Quadratic Inequality Constraints Research Articles

A Binary Characterization Method for Shape Convexity and Applications

Convexity prior is one of the main cues for human vision and shape completion with important applications in image processing and computer vision. This paper provides a computable characterization method for shape convexity, and illustrates its applications in image segmentation and convex hull computation. We prove that the convexity of a region is equivalent to a series of quadratic inequality constraints on its indicator function. By incorporating this result, models are proposed for image segmentation with convexity prior and convex hull computation of clean and noisy data sets, respectively. Then, these models are summarized into a unified optimization problem on binary function(s) with the quadratic inequality constraints. Numerical method is proposed by linearizing the quadratic constraints. The linearization problem is solved by a proximal alternating direction method of multipliers, the convergence of which is guaranteed under some proper conditions. Numerical experiments demonstrate the efficiency and effectiveness of the proposed methods for image segmentation and convex hull computation.

LPI Radar Waveform Design With Desired Cyclic Spectrum and Pulse Compression Properties

This paper considers the practical LPI radar waveform design problem for jointly achieving a desired cyclic spectrum and pulse compression properties. In this work, the weighted cyclic frequency sidelobe (WCFS) level is minimized to achieve the LPI purpose while the similarity constraint regarding a known radar waveform that occupies the desired detection properties is considered. The waveform design problem is then formulated as a quartic optimization problem with heterogeneous quadratic inequality constraints. These properties make the problem NP-hard in general. To solve this problem efficiently, by introducing necessary auxiliary variables, the original problem is converted into several quadratic optimization problems with respect to each independent variable under a separated linear inequality constraint, which can be efficiently solved by leveraging the alternation direction method of multipliers (ADMM) framework. Finally, the performance of the proposed method is demonstrated via numerical examples.

Sequential quadratic programming-based non-cooperative target distributed hybrid processing optimization method

The distributed hybrid processing optimization problem of non-cooperative targets is an important research direction for future networked air-defense and anti-missile firepower systems. In this paper, the air-defense anti-missile targets defense problem is abstracted as a nonconvex constrained combinatorial optimization problem with the optimization objective of maximizing the degree of contribution of the processing scheme to non-cooperative targets, and the constraints mainly consider geographical conditions and anti-missile equipment resources. The grid discretization concept is used to partition the defense area into network nodes, and the overall defense strategy scheme is described as a nonlinear programming problem to solve the minimum defense cost within the maximum defense capability of the defense system network. In the solution of the minimum defense cost problem, the processing scheme, equipment coverage capability, constraints and node cost requirements are characterized, then a nonlinear mathematical model of the non-cooperative target distributed hybrid processing optimization problem is established, and a local optimal solution based on the sequential quadratic programming algorithm is constructed, and the optimal firepower processing scheme is given by using the sequential quadratic programming method containing non-convex quadratic equations and inequality constraints. Finally, the effectiveness of the proposed method is verified by simulation examples.

A Second Order Cone Programming Approach for Simulating Biphasic Materials

AbstractStrain limiting is a widely used approach for simulating biphasic materials such as woven textiles and biological tissue that exhibit a soft elastic regime followed by a hard deformation limit. However, existing methods are either based on slowly converging local iterations, or offer no guarantees on convergence. In this work, we propose a new approach to strain limiting based on second order cone programming (SOCP). Our work is based on the key insight that upper bounds on per‐triangle deformations lead to convex quadratic inequality constraints. Though nonlinear, these constraints can be reformulated as inclusion conditions on convex sets, leading to a second order cone programming problem—a convex optimization problem that a) is guaranteed to have a unique solution and b) allows us to leverage efficient conic programming solvers. We first cast strain limiting with anisotropic bounds on stretching as a quadratically constrained quadratic program (QCQP), then show how this QCQP can be mapped to a second order cone programming problem. We further propose a constraint reflection scheme and empirically show that it exhibits superior energy‐preservation properties compared to conventional end‐of‐step projection methods. Finally, we demonstrate our prototype implementation on a set of examples and illustrate how different deformation limits can be used to model a wide range of material behaviors.

