Quadratically Constrained Quadratic Programming Research Articles

Constrained minimum variance control for discrete-time stochastic linear systems

We propose a computational scheme for the solution of the so-called minimum variance control problem for discrete-time stochastic linear systems subject to an explicit constraint on the 2-norm of the input (random) sequence. In our approach, we utilize a state space framework in which the minimum variance control problem is interpreted as a finite-horizon stochastic optimal control problem with incomplete state information. We show that if the set of admissible control policies for the stochastic optimal control problem consists exclusively of sequences of causal (non-anticipative) control laws that can be expressed as linear combinations of all the past and present outputs of the system together with its past inputs, then the stochastic optimal control problem can be reduced to a deterministic, finite-dimensional optimization problem. Subsequently, we show that the latter optimization problem can be associated with an equivalent convex program and in particular, a quadratically constrained quadratic program (QCQP), by means of a bilinear transformation. Finally, we present numerical simulations that illustrate the key ideas of this work.

Successive Lagrangian relaxation algorithm for nonconvex quadratic optimization

Optimization problems whose objective function and constraints are quadratic polynomials are called quadratically constrained quadratic programs (QCQPs). QCQPs are NP-hard in general and are important in optimization theory and practice. There have been many studies on solving QCQPs approximately. Among them, a semidefinite program (SDP) relaxation is a well-known convex relaxation method. In recent years, many researchers have tried to find better relaxed solutions by adding linear constraints as valid inequalities. On the other hand, the SDP relaxation requires a long computation time, and it has high space complexity for large-scale problems in practice; therefore, the SDP relaxation may not be useful for such problems. In this paper, we propose a new convex relaxation method that is weaker but faster than SDP relaxation methods. The proposed method transforms a QCQP into a Lagrangian dual optimization problem and successively solves subproblems while updating the Lagrange multipliers. The subproblem in our method is a QCQP with only one constraint for which we propose an efficient algorithm. Numerical experiments confirm that our method can quickly find a relaxed solution with an appropriate termination condition.

Constant Envelope Precoding for MIMO Systems

Constant envelope (CE) precoding is an appealing transmission technique, which enables highly efficient power amplification, and is realizable with a single radio frequency (RF) chain at the multi-antenna transmitter. In this paper, we study the transceiver design for a point-to-point multiple-input multiple-output (MIMO) system with CE precoding. Both single-stream transmission (i.e., beamforming) and multi-stream transmission (i.e., spatial multiplexing) are considered. For single-stream transmission, we optimize the receive beamforming vector to minimize the symbol error rate (SER) for any given channel realization and desired constellation at the combiner output. By reformulating the problem as an equivalent quadratically constrained quadratic program (QCQP), we propose an efficient semi-definite relaxation (SDR) based algorithm to find an approximate solution. Next, for multi-stream transmission, we propose a new scheme based on antenna grouping at the transmitter and minimum mean squared error (MMSE) or zero-forcing (ZF) based beamforming at the receiver. The transmit antenna grouping and receive beamforming vectors are then jointly designed to minimize the maximum SER over all data streams. Finally, the error-rate performance of single- versus multi-stream transmission is compared via simulations under different setups.

Optimization of spatiotemporally fractionated radiotherapy treatments with bounds on the achievable benefit

Spatiotemporal fractionation schemes, that is, treatments delivering different dose distributions in different fractions, can potentially lower treatment side effects without compromising tumor control. This can be achieved by hypofractionating parts of the tumor while delivering approximately uniformly fractionated doses to the surrounding tissue. Plan optimization for such treatments is based on biologically effective dose (BED); however, this leads to computationally challenging nonconvex optimization problems. Optimization methods that are in current use yield only locally optimal solutions, and it has hitherto been unclear whether these plans are close to the global optimum. We present an optimization framework to compute rigorous bounds on the maximum achievable normal tissue BED reduction for spatiotemporal plans.The approach is demonstrated on liver tumors, where the primary goal is to reduce mean liver BED without compromising any other treatment objective. The BED-based treatment plan optimization problems are formulated as quadratically constrained quadratic programming (QCQP) problems. First, a conventional, uniformly fractionated reference plan is computed using convex optimization. Then, a second, nonconvex, QCQP model is solved to local optimality to compute a spatiotemporally fractionated plan that minimizes mean liver BED, subject to the constraints that the plan is no worse than the reference plan with respect to all other planning goals. Finally, we derive a convex relaxation of the second model in the form of a semidefinite programming problem, which provides a rigorous lower bound on the lowest achievable mean liver BED.The method is presented on five cases with distinct geometries. The computed spatiotemporal plans achieve 12–35% mean liver BED reduction over the optimal uniformly fractionated plans. This reduction corresponds to 79–97% of the gap between the mean liver BED of the uniform reference plans and our lower bounds on the lowest achievable mean liver BED. The results indicate that spatiotemporal treatments can achieve substantial reductions in normal tissue dose and BED, and that local optimization techniques provide high-quality plans that are close to realizing the maximum potential normal tissue dose reduction.

