Quadratic Programming Problem Research Articles

Zhang equivalency of inequation-to-inequation type for constraints of redundant manipulators

In solving specific problems, physical laws and mathematical theorems directly express the connections between variables with equations/inequations. At times, it could be extremely hard or not viable to solve these equations/inequations directly. The PE (principle of equivalence) is a commonly applied pragmatic method across multiple fields. PE transforms the initial equations/inequations into simplified equivalent equations/inequations that are more manageable to solve, allowing researchers to achieve their objectives. The problem-solving process in many fields benefits from the use of PE. Recently, the ZE (Zhang equivalency) framework has surfaced as a promising approach for addressing time-dependent optimization problems. This ZEF (ZE framework) consolidates constraints at different tiers, demonstrating its capacity for the solving of time-dependent optimization problems. To broaden the application of ZEF in time-dependent optimization problems, specifically in the domain of motion planning for redundant manipulators, the authors systematically investigate the ZEF-I2I (ZEF of the inequation-to-inequation) type. The study concentrates on transforming constraints (i.e., joint constraints and obstacles avoidance depicted in different tiers) into consolidated constraints backed by rigorous mathematical derivations. The effectiveness and applicability of the ZEF-I2I are verified through two optimization motion planning schemes, which consolidate constraints in the velocity-tier and acceleration-tier. Schemes are required to accomplish the goal of repetitive motion planning within constraints. The firstly presented optimization motion planning schemes are then reformulated as two time-dependent quadratic programming problems. Simulative experiments conducted on the basis of a six-joint redundant manipulator confirm the outstanding effectiveness of the firstly presented ZEF-I2I in achieving the goal of motion planning within constraints.

Unification and comparison of zeroing neural networks based on nonlinear complementary problem functions applied to serial and parallel robots

Quadratic programming (QP) problems are frequently encountered in science, engineering, and robotic applications. Zeroing neural network (ZNN), as a special type of recurrent neural network (RNN), is an effective solver for solving QP. However, conventional ZNNs cannot solve QP with inequality and/or bound constraints. To obtain the optimal solution of QP with such constraints, the design of ZNN models roughly resorts to three techniques: slack variable, penalty function, and nonlinear complementary problem (NCP) function. Comparative results demonstrated that the NCP function-based ZNN (NCP-ZNN) has superior performance compared to ZNN models based on the other two techniques. Note that there exist various NCP functions that can be used to design different ZNN models by following similar procedures. One research gap is that these ZNN models have not been designed, presented and compared. To fill such a gap, this paper aims to design and investigate various NCP-ZNN models employing different NCP functions. Specifically, six NCP functions are used in this paper: the perturbed Fisher-Burmeister (pFB) function, Mangasarian and Solodov (MS) function, Evtushenko and Purtov I (EPI) function, EPII function, “discrete generalization” to the FB (D-FB) function, and Ranjbar, Effati, and Miri (REM) function. The performance of six NCP-ZNN models is validated by employing them to solve two QP problems in numerical comparisons. All NCP-ZNN models are successfully applied to parallel and serial robots with joint constraints to accomplish path-tracking tasks. Both simulative and experimental results demonstrate that six NCP-ZNN models can handle the joint constraints, with the best performance achieved by the MS-ZNN model.

A Communication Anti-Jamming Scheme Assisted by RIS with Angular Response.

By optimizing the reconfigurable intelligent surface (RIS) reflection coefficients, the channel capacity of legitimate users can be increased, thereby enhancing the anti-jamming performance of communication systems. However, existing studies on RIS-assisted anti-jamming assume that there is no coupling between the RIS reflection coefficients and the incident angle of electromagnetic (EM) waves, which is quite unreasonable. Therefore, we consider the effect of the incident angle of EM waves on the reflection coefficients of the RIS and propose a communication anti-jamming scheme assisted by an RIS with angular response. Specifically, a problem is formulated to optimize the RIS reflection coefficients so that the legitimate signal is amplified, but the jamming signal is attenuated, thus enhancing the legitimate channel capacity. However, the coupling of the EM incident angle and the RIS reflection coefficients causes the problem to be non-convex. To tackle this problem, we equivalently transform the RIS reflection coefficients optimization problem into a quadratically constrained quadratic programming (QCQP) problem using the complex Taylor expansion and the multidimensional complex quadratic transform (MCQT) and solve it utilizing the alternating direction method of the multipliers (ADMM) algorithm. The simulation results reveal that, compared to other schemes supported by RIS without angular response, the proposed scheme is able to achieve a significant improvement in the anti-jamming performance.

