient scheme is presented for solving the large-scale inverse problem. We more precisely update the rank of an inverse matrix, so that the optimal solution can be easily and quickly obtained by our proposed scheme.</p> <p>Various numerical and practical experiments are presented to demonstrate the validity and reliability of our scheme. Our empirical application to the U.S. and South Korea stock markets is tested and highlighted. Moreover, comparisons of a random allocation strategy and our proposed scheme reveal the better performance in the lower risks and higher expected returns obtained by our scheme.</p></abstract>

Sampled-Data Stabilization With Control Lyapunov Functions via Quadratically Constrained Quadratic Programs

Controller design for nonlinear systems with Control Lyapunov Function (CLF) based quadratic programs has recently been successfully applied to a diverse set of difficult control tasks. These existing formulations do not address the gap between design with continuous time models and the discrete time sampled implementation of the resulting controllers, often leading to poor performance on hardware platforms. We propose an approach to close this gap by synthesizing sampled-data counterparts to these CLF-based controllers, specified as quadratically constrained quadratic programs (QCQPs). Assuming feedback linearizability and stable zero-dynamics of a system's continuous time model, we derive practical stability guarantees for the resulting sampled-data system. We demonstrate improved performance of the proposed approach over continuous time counterparts in simulation.

Robust Reconstruction Method Based on QCQP Optimization for Multichannel SAR With Closely Singular Sampling

The conventional reconstruction algorithm allows for unambiguous recovery of the Doppler spectrum of multichannel synthetic aperture radar (SAR) system in azimuth with an accurately known channel response matrix. However, the available knowledge of the actual channel response matrix is imprecise due to channel errors. In this case, the SAR image will suffer from severe error-induced azimuth ambiguities if reconstructed by the conventional reconstruction algorithm, especially for the case of closely singular sampling. To this end, a novel method based on a quadratically constrained quadratic program (QCQP) optimization is proposed to increase the robustness to the channel errors. Accordingly, the error-induced azimuth ambiguities can be greatly suppressed by the proposed method. Furthermore, performance comparisons of the conventional reconstruction algorithm and the proposed method via numerical simulation are presented.

Grid Services Optimization From Multiple Microgrids

We propose a multi-case, multi-period AC model to coordinate the procurement of grid services from an aggregation of connected distributed energy resources and microgrids. The proposed model aggregates the capabilities of all resources in the system to manage the network’s power flow and provide services to the power grid at the point of common coupling. The modeling approach enables incorporating a number of services (e.g., reserve, regulation, and reactive power capacity) with simple addition of service defining equations. We also propose a solution methodology for the resultant mixed-integer quadratically-constrained programming (MICQP) problem based on successive linear programming (SLP). The SLP is augmented with a fix-binary algorithm that promises efficient handling of discrete variables. The paper also provides a decentralized formulation that decomposes the problem into an aggregator problem and multiple small <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> Gs’ problems. We integrate the SLP method with the analytical target cascading (ATC) algorithm to optimize the overall system with minimal shared information between the aggregator and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> Gs. Case studies analyze the convergence of the centralized and the decentralized approaches, present the optimal power and service schedules of the tested system, and study the network conditions under service calls.

Real-Time Control Allocation for Autonomous Surface Vehicle Using Constrained Quadratic Programming

Control system is very important for each autonomous surface vehicle (ASV), which involves the problem of maintaining the vehicle’s position and heading using feedback controller and achieving the desired forces through thrust allocation. In this paper, we present a practical thrust allocator for under-actuated and fully-actuated vehicles, which can be represented as a quadratic programming (QP) problem with linear constraints. Such an optimization method allows us to consider common propulsion system, including tunnel thruster, azimuth thruster, and fixed propeller with rudder. These linear constraints enable us to explicitly account for the rate, amplitude and azimuth constraints of each propeller on the vessel. The proposed methods have been illustrated by simulated and experimental maneuvers for different thruster layout of a vehicle.

Scalable Optimization Methods for Incorporating Spatiotemporal Fractionation into Intensity-Modulated Radiotherapy Planning

It has been recently shown that an additional therapeutic gain may be achieved if a radiotherapy plan is altered over the treatment course using a new treatment paradigm referred to in the literature as spatiotemporal fractionation. Because of the nonconvex and large-scale nature of the corresponding treatment plan optimization problem, the extent of the potential therapeutic gain that may be achieved from spatiotemporal fractionation has been investigated using stylized cancer cases to circumvent the arising computational challenges. This research aims at developing scalable optimization methods to obtain high-quality spatiotemporally fractionated plans with optimality bounds for clinical cancer cases. In particular, the treatment-planning problem is formulated as a quadratically constrained quadratic program and is solved to local optimality using a constraint-generation approach, in which each subproblem is solved using sequential linear/quadratic programming methods. To obtain optimality bounds, cutting-plane and column-generation methods are combined to solve the Lagrangian relaxation of the formulation. The performance of the developed methods are tested on deidentified clinical liver and prostate cancer cases. Results show that the proposed method is capable of achieving local-optimal spatiotemporally fractionated plans with an optimality gap of around 10%–12% for cancer cases tested in this study. Summary of Contribution: The design of spatiotemporally fractionated radiotherapy plans for clinical cancer cases gives rise to a class of nonconvex and large-scale quadratically constrained quadratic programming (QCQP) problems, the solution of which requires the development of efficient models and solution methods. To address the computational challenges posed by the large-scale and nonconvex nature of the problem, we employ large-scale optimization techniques to develop scalable solution methods that find local-optimal solutions along with optimality bounds. We test the performance of the proposed methods on deidentified clinical cancer cases. The proposed methods in this study can, in principle, be applied to solve other QCQP formulations, which commonly arise in several application domains, including graph theory, power systems, and signal processing.