Unified Power Allocation for Receive Spatial Modulation Based on Approximate Optimization

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

First-Order Methods for Fast Feasibility Pursuit of Non-convex QCQPs

Quadratically constrained quadratic programming (QCQP) is NP-hard in its general non-convex form, yet it frequently arises in various engineering applications. Several polynomial-time approximation algorithms exist for non-convex QCQP problems (QCQPs), but their success hinges upon the ability to find at least one feasible point—which is also hard for a general problem instance. In this paper, we present a heuristic framework for computing feasible points of general non-convex QCQPs using simple first-order methods. Our approach features low computational and memory requirements, which makes it well suited for application to large-scale problems. While a priori it may appear that these benefits come at the expense of technical sophistication, rendering our approach too simple to even merit consideration for a non-convex and NP-hard problem, we provide compelling empirical evidence to the contrary. Experiments on synthetic as well as real-world instances of non-convex QCQPs reveal the surprising effectiveness of first-order methods compared to more established and sophisticated alternatives.

Multigroup Multicast Beamformer Design for MISO-OFDM With Antenna Selection

We study the problem of designing transmit beamformers for a multigroup multicasting by considering a multiple-input single-output orthogonal frequency-division multiplexing framework. The design objective involves either minimizing the total transmit power for certain guaranteed quality of service or maximizing the minimum achievable rate among the users for a given transmit power budget. The problem of interest can be formulated as a nonconvex quadratically constrained quadratic programming (QCQP) for which the prevailing semidefinite relaxation (SDR) technique is inefficient for at least two reasons. At first, the relaxed problem cannot be reformulated as a semidefinite programming. Second, even if the relaxed problem is solved, the so-called randomization procedure should be used to generate a feasible solution to the original QCQP, which is difficult to derive for the considered problem. To overcome these shortcomings, we adopt successive convex approximation framework to find multicast beamformers directly. The proposed method not only avoids the need of randomization search, but also incurs less computational complexity compared to an SDR approach. In addition, we also extend multicasting beamformer design problem with an additional constraint on the number of active elements, which is particularly relevant when the number of antennas is larger than that of radio frequency chains. Numerical results are used to demonstrate the superior performance of our proposed methods over the existing solutions.

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

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

A branch-and-cut algorithm using polar cuts for solving nonconvex quadratic programming problems

In this paper, we propose a branch-and-cut algorithm for solving a nonconvex quadratically constrained quadratic programming (QCQP) problem with a nonempty bounded feasible domain. The problem is first transformed into a linear conic programming problem, and then approximated by semidefinite programming problems over different intervals. In order to improve the lower bounds, polar cuts, generated from corresponding cut-generation problems, are embedded in a branch-and-cut algorithm to yield a globally --optimal solution (with respect to feasibility and optimality, respectively) in a finite number of iterations. To enhance the computational speed, an adaptive branch-and-cut rule is adopted. Numerical experiments indicate that the number of explored nodes required for solving QCQP problems can be significantly reduced by employing the proposed polar cuts.

Secure Communication for Two-Way Relay Networks with Imperfect CSI

This paper considers a two-way relay network, where two legitimate users exchange messages through several cooperative relays in the presence of an eavesdropper, and the Channel State Information (CSI) of the eavesdropper is imperfectly known. The Amplify-and-Forward (AF) relay protocol is used. We design the relay beamforming weights to minimize the total relay transmit power, while requiring the Signal-to-Noise-Ratio (SNRs) of the legitimate users to be higher than the given thresholds and the achievable rate of the eavesdropper to be upper-bounded. Due to the imperfect CSI, a robust optimization problem is summarized. A novel iterative algorithm is proposed, where the line search technique is applied, and the feasibility is preserved during iterations. In each iteration, two Quadratically-Constrained Quadratic Programming (QCQP) subproblems and a one-dimensional subproblem are optimally solved. The optimality property of the robust optimization problem is analyzed. Simulation results show that the proposed algorithm performs very close to the non-robust model with perfect CSI, in terms of the obtained relay transmit power; it~achieves higher secrecy rate compared to the existing work. Numerically, the proposed algorithm converges very quickly, and more than 85% of the problems are solved optimally.

