Quadratic Programming Solution Research Articles

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.

DC algorithms in Hilbert spaces and the solution of indefinite infinite-dimensional quadratic programs

DC algorithms in Hilbert spaces and the solution of indefinite infinite-dimensional quadratic programs

Alpha-factor integrated risk parity portfolio strategy in global equity fund of funds

This study proposes a principal alpha-style factor integrated risk parity strategy that can diversify style risk factors and the stock selection risk of external managers in Fund-of-Funds (FoFs) portfolios. First, we separated the style risk factors and stock-specific sources held by each individual fund. Stock-specific sources, referred to as principal alpha portfolios, are extracted through principal component analysis, where the sources are utilized for risk parity in the alpha division. As the parity portfolio was integrated into both the alpha and style factor divisions, we used a Basin-Hopping two-phase optimization technique, which can mitigate the local optimal trap by exploring the surroundings of the sequential quadratic programming solution secondarily. Through this, a more stable integrated risk parity portfolio can be realized. Finally, the suggested integrated risk parity portfolios were simulated with a global fund dataset. The simulation results from 2006 through June 2022 show a more stable risk-return profile than an independently constructed strategy using style risk factors or principal alpha sources, especially in high volatility and down-market periods, such as a global financial crisis or unexpected events like COVID-19. This study can be applied to various areas covering other FoFs and asset allocation strategies by integrating alpha and factor divisions.

A Privacy-preserving mean–variance optimal portfolio

Following strong regulations such as the European General Data Protection Regulation (GDPR), privacy protection in the financial sector has recently emerged as an urgent issue. To manage the privacy risk in robo-advisor, a representative fintech service, we propose a novel framework that allows robo-advisors to offer the optimal portfolio while complying with the privacy of their customers by encrypting individual risk aversion with homomorphic encryption (HE). By introducing an HE-friendly method for constrained optimization, our model can find a mean–variance quadratic programming solution even with inequality constraints. This study makes two main findings through empirical evaluation (i) our model can approximate optimal solution at an acceptable level of accuracy loss and the cost of preserving privacy, and (ii) the number of assets and the degree of correlation between assets affect the accuracy loss.

Maximum Margin Multi-Dimensional Classification.

Multi-dimensional classification (MDC) assumes heterogeneous class spaces for each example, where class variables from different class spaces characterize semantics of the example along different dimensions. The heterogeneity of class spaces leads to incomparability of the modeling outputs from different class spaces, which is the major difficulty in designing MDC approaches. In this article, we make a first attempt toward adapting maximum margin techniques for MDC problem and a novel approach named M3MDC is proposed. Specifically, M3MDC maximizes the margins between each pair of class labels with respect to individual class variable while modeling relationship across class variables (as well as class labels within individual class variable) via covariance regularization. The resulting formulation admits convex objective function with nonlinear constraints, which can be solved via alternating optimization with quadratic programming (QP) or closed-form solution in either alternating step. Comparative studies on the most comprehensive real-world MDC datasets to date are conducted and it is shown that M3MDC achieves highly competitive performance against state-of-the-art MDC approaches.

A decision support system to evaluate the quality and warranty policies for remanufactured products

PurposeThis paper proposes optimisation models to evaluate and examine the selling of extended warranty policies in terms of improved profits in producing/marketing remanufactured products. These models are numerically solved using a quadratic programming solution approach and implemented in the decision support system (DSS).Design/methodology/approachThe purpose of this paper is to develop the optimisation models for a DSS and evaluate different warranty policies for buyers.FindingsThis study has demonstrated the flexibility and usefulness of a model-driven DSS for the quality and warranty management, which is applied to examine and evaluate different configurations (i.e. component reuse, rebuild and recycle) for remanufactured products and propose the selling of extended warranty policies for buyers.Research limitations/implicationsThe developed model-driven DSS can assist manufacturers to select and increase the number of components, e.g. to be reused, rebuilt, and recycled for producing a remanufactured product and propose suitable warranty policies for buyers. However, this study focusses only on the evaluation of warranty policies for specific remanufactured products in a DSS, i.e. types of air compressors for production operations in manufacturing industry.Originality/valueThis study developed optimisation models to be used in a DSS for proposing the selling of extended warranty of a remanufactured product to improve customer satisfaction and maximise the gained profits for manufacturers.