QPPAL: A Two-phase Proximal Augmented Lagrangian Method for High-dimensional Convex Quadratic Programming Problems

In this article, we aim to solve high-dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality, and inequality constraints. To solve the targeted QP problem to a desired accuracy efficiently, we consider the restricted-Wolfe dual problem and develop a two-phase Proximal Augmented Lagrangian method (QPPAL), with Phase I to generate a reasonably good initial point to warm start Phase II to obtain an accurate solution efficiently. More specifically, in Phase I, based on the recently developed symmetric Gauss-Seidel (sGS) decomposition technique, we design a novel sGS-based semi-proximal augmented Lagrangian method for the purpose of finding a solution of low to medium accuracy. Then, in Phase II, a proximal augmented Lagrangian algorithm is proposed to obtain a more accurate solution efficiently. Extensive numerical results evaluating the performance of QPPAL against existing state-of-the-art solvers Gurobi, OSQP, and QPALM are presented to demonstrate the high efficiency and robustness of our proposed algorithm for solving various classes of large-scale convex QP problems. The MATLAB implementation of the software package QPPAL is available at https://blog.nus.edu.sg/mattohkc/softwares/qppal/ .

Global optimality condition for quadratic optimization problems under data uncertainty

In this paper, we establish a robust version of the S-lemma that we use to characterize robust solutions for classes of homogeneous and non-homogeneous quadratic problems with a quadratic inequality constraint under interval uncertainty and a linear equality constraint. Necessary and sufficient conditions of global optimality of robust solution of these problems are given.

Convex Polar Second-Order Taylor Approximation of AC Power Flows: A Unit Commitment Study

Modern mixed-integer quadratic solvers generally handle binary variables more efficiently than nonlinear mixed-integer solvers. This is relevant to the power system operation models as the unit commitment formulations typically contain a large number of binary variables. This paper investigates how to achieve the accuracy level close to the one of the exact nonlinear models, but by utilising convex models and solvers. The presented unit commitment model is based on a Taylor-series expansion where both the voltage magnitude and angle are quadratically constrained. To achieve high accuracy, the model takes advantage of the meshed transmission network structure that enables replacement of the quadratic inequality constraints that cause constraint relaxation errors with the linear equality constraints. Quadratic constraints to be replaced as well as the operating point parameters are determined based on the presolve. The first presented case study validates the model's accuracy and the convergence of the iterative algorithm, while the second is a non-iterative full unit commitment problem. Unit commitment results show superior accuracy and similar computation times to the existing quadratic formulations on one hand and faster computation times than the exact nonlinear polar formulation on the other.

Robust Constrained MFMVDR Filters for Single-Channel Speech Enhancement Based on Spherical Uncertainty Set

Aiming at exploiting speech correlation across consecutive time-frames in the short-time Fourier transform domain, the multi-frame minimum variance distortionless response (MFMVDR) filter for single-channel speech enhancement has been proposed. The MFMVDR filter requires an accurate estimate of the normalized speech correlation vector in order to avoid speech distortion and artifacts. In this paper we investigate the potential of using robust MVDR filtering techniques to estimate the normalized speech correlation vector as the vector maximizing the total signal output power within a spherical uncertainty set, which corresponds to imposing a quadratic inequality constraint. Whereas the singly-constrained (SC) MFMVDR filter only considers the quadratic inequality constraint to estimate the (non-normalized) speech correlation vector, the doubly-constrained (DC) MFMVDR filter integrates a linear normalization constraint into the optimization problem to directly estimate the normalized speech correlation vector. To set the upper bound of the quadratic inequality constraint for each time-frequency point, we propose to use a trained non-linear mapping function that depends on the a-priori signal-to-noise ratio (SNR). Experimental results for different speech signals, noise types and SNRs show that the proposed constrained approaches yield a more accurate estimate of the normalized speech correlation vector than a state-of-the-art maximum-likelihood (ML) estimator. An instrumental and a perceptual evaluation show that both constrained MFMVDR filters lead to less speech and noise distortion but a lower noise reduction than the ML-MFMVDR filter, where the DC-MFMVDR filter is preferred in terms of overall quality compared to the SC-MFMVDR and ML-MFMVDR filters.