Sum-Rate Maximization for Multi-Reconfigurable Intelligent Surface-Assisted Device-to-Device Communications

This paper proposes to deploy multiple reconfigurable intelligent surfaces (RISs) in device-to-device (D2D)-underlaid cellular systems. The uplink sum-rate of the system is maximized by jointly optimizing the transmit powers of the users, the pairing of the cellular users (CUs) and D2D links, the receive beamforming of the base station (BS), and the configuration of the RISs, subject to the power limits and quality-of-service (QoS) of the users. To address the non-convexity of this problem, we develop a new block coordinate descent (BCD) framework which decouples the D2D-CU pairing, power allocation and receive beamforming, from the configuration of the RISs. Specifically, we derive closed-form expressions for the power allocation and receive beamforming under any D2D-CU pairing, which facilitates interpreting the D2D-CU pairing as a bipartite graph matching solved using the Hungarian algorithm. We transform the configuration of the RISs into a quadratically constrained quadratic program (QCQP) with multiple quadratic constraints. A low-complexity algorithm, named Riemannian manifold-based alternating direction method of multipliers (RM-ADMM), is developed to decompose the QCQP into simpler QCQPs with a single constraint each, and solve them efficiently in a decentralized manner. Simulations show that the proposed algorithm can significantly improve the sum-rate of the D2D-underlaid system with a reduced complexity, as compared to its alternative based on semidefinite relaxation (SDR).

Dynamic Economic Dispatch with Integration of Compressed Air Energy Storage Considering Large Penetration of Photovoltaic Generation Systems

The large penetration of Photovoltaic (PV) systems is reshaping the load demands curve. PV makes the load demand gaps very large, lower during the day, and higher at night. It creates a duck curve, which affects significantly the power system operation. In a duck curve, thermal generators' efficiency has decreased because their operations have not been optimal, thereby increasing fuel costs. Therefore, this research proposes Compressed Air Energy Storage (CAES) as energy storage for balancing the load demand. CAES saves the surplus power from the system and releases the saved power when the system needs it. The optimal Dynamic Economic Dispatch (DED) is carried out on CAES and thermal generators to determine optimal CAES operation and thermal generators output to meet the 24 hours load demand for saving thermal generators' fuel cost. Furthermore, the Quadratic Constrained Programming (QCP) on the DED is solved using the CPLEX solver. It has been applied to solve the 24-hour dispatch periods operation of IEEE 30 and IEEE 118-bus test system in order to verify the proposed system's effectiveness.

Denoising modulo samples: k-NN regression and tightness of SDP relaxation

Abstract Many modern applications involve the acquisition of noisy modulo samples of a function $f$, with the goal being to recover estimates of the original samples of $f$. For a Lipschitz function $f:[0,1]^d \to {{\mathbb{R}}}$, suppose we are given the samples $y_i = (f(x_i) + \eta _i)\bmod 1; \quad i=1,\dots ,n$, where $\eta _i$ denotes noise. Assuming $\eta _i$ are zero-mean i.i.d Gaussian’s, and $x_i$’s form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples $f(x_i)$ with a uniform error rate $O\big(\big(\frac{\log n}{n}\big)^{\frac{1}{d+2}}\big)$ holding with high probability. The first stage involves embedding the points on the unit complex circle, and obtaining denoised estimates of $f(x_i)\bmod 1$ via a $k$NN (nearest neighbor) estimator. The second stage involves a sequential unwrapping procedure which unwraps the denoised mod $1$ estimates from the first stage. The estimates of the samples $f(x_i)$ can be subsequently utilized to construct an estimate of the function $f$, with the aforementioned uniform error rate. Recently, Cucuringu and Tyagi proposed an alternative way of denoising modulo $1$ data, which works with their representation on the unit complex circle. They formulated a smoothness regularized least squares problem on the product manifold of unit circles, where the smoothness is measured with respect to the Laplacian of a proximity graph $G$ involving the $x_i$’s. This is a nonconvex quadratically constrained quadratic program (QCQP) hence they proposed solving its semidefinite program (SDP) based relaxation. We derive sufficient conditions under which the SDP is a tight relaxation of the QCQP. Hence under these conditions, the global solution of QCQP can be obtained in polynomial time.