Deep Reinforcement Scheduling of Energy Storage Systems for Real-Time Voltage Regulation in Unbalanced LV Networks With High PV Penetration

The ever-growing higher penetration of distributed energy resources (DERs) in low-voltage (LV) distribution systems brings both opportunities and challenges to voltage support and regulation. This paper proposes a deep reinforcement learning (DRL)-based scheduling scheme of energy storage systems (ESSs) to mitigate system voltage deviations in unbalanced LV distribution networks. The ESS-based voltage regulation problem is formulated as a multi-stage quadratic stochastic program, with the objective of minimizing the expected total daily voltage regulation cost while satisfying operational constraints. While existing voltage regulation methods are mostly focused on one-time-step control, this paper explores a day-horizon system-wide voltage regulation problem. In other words, the size of action and state spaces are extremely high-dimensional and need to be delicately handled. Furthermore, in order to overcome the difficulty of modeling uncertainties and develop a real-time solution, a learn-to-schedule feedback control framework is proposed by adapting the problem to a model-free DRL setting. The proposed algorithm is tested on a customized 6-bus system and a modified IEEE 34-bus system. Simulation results validate the effectiveness and near-optimality of voltage regulation by ESS in comparison with a deterministic quadratic program solution.

Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

On the minimum-norm solution of convex quadratic programming

We discuss some basic concepts and present a numerical procedure for finding the minimum-norm solution of convex quadratic programs (QPs) subject to linear equality and inequality constraints. Our approach is based on a theorem of alternatives and on a convenient characterization of the solution set of convex QPs. We show that this problem can be reduced to a simple constrained minimization problem with a once-differentiable convex objective function. We use finite termination of an appropriate Newton’s method to solve this problem. Numerical results show that the proposed method is efficient.

An Adaptive Predictive Control Method Based on State-space Model

An adaptive state-space model predictive control strategy is proposed for complex industrial processes with nonlinear, time-varying and constrained characteristics. The state-space model obtained by on-line identification algorithm is used as the system model, and the indirect form is used to design the adaptive predictive controller. The controller includes quadratic programming solution to the constraint problem. The effectiveness of the proposed control strategy is verified by the simulation experiment of 2-CSTR process control.

Constrained discrete model predictive control of an arm‐manipulator using Laguerre function

SummaryThis work presents a multivariable predictive controller applied on a redundant robotic manipulator with three degrees of freedom. The article focuses on the design of a discrete model‐based predictive controller (DMPC) using the Laguerre function as a control effort weighting method to enhance the solution of Hildreth's quadratic programming and to minimize the trade‐off problem in constrained case. The Laguerre functions are used to simplify and enhance the control horizon effect through parsimonious control trajectory, thus reducing the computational load required to find the optimal control solution. Furthermore, these results can be confirmed by simulations and experimental tests on the manipulator and comparing it to the traditional DMPC approach and the discrete linear quadratic regulator.

Exponential Decay in the Sensitivity Analysis of Nonlinear Dynamic Programming

In this paper, we study the sensitivity of discrete-time dynamic programs with nonlinear dynamics and objective to perturbations in the initial conditions and reference parameters. Under uniform controllability and boundedness assumptions for the problem data, we prove that the directional derivative of the optimal state and control at time $k$, $x^*_k$, and $u^*_k$, with respect to the reference signal at time $i$, $d_i$, will have exponential decay in terms of $|k-i|$ with a decay rate $\rho$ independent of the temporal horizon length. The key technical step is to prove that a version of the convexification approach proposed by Verschueren et al. can be applied to the KKT conditions and results in a convex quadratic program with uniformly bounded data. In turn, Riccati techniques can be further employed to obtain the sensitivity result, borne from the observation that the directional derivatives are solutions of quadratic programs with structure similar to the KKT conditions themselves. We validate our findings with numerical experiments on a small nonlinear, nonconvex, dynamic program.