Portfolio Choice Based on Third-Degree Stochastic Dominance

We develop an optimization method for constructing investment portfolios that dominate a given benchmark portfolio in terms of third-degree stochastic dominance. Our approach relies on the properties of the semivariance function, a refinement of an existing “superconvex” dominance condition, and quadratic constrained programming. We apply our method to historical stock market data using an industry momentum strategy. Our enhanced portfolio generates important performance improvements compared with alternatives based on mean-variance dominance and second-degree stochastic dominance. Relative to the Center for Research in Security Prices all-share index, our portfolio increases average out-of-sample return by almost seven percentage points per annum without incurring more downside risk, using quarterly rebalancing and without short selling. This paper was accepted by Lauren Cohen, finance.

Global solution of non-convex quadratically constrained quadratic programs

The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let be a QCQP. Our approach to solve is first to build an equivalent mixed-integer quadratic problem . This equivalent problem has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints , where x are the initial variables of problem . We then propose to solve by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON andGloMIQO.

Fast Robust Model Predictive Control for Nonlinear Jump Markov Systems

This paper proposes a scheme of robust model predictive control (MPC) for discrete-time nonlinear jump Markov systems (JMS) considering polytopic state and input constraints. The approach consists of an offline part and the online optimization: First, a time-varying linear JMS representation of the nonlinear JMS is used to determine time-varying feedback controllers and robust control invariant sets (RCIS) offline, solving a semi-definite program (SDP). Then, an MPC formulation uses online linearization, the determined RCIS, and Lyapunov functions to guarantee constraint satisfaction, mean square stability, and recursively feasibility. The proposed formulation is robust against bounded disturbances and the linearization errors. A simulation study demonstrates that the involved quadratically constrained quadratic programs (QCQP) can be solved quite efficiently, rendering this approach applicable for high-dimensional systems.

Fast Approximation Algorithms for a Class of Non-convex QCQP Problems Using First-Order Methods

A number of important problems in engineering can be formulated as non-convex quadratically constrained quadratic programming (QCQP). The general QCQP problem is NP-Hard. In this paper, we consider a class of non-convex QCQP problems that are expressible as the maximization of the point-wise minimum of homogeneous convex quadratics over a “simple” convex set. Existing approximation strategies for such problems are generally incapable of achieving favorable performance-complexity tradeoffs. They are either characterized by good performance but high complexity and lack of scalability, or low complexity but relatively inferior performance. This paper focuses on bridging this gap by developing high performance, low complexity successive non-smooth convex approximation algorithms for problems in this class. Exploiting the structure inherent in each subproblem, specialized first-order methods are used to efficiently compute solutions. Multicast beamforming is considered as an application example to showcase the effectiveness of the proposed algorithms, which achieve a very favorable performance-complexity tradeoff relative to the existing state of the art.

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

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

ADMM-Based Fast Algorithm for Multi-Group Multicast Beamforming in Large-Scale Wireless Systems

Multi-group multicast beamforming in wireless systems with large antenna arrays and massive audience is investigated in this paper. Multicast beamforming design is a well-known non-convex quadratically constrained quadratic programming (QCQP) problem. A conventional method to tackle this problem is to approximate it as a semi-definite programming problem via semi-definite relaxation, whose performance, however, deteriorates considerably as the number of per-group users goes large. A recent attempt is to apply convex-concave procedure (CCP) to find a stationary solution by treating it as a difference of convex programming problem, whose complexity, however, increases dramatically as the problem size increases. In this paper, we propose a low-complexity high-performance algorithm for multi-group multicast beamforming design in large-scale wireless systems by leveraging the alternating direction method of multipliers (ADMM) together with CCP. In specific, the original non-convex QCQP problem is first approximated as a sequence of convex subproblems via CCP. Each convex subproblem is then reformulated as a novel ADMM form. Our ADMM reformulation enables that each updating step is performed by solving multiple small-size subproblems with closed-form solutions in parallel. Numerical results show that our fast algorithm maintains the same favorable performance as state-of-the-art algorithms but reduces the complexity by orders of magnitude.

Power System State Estimation via Feasible Point Pursuit: Algorithms and Cramér-Rao Bound