Robustness D3LS beamforming algorithm against the uncertainties in a uniform linear array radar of dipole antenna

SummaryIn practical terms, direct data domain least squares (D3LS) beamforming algorithm performance is severely decreased by the array steering vector uncertainty. In this paper, we investigate the performance of D3LS approach for beamforming. Uniform linear antenna array radar of dipole antenna is considered in the presence of mutual coupling between the elements of the array. The performance of D3LS beamforming algorithm is severely affected by uncertainty caused by mutual coupling. In order to test the robustness of the D3LS approach against uncertainty, a quadratic inequality constraint is considered to optimize weight vector of D3LS beamforming algorithm. The convex optimization method is used to calculate optimized weight vector. The proposed convex optimization problem is solved by barrier method. It is a subset of Interior‐point numerical analysis methods. Several different scenarios are used to evaluate the proposed algorithm in MATLAB.

Robust Downlink Transmit Optimization Under Quantized Channel Feedback via the Strong Duality for QCQP

Consider a robust multiple-input single-output downlink beamforming optimization problem in a frequency division duplexing system. The base station (BS) sends training signals to the users, and every user estimates the channel coefficients, quantizes the gain and the direction of the estimated channel and sends them back to the BS. Suppose that the channel state information at the transmitter is imperfectly known mainly due to the channel direction quantization errors, channel estimation errors and outdated channel effects. The actual channel is modeled as in an uncertainty set composed of two inequality homogeneous and one equality inhomogeneous quadratic constraints, in order to account for the aforementioned errors and effects. Then the transmit power minimization problem is formulated subject to robust signal-to-noise-plus-interference ratio constraints. Each robust constraint is transformed equivalently into a quadratic matrix inequality (QMI) constraint with respect to the beamforming vectors. The transformation is accomplished by an equivalent phase rotation process and the strong duality result for a quadratically constrained quadratic program. The minimization problem is accordingly turned into a QMI problem, and the problem is solved by a restricted linear matrix inequality relaxation with additional valid convex constraints. Simulation results are presented to demonstrate the performance of the proposed method, and show the efficiency of the restricted relaxation.

The exact solution of multiparametric quadratically constrained quadratic programming problems

In this paper, we present a strategy for the exact solution of multiparametric quadratically constrained quadratic programs (mpQCQPs). Specifically, we focus on multiparametric optimization problems with a convex quadratic objective function, quadratic inequality and linear equality constraints, described by constant matrices. The proposed approach is founded on the expansion of the Basic Sensitivity Theorem to a second-order Taylor approximation, which enables the derivation of the exact parametric solution of mpQCQPs. We utilize an active set strategy to implicitly explore the parameter space, based on which (i) the complete map of parametric solutions for convex mpQCQPs is constructed, and (ii) the determination of the optimal parametric solution for every feasible parameter realization reduces to a nonlinear function evaluation. Based on the presented results, we utilize the second-order approximation to the Basic Sensitivity Theorem to expand to the case of nonconvex quadratic constraints, by employing the Fritz John necessary conditions. Example problems are provided to illustrate the algorithmic steps of the proposed approach.

Tilt Stability for Quadratic Programs with One or Two Quadratic Inequality Constraints

This paper examines tilt stability for quadratic programs with one or two quadratic inequality constraints. Exploiting specific features of these problems and using some known results on tilt stability in nonlinear programming, we establish quite simple characterizations of tilt-stable local minimizers for quadratic programs with one quadratic inequality constraint under metric subregularity constraint qualification. By the same way, we also derive various tilt stability conditions for quadratic programs with two quadratic inequality constraints and satisfying certain suitable assumptions. Especially, the obtained results show that some tilt stability conditions only known to be sufficient in nonlinear programming become the necessary ones when the considered problems are quadratic programs with one or two quadratic inequality constraints.

Natural Gas Flow Solvers Using Convex Relaxation

The vast infrastructure development, gas flow (GF) dynamics, and complex interdependence of gas with electric power networks call for advanced computational tools. Solving the equations relating gas injections to pressures and pipeline flows lies at the heart of natural gas network (NGN) operation, yet existing solvers that require careful initialization and uniqueness has been an open question. In this context, this article considers the nonlinear steady-state version of the GF problem. It first establishes that the solution to the GF problem is unique under arbitrary NGN topologies, compressor types, and sets of specifications. For GF setups where pressure is specified on a single (reference) node and compressors do not appear in cycles, the GF task is posed as n convex minimization. To handle more general setups, a GF solver relying on a mixed-integer quadratically constrained quadratic program (MI-QCQP) is also devised. This solver can be used for any GF setup at any NGN. It introduces binary variables to capture flow directions, relaxes the pressure drop equations to quadratic inequality constraints, and uses a carefully selected objective to promote the exactness of this relaxation. The relaxation is probably exact in NGNs with nonoverlapping cycles and a single fixed-pressure node. The solver handles efficiently the involved bilinear terms through McCormick linearization. Numerical tests validate our claims, demonstrate that the MI-QCQP solver scales well, and that the relaxation is exact even when the sufficient conditions are violated, such as in NGNs with overlapping cycles and multiple fixed-pressure nodes.