On Pareto optimality using novel goal programming approach for fully intuitionistic fuzzy multiobjective quadratic problems

Nowadays, intuitionistic fuzzy theory is gaining rapid acceptability as it emphasizes non-membership degree also, in addition to membership degree. In this paper, a simplified novel goal programming method under intuitionistic fuzzy environment (analogous to Angelov’s approach) using both membership/non-membership functions is developed. It is then applied to obtain a Pareto optimal solution to a multiobjective quadratic programming problem (QPP) with all the parameters and variables as intuitionistic fuzzy numbers. The relations between the multiobjective intuitionistic fuzzy QPP and its equivalent crisp problems under various membership/ non-membership functions using goal programming are also established by proving theorems at appropriate places. Moreover, a numerical example and a practical problem in production planning have been projected to illustrate the steps involved in the proposed approach.

On fuzzy fractional quadratic programming problems with an application in the tourism sector

In the corporations working with logistics, many management issues involving optimization require fractional quadratic programming. Further, crisp parameters are unable to model the real life problems, where impreciseness is involved. In the present study, fuzzy fractional quadratic programming problem is studied using a hybrid method that combines analytic and numerical approaches. The original problem is converted to a crisp multiobjective problem, wherein the constraint coefficients are handled by variation of parameter technique through an interactive manner. The proposed method attempts to alleviate existing problems of straightforward conversion to a deterministic model by making constraint parameters vary as per αr∈(0,1]. The flexibility of choices in constraint parameters with αr=0 leading to best solution and αr=1 resulting in worst solution, indicates the robust nature of proposed algorithm. The theorems have been constructed and proved to show the equivalence between the original and transformed problem. A numerical example and a transportation problem in tourism sector switching between both balanced and unbalanced cases is modelled and later solved using the proposed methodology.

Robust Radar Waveform Optimization Under Target Interpulse Fluctuation and Practical Constraints via Sequential Lagrange Dual Approximation

This correspondence considers the problem of robust radar waveform design against target interpulse fluctuation with constraints on the transmit energy, similarity, and signal dynamic range. We aim to make robust the transmit waveform and receive filter with respect to the Target Amplitude Vector (TAV) uncertainties by enhancing the worst-case Signal-to-Noise-Ratio (SNR) over the modeled ellipsoidal set. First of all, the filter is directly solved leveraging on the existence of saddle point condition, which leads to a suitable simplification of the original non-convex problem. Then, to handle the resultant max-min problem, the Sequential Lagrange Dual Approximation (SLDA) procedure is devised. Specifically, we update the transmit waveform via solving a sequence of max-min approximate problems, which are proven to be hidden convex relying on the theory of Lagrange duality. Finally, the filter is synthesized after solving a Quadratical Constraint Quadratic Programming (QCQP) problem based on the optimized waveform. Numerical results highlight the robustness of the proposed design which guarantees an enhanced worst-case SNR with respect to TAV uncertainties.

Joint group and pairwise localities embedding for feature extraction

Many practical applications generate high-dimensional data, which poses challenges in terms of computational time and storage. To address this issue, feature extraction has become a popular research topic. Learning by way of graph embedding is useful for discovering potential intrinsic low-dimensional structures and has thus gained wide attention. However, traditional embedding models typically utilize only one graph or one type of multiple graphs to capture local relationships, which may be insufficient for high-dimensional data that may exhibit different kinds of local relationships. To overcome this drawback, we developed a joint embedding framework that incorporates multiple types of graphs. Under this framework, a novel joint group and pairwise locality embedding model (GPE) is proposed. The GPE model has the following distinctive merits: (1) it simultaneously incorporates simple graphs, hypergraphs, and probabilistic hypergraphs, enabling the use of not only one type of multiple graphs but also multiple types of multiple graphs; (2) it can leverage both group relationships and pairwise graphs to discover local information during feature extraction; and (3) its objective function can be solved using an alternating optimization strategy, which involves solving eigenvalue problems and quadratic programming problems, resulting in very fast convergence. Finally, we arranged classification tasks and clustering tasks on several high-dimensional real-world datasets, and the experimental results prove that the identification capability of GPE is encouraging.