Ex-post Analysis of the WEC Control Competition Results Using a Fourier Spectral Control Approach

The wave energy converter control competition (WECCCOMP) allowed several real-time control approaches to be assessed, both in numerical and physical experiments. The solution retained by IFP Énergies Nouvelles (IFPeN), which won the numerical simulation and experimental evaluation phases, consists of a receding-horizon MPC algorithm, including an estimator and a predictor for the wave excitation torque. The control objective function, solved by a quadratic programming (QP) optimiser in the real-time implementation, is weighted over the receding time horizon by means of weighting coefficients, which are optimised off-line for each sea state, in order to take into account the non-ideal power take-off (PTO) efficiency. Given the potential complexity of the interaction between the different components involved in the control implementation (estimation, prediction, QP solution, choice of weightings), it is useful to carry out an ex-post analysis, in order to understand if, and how, the solution proposed by IFPEN could have been improved. To that end, a Fourier spectral control algorithm is implemented, which is able to calculate the optimal trajectory and control torque for the totality of a signal, simulated from WeCCCOMP sea states, taking the non-ideal PTO efficiency into account. By comparing MPC results with the theoretically optimal solutions provided by the spectral method, it is found that, in the studied WECCCOMP sea states, the IFPeN MPC algorithm performance lies within approximately 10% of the optimal solution, in terms of electric power.The influence of the MPC forecast accuracy and prediction horizon is examined. Finally, somechallenges associated with the offline MPC weighting optimisation are identified.

Strategic evaluation of different direct nucleation control approaches for controlling batch cooling crystallisation via simulation and experimental case studies

This paper presents a systematic evaluation of direct nucleation control (DNC) approach of controlling batch cooling crystallization systems. DNC, as named is aimed to control nucleation events to achieve the desired crystal size distribution (CSD) in a model free approach that requires no prior knowledge on nucleation or growth kinetics of the system. The performance and robustness of proposed DNC approaches were examined through computer simulations and laboratory experiments. Computer simulations were done in two phases. In the first phase, detailed mathematical models covering reaction kinetics and heat mass balances were developed for a batch cooling crystallization system of Paracetamol in water. Based on these models, rigorous simulation programs were developed in MATLAB®. This mechanistic model was used to solve an open loop optimal control problem of maximizing the mean crystal length within a fixed batch time. The successive quadratic programming solution provides the optimal temperature trajectory, corresponding number of counts and the total length. In the second phase, treating the MATLAB® model as the real processes, the performance of DNC was evaluated by setting the optimal number of counts as the target and applying different DNC strategies: changing DNC structures/algorithm; varying the acceptable range of counts limits; increasing and decreasing the limit of desired number of counts; and changing the heating and cooling rates. The results demonstrate that the fast/slow cooling-fast/slow heating approach yields the best result in terms of reduced number of temperature loops that produced the desired counts in smaller batch duration than the optimal one. Later, the simulated results were justified through laboratory experiments.

An Adaptive Partial Sensitivity Updating Scheme for Fast Nonlinear Model Predictive Control

In recent years, efficient optimization algorithms for Nonlinear Model Predictive Control (NMPC) have been proposed, that significantly reduce the on-line computational time. In particular, direct multiple shooting and Sequential Quadratic Programming (SQP) are used to efficiently solve Nonlinear Programming (NLP) problems arising from continuous-time NMPC applications. One of the computationally demanding steps for on-line optimization is the computation of sensitivities of the nonlinear dynamics at every sampling instant, especially for systems of large dimensions, strong stiffness, and when using long prediction horizons. In this paper, within the algorithmic framework of the Real-Time Iteration (RTI) scheme based on multiple shooting, an inexact sensitivity updating scheme is proposed, that performs a partial update of the Jacobian of the constraints in the NLP. Such update is triggered by using a Curvature-like Measure of Nonlinearity (CMoN), so that only sensitivities exhibiting highly nonlinear behaviour are updated, thus adapting to system operating conditions and possibly reducing the computational burden. An advanced tuning strategy for the updating scheme is provided to automatically determine the number of sensitivities being updated, with a guaranteed bounded error on the Quadratic Programming (QP) solution. Numerical and control performance of the scheme is evaluated by means of two simulation examples performed on a dedicated implementation. Local convergence analysis is also presented and a tunable convergence rate is proven, when applied to the SQP method.