A Lifting-Penalty Method for Quadratic Programming with a Quadratic Matrix Inequality Constraint

In this paper, a lifting-penalty method for solving the quadratic programming with a quadratic matrix inequality constraint is proposed. Additional variables are introduced to represent the quadratic terms. The quadratic programming is reformulated as a minimization problem having a linear objective function, linear conic constraints and a quadratic equality constraint. A majorization–minimization method is used to solve instead a l 1 penalty reformulation of the minimization problem. The subproblems arising in the method can be solved by using the current semidefinite programming software packages. Global convergence of the method is proven under some suitable assumptions. Some examples and numerical results are given to show that the proposed method is feasible and efficient.

Optimal design of window functions for filter window bank

This paper considers the designs of the periodic window functions in the filter window banks. First, the filter window bank with the constant synthesis periodic window functions is considered. The total number of the nonzero coefficients in the impulse responses of the analysis periodic window functions is minimized subject to the near perfect reconstruction condition. This is an $ L_0 $ norm optimization problem. To find its solution, the $ L_0 $ norm optimization problem is approximated by the $ L_1 $ norm optimization problem. Then, the column of the constraint matrix corresponding to the element in the solution with the smallest magnitude is removed. Next, it is tested whether the feasible set corresponding to the new $ L_0 $ norm optimization problem is empty or not. By repeating the above procedures, a solution of the $ L_0 $ norm optimization problem is obtained. Second, the filter window bank with the time varying synthesis periodic window functions is considered. Likewise, the design of the periodic window functions in both the analysis periodic window functions and the synthesis periodic window functions is formulated as an $ L_0 $ optimization problem. However, this $ L_0 $ norm optimization problem is subject to a quadratic matrix inequality constraint. To find its solution, the set of the synthesis periodic window functions is initialized. Then, the set of the analysis periodic window functions is optimized based on the initialized set of the synthesis periodic window functions. Next, the set of the synthesis periodic window functions is optimized based on the found set of the analysis periodic window functions. Finally, these two procedures are iterated. It is shown that the iterative algorithm converges. A design example of a filter window bank with the constant synthesis periodic window functions and a design example of a filter window bank with the time varying synthesis periodic window functions are illustrated. It is shown that the near perfect reconstruction condition is satisfied, whereas this is not the cases for the nonuniform filter banks with the conventional samplers and the conventional block samplers.

Set-Membership Filtering for Nonlinear Dynamic Systems With Quadratic Inequality Constraints

The present study investigates the problem of set-membership filtering for nonlinear dynamic systems with general nonconvex inhomogeneous quadratic inequality constraints. The investigators propose an ellipsoidal state bounding estimation in the setting of unknown but bounded noise. In order to guarantee the on-line usage, the nonlinear function is linearized by Taylor expansion at each time step, where the bounding ellipsoid of the remainder is updated on-line based on the current state bounding ellipsoid. Furthermore, based on the remainder bounds and the constraints, both the state prediction and measurement update of the filtering can be transformed into a semidefinite programming problem that can be efficiently solved. In order to further reduce the computational complexity, a part-analytical formula of the shape matrix and the center of the bounding ellipsoid is derived using a decoupled technique, which is also helpful to clarify how these constraints affect the state estimation. Finally, typical numerical examples demonstrate the effectiveness of this filtering.

Performance analysis for time-delay systems: application to the control of an active mass damper

This paper proposes a method to analyze, beyond stability, the performances of linear time-delay systems. Using robust analysis techniques, a sufficient condition that analyzes the location of eigenvalues in the complex plane is presented. More precisely, a set of quadratic inequality constraints are designed to define an admissible region for the infinitely many eigenvalues of a time-delay system and the quadratic separation theorem is applied to assess that the eigenvalues are effectively belonging to that stability region. This method is then used for the control of an active mass damper. A standard state feedback control is replaced with a static output feedback plus a static delayed output feedback. This strategy avoids the full measurement of the state and shows that delays in the dynamic may be helpful for stabilization. The closed-loop system is then expressed as a time-delay system and the performance criterion is exploited to analyze the stability and the damping properties. Simulations and experimental tests support the approach.