Control barrier function based trajectory generation and tracking control for spacecraft inspection mission under multiple safety constraints

In this paper, the safety control problem for spacecraft inspection mission is investigated, which means that there is no position and attitude obstacle collisions between the inspection spacecraft and the defined forbidden zones. We propose a control strategy that ensures safety, utilizing the control barrier function to address multiple safety constraints with logical relationships among them. Firstly, the position and attitude constraints are defined, and the hybrid logical relationship among different attitude constraints is analyzed. To handle complex position and attitude constraints, the Kreisselmeier-Steinhauser function is introduced to approximate the max/min function, thereby expressing them as continuous and differentiable functions. Subsequently, the control barrier function-based quadratic programming (CBF-QP) problems are solved to generate safe velocity and angular velocity. These problems have explicit solutions and do not require extensive computational resources. The safe position and attitude are then obtained by integrating the safe velocity and angular velocity. Notably, the forbidden regions are expanded by a constant in the construction of CBF-QP problems, which acts as a margin for the tracking controller’s performance. Finally, attitude and position tracking controllers based on the control Lyapunov barrier function (CLBF) are designed respectively, taking into account the lumped disturbance in the system, which means that the attitude and position control are computed independently. Numerical simulation results demonstrate that the proposed control strategy effectively handles complex constraints in inspection missions, ensuring system stability and safety, even under severe conditions.

Prescribed performance formation control for second-order multi-agent systems with connectivity and collision constraints

This paper studies the distributed formation control problem of second-order multi-agent systems (MASs) with limited communication ranges and collision avoidance constraints. A novel connectivity preservation and collision-free distributed control algorithm is proposed by combining prescribed performance control (PPC) and exponential zeroing control barrier Lyapunov functions (EZCBFs). In particular, we impose the time-varying performance constraints on the relative position and velocity errors between the neighboring agents, and then a PPC-based formation control algorithm is developed such that the connectivity of the communication graph can be preserved at all times, and the prescribed transient and steady performance on the relative position and velocity error can be achieved. Subsequently, by introducing the EZCBFs method, an inequality constraint condition on the control input is derived to guarantee the collision-free formation motion. By regarding the PPC-based formation controller as a nominal input, an actual formation control input is given by solving the quadratic programming (QP) problem such that each agent achieves collision-free formation motion while guaranteeing the connectivity and prescribed performance as much as possible. Finally, numerical simulation is carried out to validate the effectiveness of the proposed algorithm.

Off‐policy model‐based end‐to‐end safe reinforcement learning

AbstractSafety and stability considerations play a crucial role in the development of learning based strategies for control design of systems that require high levels of safety. Safe reinforcement learning (RL) based approaches traditionally seek learning of the control laws that are optimal with respect to system performance whilst ensuring system stability and safety. In this article, an off‐policy safe RL based approach is proposed for nonlinear systems affine in control in continuous time. In this novel work, safety and stability are guaranteed during initialization and exploration phases by adjusting the control input with the solution of a quadratic programming problem combining both input to state stable‐control Lyapunov function and robust control barrier function (R‐CBF) conditions. Moreover, the safety of the learned policy is assured by augmenting the cost function with a CBF to maintain safety and optimize performance simultaneously. Novel mathematically rigorous proofs are provided to establish the stability and safety guarantees, offering a sound theoretical foundation for the approach. To demonstrate the effectiveness of the algorithm, two examples are presented: engine surge and stall dynamics, and an unstable nonlinear system.

Modeling of suspended weightlessness on the cables of the beam system, by changing the tension forces

The problem of weightlessness simulation of beam systems suspended on inextensible cables is considered. Imitation of weightlessness means zeroing or reducing any selected force factor (for example, the reaction of the support or the moment in the support or joint), and the kinematic factor (deflection or angle of rotation). It is required to select the forces in the cables such that the sum of the squares of the deflections at the points of the elastic line of the beam is minimal. The problem is formulated as a nonlinear programming problem; the search for the minimum of the objective function with constraints, in the form of equilibrium equations, is carried out. In general, all equations written out for a geometrically variable system are linearly dependent. Parameters are selected from the system of equations, the vectors at which are entered into the basis, and the remaining parameters are considered free and are the coordinates of the objective function. The problem was reduced to the problem of quadratic programming without restrictions. Partial derivatives of coordinates give a system of linear algebraic equations that allows you to determine the coordinates taken as free parameters, and then calculate the coordinates entered into the basis. The reference plan of nonlinear optimization problems can have local minima; it is shown that for any initial basis, the optimal plan is the only one. To calculate the deflections of the beam, the method of initial parameters is used. Deflection, angle of rotation, additional angles of rotation in articulated joints are considered as initial parameters; as well as the reaction and bending moment. The continuum problem is transformed into a discrete one by limiting the number of points at which deflections are calculated. The objective function has a finite number of variables. It is determined which number of selected points on the elastic line of the beams is sufficient to ensure the convergence of the functions of deflections, angles of rotation, bending moments and transverse forces for the purpose of application to practical calculations. Optimization of deflections of a beam pivotally fixed, suspended on two cables with verification of solutions, change of basic variables and convergence study depending on the choice of the number of points at which deflections are calculated is performed. The deformation of systems of I-beams connected by hinges to each other, having linear weight in gravity conditions, is analyzed. To simulate weightlessness, the system is supported by six cables. The boundary conditions are considered: – rigid pinching; – hinge-fixed support, – sliding sealing; – free edge. Models of three-beam systems in the simulation of weightlessness, to a certain extent equivalents. The type of boundary condition affects the first beam to a greater extent; the tension forces of the cables equalize the deformed and stressed state in subsequent beams. Any of the considered systems with the presented boundary conditions can be converted into an equivalent one by changing the boundary force factors, setting torques or installing a spring with a given stiffness and adjusting the tension of the cables.