Research on SVM environment performance of parallel computing based on large data set of machine learning

The support vector machine (SVM) algorithm is widely used in various fields because of its good classification effect, simplicity and practicability. However, the support vector machine calculates the support vector by quadratic programming, and the solution of quadratic programming will calculate the n-order matrix. When the amount of data is large, the calculation and storage of the n-order matrix will make the optimization speed very slow, even lead to memory overflow and interrupt operation. Using the big data computing platform Spark to improve the support vector machine algorithm can solve the above problems, but it’s not competent for multi-classification problems. Therefore, this paper starts with constructing multiple classifiers, combines the Spark framework of big data programming model and the classification characteristics of support vector machine to realize a parallel one-to-many SVM optimization algorithm based on large data sets and compares them through UCI data sets. In the experiments, the one-to-many support vector machine improved by Spark is obviously better than the one-to-many support vector machine in the single-machine environment. The simulation results show that the proposed algorithm has better performance.

Modelling, simulation and control of a class of hybrid dynamic systems: using hybrid automaton, APROS and mixed integer quadratic optimisation algorithm

This paper presents modelling, simulation and control of a class of hybrid dynamic systems. Hybrid automaton is used for modelling a transition system with continuous dynamics. The framework consists of a finite set of state and transition for modelling a discrete dynamics called control mode. Each control mode describes continuous dynamics. Using APROS tools, it permits to simulate the behaviour of the hybrid system approaching in the experiment case: actuators, pumps and valves. Mixed logic dynamics formalism allows to describe the dynamics defined by logic rules, continuous dynamics and constraints. These are described by linear dynamic equations subject to linear inequalities involving continuous, discrete and auxiliary variables. This model is used to synthesise a predictive control law under constraints. The controller requires online mixed-integer quadratic programming solution to an optimisation problem. Simulation was performed to illustrate performances and efficiency of these methods and tools.

QPLIB: a library of quadratic programming instances

This paper describes a new instance library for quadratic programming (QP), i.e., the family of continuous and (mixed)-integer optimization problems where the objective function and/or the constraints are quadratic. QP is a very diverse class of problems, comprising sub-classes ranging from trivial to undecidable. This diversity is reflected in the variety of QP solution methods, ranging from entirely combinatorial approaches to completely continuous algorithms, including many methods for which both aspects are fundamental. Selecting a set of instances of QP that is at the same time not overwhelmingly onerous but sufficiently challenging for the different, interested communities is therefore important. We propose a simple taxonomy for QP instances leading to a systematic problem selection mechanism. We then briefly survey the field of QP, giving an overview of theory, methods and solvers. Finally, we describe how the library was put together, and detail its final contents.

Solution of Quadratic Programming with Interval Variables Using a Two-Level Programming Approach

Quadratic programming with interval variables is developed from quadratic programming with interval coefficients to obtain optimum solution in interval form, both the optimum point and optimum value. In this paper, a two-level programming approach is used to solve quadratic programming with interval variables. Procedure of two-level programming is transforming the quadratic programming model with interval variables into a pair of classical quadratic programming models, namely, the best optimum and worst optimum problems. The procedure to solve the best and worst optimum problems is also constructed to obtain optimum solution in interval form.

Improved row-by-row method for binary quadratic optimization problems

The research presented here is an improved row-by-row (RBR) algorithm for the solution of boolean quadratic programming (BQP) problems. While a faster and implementable RBR method has been widely used for semidefinite programming (SDP) relaxed BQPs, it can be challenged by SDP relaxations because of the fact that it produce a tighter lower bounds than RBR on BQPs. On the other hand, solving SDP by interior point method (IPM) is computationally expensive for large scale problems. Departing from IPM, our methods provides better lower bound than the RBR algorithm by Wai et al. (IEEE international conference on acoustics, speech and signal processing ICASSP, 2011) and competitive with SDP solved by IPM. The method includes the SDP cut relaxation on the SDP and is solved by a modified RBR method. The algorithm has been tested on MATLAB platform and applied to several BQPs from BQPLIB (a library by the authors). Numerical experiments show that the proposed method outperform the previous RBR method proposed by several authors and the solution of BQP by IPM as well.