A quadratic matrix inequality based PID controller design for LPV systems

This paper develops a robust gain-scheduled proportional–integral–derivative (PID) controller design method for a linear-parameter-varying (LPV) system with parametric uncertainty. It is recognized in the literature that the robust fixed-order controller design can be formulated as a feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is an NP hard problem in general. Previous researchers have applied a linearization method, such as a variable change technique or a congruence transformation, to transform the BMI into a LMI. The applicability of the linearization method depends on the specific structure of the problem at hand and cannot be generalized. This paper instead formulates the gain-scheduled PID controller design as a feasibility problem of a quadratic matrix inequality (QMI) constraint, which covers the BMI constraint as a special case. An augmented sequential LMI optimization method is proposed to search for a feasible solution of the QMI constraint iteratively. As an illustrative application, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real-world output feedback control design system.

Minimax and WLS Designs of Digital FIR Filters Using SOCP for Aliasing Errors Reduction in BI-DAC

This paper presented the optimal minimax and weighted least squares (WLS) methods for designing digital finite impulse response (FIR) filters to reduce the aliasing errors generated by the non-ideality of analog filters and mixers in bandwidth interleaving digital-to-analog converter (BI-DAC). To satisfy the given expected spurious free dynamic range (SFDR), we formulated these optimal designs of digital FIR filters in BI-DAC as a convex optimization problem-second-order cone programming (SOCP) that allowed the linear equality and convex quadratic inequality constraints including the magnitude flatness and the peak aliasing errors constraints to be merged. Furthermore, we derived the computational complexity of our presented optimal design. Several design examples were given to evaluate the performance of our presented unconstrained and constrained minimax and WLS designs using SOCP including their effectiveness and computational complexity. The simulation results showed that, in our presented unconstrained minimax and WLS designs using SOCP, the maximum distortion errors were all around 0.02 dB. The maximum aliasing errors were-73.9 and -80.5 dB, which satisfied the expected SFDR of a 12-bit BI-DAC system. In addition, we analyzed the influence of different values of the nonnegative weighting function on our presented unconstrained minimax and WLS designs using SOCP, and we found that there was a tradeoff among the nonnegative weighting function's value, and the distortion and aliasing errors. Moreover, when the constraints were imposed in our presented constrained minimax and WLS designs using SOCP in the selected frequency bands, the distortion errors were equal to zero and the aliasing errors were reduced below -110 dB, but the expense was that the larger distortion and aliasing errors achieved out of these selected frequency bands. Finally, we gave the computational complexity comparisons among our presented unconstrained and constrained minimax and WLS design using SOCP, we also compared the influence of the digital FIR filters' length on our presented designs' worst-case passband ripple and stopband roll-off, and we found that there was a tradeoff among the digital FIR filters' length, the passband ripple, the stopband roll-off, the computational complexity, and the actual hardware cost.

Secure Transmission for Heterogeneous Cellular Networks With Wireless Information and Power Transfer

In this paper, we investigate an artificial-noise-aided secure beamforming design for simultaneous wireless information and power transfer in a two-tier downlink heterogeneous cellular network, in which each energy receiver in a femtocell is seen as a potential eavesdropper to wiretap the confidential message intended for the information receiver. Our design objective is to maximize the secrecy rate at the information receiver, while satisfying the signal-to-interference-plus-noise ratio requirement of each macrouser and the energy harvesting and transmit power constraints. Both the scenarios of perfect and imperfect channel state information (CSI) are considered. With perfect CSI, the formulated optimization problem constitutes a difference of a convex function programming problem, which is hard to directly solve. To tackle this challenge, we transform it into a series of semidefinite programs by using successive convex approximation, and an iterative algorithm is proposed to arrive at a provably convergent solution. With imperfect CSI, we address robust secure beamforming relying on the worst-case design philosophy. To circumvent this predicament, we resort to the S-procedure to reformulate the robust quadratic matrix inequality (QMI) constraints and then obtain the linear matrix inequality representations for these QMIs. Numerical results are finally presented to demonstrate the performance of our proposed schemes in improving the secrecy rate of heterogeneous cellular networks.