Energy-based structural least squares twin support vector clustering

Clustering is an unsupervised learning algorithm and it is widely used in machine learning. Twin support vector clustering (TWSVC) is a new plane-based clustering algorithm, which exploits information from both within and between clusters to generate plane for each cluster. However, TWSVC suffered from relatively poor performance because it ignores the intrinsic structural information of data and solves a series of quadratic programming problems (QPPs). In order to address these problems, in this paper, we propose a novel energy-based structural least squares twin support vector clustering, termed as ESLSTWSVC. Firstly, based on least squares twin support vector clustering (LSTWSVC), we introduce within-class covariance matrix into the objective function of LSTWSVC to obtain more intrinsic structural information. ESLSTWSVC solves a series of system of linear equations rather than to solve QPPs in TWSVC, it leads to simple algorithm and less computation time. In addition, ESLSTWSVC converts the constraints of LSTWSVC into energy-based model by introducing an energy parameter for each cluster that makes ESLSTWSVC more robust. Experiments are performed on artificial datasets as well as UCI datasets, and the experimental results illustrate the effectiveness of the proposed algorithm.

Certification of the proximal gradient method under fixed-point arithmetic for box-constrained QP problems

In safety-critical applications that rely on the solution of an optimization problem, the certification of the optimization algorithm is of vital importance. Certification and suboptimality results are available for a wide range of optimization algorithms. However, a typical underlying assumption is that the operations performed by the algorithm are exact, i.e., that there is no numerical error during the mathematical operations, which is hardly a valid assumption in a real hardware implementation. This is particularly true in the case of fixed-point hardware, where computational inaccuracies are not uncommon. This article presents a certification procedure for the proximal gradient method for box-constrained QP problems implemented in fixed-point arithmetic. The procedure provides a method to select the minimal fractional precision required to obtain a certain suboptimality bound, indicating the maximum number of iterations of the optimization method required to obtain it. The procedure makes use of formal verification methods to provide arbitrarily tight bounds on the suboptimality guarantee. We apply the proposed certification procedure on the implementation of a non-trivial model predictive controller on 32-bit fixed-point hardware.

Identification and high-precision trajectory tracking control for space robotic manipulator

In this paper, an effective control and parameters identification scheme was proposed for high-precision trajectory tracking of space robotic manipulators. Our proposed control method employed a linear-extended-state-observer (LESO) based model predictive control (MPC) scheme. The feedback linearization approach was utilized to develop a model predictive controller. The optimization objective function was simplified to a standard quadratic programming (QP) problem for solution. To minimize the impact of the inaccuracy in the dynamics model, a LESO was designed to estimate the disturbance, and its convergence was demonstrated. In order to improve the accuracy of dynamic model, the physical consistency of inertial parameters was constrained using linear matrix inequality (LMI) techniques. Moreover, the dynamics parameters were estimated through semidefinite programming (SDP) techniques. To validate the effectiveness of the proposed control method, it was compared with existing controllers through simulations and experiments. The results demonstrated the superior performance of the proposed scheme, highlighting its potential for achieving high-precision trajectory tracking control in space robotic manipulators.

An efficient and provable sequential quadratic programming method for American and swing option pricing

A sequential quadratic programming numerical method is proposed for American option pricing based on the variational inequality formulation. The variational inequality is discretized using the θ-method in time and the finite element method in space. The resulting system of algebraic inequalities at each time step is solved through a sequence of box-constrained quadratic programming problems, with the latter being solved by a globally and quadratically convergent, large-scale suitable reflective Newton method. It is proved that the sequence of quadratic programming problems converges with a constant rate under a mild condition on the time step size. The method is general in solving the variational inequalities for the option pricing with many styles of optimal stopping and complex underlying asset models. In particular, swing options and stochastic volatility and jump diffusion models are studied. Numerical examples are presented to confirm the effectiveness of the method.