Accurately monitoring the system's operating point is central to the reliable and economic operation of an electric power grid. Power system state estimation (PSSE) aims to obtain complete voltage magnitude and angle information at each bus given a number of system variables at selected buses and lines. Power flow analysis is a special case of PSSE, and amounts to solving a set of noise-free power flow equations. Physical laws dictate quadratic relationships between available quantities and unknown voltages, rendering general instances of power flow and PSSE nonconvex and NP-hard. Past approaches are largely based on gradient-type iterative procedures or semidefinite relaxation (SDR). Due to nonconvexity, the solution obtained via gradient-type schemes depends on initialization, while SDR methods do not perform as desired in challenging scenarios. This paper puts forth novel \emph{feasible point pursuit} (FPP)-based solvers for power flow and PSSE, which iteratively seek feasible solutions for a nonconvex quadratically constrained quadratic programming (QCQP) reformulation of the weighted least-squares (WLS) problem. Relative to the prior art, the developed solvers offer superior performance at the cost of higher complexity. Furthermore, they converge to a stationary point of the WLS problem. As a baseline for comparing different estimators, the Cram{\' e}r-Rao lower bound (CRLB) is derived for the fundamental PSSE problem in this paper. Judicious numerical tests on several IEEE benchmark systems showcase markedly improved performance of our FPP-based solvers for both power flow and PSSE tasks over popular WLS-based Gauss-Newton iterations and SDR approaches.

Semidefinite Relaxation and Approximation Analysis of a Beamformed Alamouti Scheme for Relay Beamforming Networks

In this paper, we study the amplify-and-forward (AF) schemes in two-hop one-way relay networks. In particular, we consider the multigroup multicast transmission between long-distance users. Given that perfect channel state information is perceived, our goal is to design the AF process so that the max-min-fair (MMF) signal-to-interference-plus-noise ratio (SINR) is optimized subject to generalized power constraints. We propose a rank-two beamformed Alamouti (BFA) AF scheme and formulate the corresponding AF design problem as a \emph{two-variable} fractional quadratically-constrained quadratic program (QCQP), which is further tackled by the semidefinite relaxation (SDR) technique. We analyze the approximation quality of two-variable fractional SDRs under the Gaussian randomization algorithm. These results are fundamentally new and reveal that the proposed BFA AF scheme can outperform the traditional BF AF scheme, especially when there are many users in the system or many generalized power constraints in the problem formulation. From a practical perspective, the BFA AF scheme offers two degrees of freedom (DoFs) in beamformer design, as opposed to the one DoF offered by the BF AF scheme, to improve the receivers' SINR. In the latter part of this paper, we demonstrate how this extra DoF leads to provable performance gains by considering two special cases of multicasting, where the AF process is shown to employ a special structure. The numerical simulations further validate that the proposed BFA AF scheme outperforms the BF AF scheme and works well for large-scale relay systems.

A Multiobjective Minimax Regret Robust VAr Planning Model

This paper proposes a risk-based mixed integer quadratically-constrained programming model for the long-term VAr planning problem. Risk aversion is included in the proposed model by means of regret-based optimization to quantify the load shedding risk because of a reactive power deficit. The expected operation and expansion costs of new installed reactive power sources and load shedding risk are jointly minimized. Uncertainty in the active and reactive load demands has been included in the model. An $\epsilon$ -constraint approach is used to characterize the optimal efficient frontier. Also, discrete tap settings of tap-changing transformers are modeled as a set of mixed integer linear equations which are embedded into an ac optimal convex power flow. Computational results are obtained from a realistic South and South–East Brazilian power system to illustrate the proposed methodology. Finally, conclusions are duly drawn.

Magnetic MIMO Signal Processing and Optimization for Wireless Power Transfer

In magnetic resonant coupling (MRC) enabled multiple-input multiple-output (MIMO) wireless power transfer (WPT) systems, multiple transmitters (TXs) each with one single coil are used to enhance the efficiency of simultaneous power transfer to multiple single-coil receivers (RXs) by constructively combining their induced magnetic fields at the RXs, a technique termed "magnetic beamforming". In this paper, we study the optimal magnetic beamforming design in a multi-user MIMO MRC-WPT system. We introduce the multi-user power region that constitutes all the achievable power tuples for all RXs, subject to the given total power constraint over all TXs as well as their individual peak voltage and current constraints. We characterize each boundary point of the power region by maximizing the sum-power deliverable to all RXs subject to their minimum harvested power constraints. For the special case without the TX peak voltage and current constraints, we derive the optimal TX current allocation for the single-RX setup in closed-form as well as that for the multi-RX setup. In general, the problem is a non-convex quadratically constrained quadratic programming (QCQP), which is difficult to solve. For the case of one single RX, we show that the semidefinite relaxation (SDR) of the problem is tight. For the general case with multiple RXs, based on SDR we obtain two approximate solutions by applying time-sharing and randomization, respectively. Moreover, for practical implementation of magnetic beamforming, we propose a novel signal processing method to estimate the magnetic MIMO channel due to the mutual inductances between TXs and RXs. Numerical results show that our proposed magnetic channel estimation and adaptive beamforming schemes are practically effective, and can significantly improve the power transfer efficiency and multi-user performance trade-off in MIMO MRC-WPT systems.