Optimization of a quadratic programming problem over an integer efficient set

Multi-objective programming problem often contains numerous efficient solutions, which confuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms.

Improved fuzzy sparse multi-class least squares support vector machine

The classification problem is a key area of research in machine learning. The Least Squares Support Vector Machine (LSSVM) is an important classifier that is commonly used to solve classification problems. Its widespread use stems from its replacement of the inequality constraint in the Support Vector Machine (SVM) with the equality constraint, which transforms the convex quadratic programming (QP) problem of SVM into the solution of linear equations. However, when dealing with multi-class classification problems, LSSVM faces the challenges of lack of sparsity and sample noises, which can negatively impact its performance. Based on the modeling characteristics and data distribution of the multi-class LSSVM model, this paper proposes two improvements and establishes an improved fuzzy sparse multi-class least squares support vector machine (IF-S-M-LSSVM). The first improvement adopts a non-iterative sparse algorithm, which can delete training sample points to different degrees by adjusting the sparse ratio. The second improvement addresses the impact of sample noise on determining the optimal hyperplane by adding a fuzzy membership degree based on sample density. The advantages of the new model, in terms of training speed and classification accuracy, are verified through UCI machine learning standard data set experiments. Finally, the statistical significance of the IF-S-M-LSSVM model is tested using the Friedman and Bonferroni-Dunn tests.

Local linearization based subvector inference in moment inequality models

This paper introduces a bootstrap-based profiling inference method for subvectors in moment inequality models following insights from Bugni et al. (2017). Compared to their paper, the new method calculates the critical value by searching over a local neighborhood of a pre-estimator, instead of the whole null parameter space, to profile out nuisance parameters. In this way, non-linear moment conditions are simplified by linear expansion and the bootstrap iterates over quadratic programming problems, which significantly simplifies and accelerates computation. This method controls asymptotic size uniformly over a large class of data generating processes. In the Monte Carlo simulations, the new procedure improves upon the computing time of Bugni et al. (2017) and Kaido et al. (2019) significantly. I provide an empirical illustration estimating an airline entry game.

Robust Adaptive Beamforming for Interference Suppression Based on SNR

Robust adaptive beamforming (RAB) can be used to suppress interference signals while retaining the desired signals received by a sensor array. However, desired signal self-cancellation and model mismatch can affect RAB performance. In this paper, a novel interference-plus-noise covariance matrix (INCM) reconstruction method is proposed for RAB to solve these problems. The proposed method divides the desired signal into two ranges according to the input signal-to-noise ratio (SNR), namely low SNR and high SNR. In the low SNR range, INCM reconstruction directly uses the same sample covariance matrix as the sample matrix inversion (SMI) beamformer to retain the advantages of the traditional SMI algorithm. In the high SNR range, the eigenvalues of the sample covariance matrix are used to estimate the interference power and noise power. The optimized interference steering vector (SV) is obtained by solving a quadratic convex optimization problem in an interference subspace. The INCM is reconstructed from the interference SVs, interference power, and noise power. The reconstructed INCM is then used to correct the desired signal SV via maximizing the beamformer output power. This is achieved by solving a quadratically constrained quadratic programming (QCQP) problem. Analysis and simulation results are presented which demonstrate that the proposed method performs well under a variety of mismatch conditions.

Estimating ensemble weights for bagging regressors based on the mean–variance portfolio framework

This paper presents a novel ensemble learning framework inspired by modern portfolio optimization to address regression problems. This formulation in the ensemble learning field allows determining the ensemble’s weights considering the error predictions of the base learners, the variability in their predictions, and the covariance among their predictions’ errors, which is completely aligned with the bias–variance–covariance theory. Under the framework proposed, four potential instantiations have also been provided. The first two ensemble models impose the non-negativity constraints on the ensemble’s weights (along with the equality constraint that the ensemble’s weights sum up to one) and are solved with the active set algorithm. The second two ensemble models do not include the non-negativity constraints on the ensemble’s weights (which in the financial literature are called the non-shorting constraints), giving rise to a convex quadratic programming (QP) problem (as the matrix included in the quadratic term is symmetric and positive definite) that is solved by the Lagrangian procedure. Extensive experiments with regression datasets evaluate the proposed ensemble framework. Comparisons with other state-of-the-art ensemble methods confirm that the ensemble framework yields the best overall performance.