Convex Quadratic Programming Research Articles

A Unifying Complexity Certification Framework for Active-Set Methods for Convex Quadratic Programming

In model-predictive control (MPC), an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved. We propose an algorithm for computing which sequence of subproblems an active-set algorithm will solve, for every parameter of interest. These sequences can be used to set worst-case bounds on how many iterations, floating-point operations, and, ultimately, the maximum solution time the active-set algorithm requires to converge. The usefulness of the proposed method is illustrated on a set of QPs originating from MPC problems, by computing the exact worst-case number of iterations primal and dual active-set algorithms require to reach optimality.

Stability analysis of proportional delayed projection neural network for quadratic programming problem

At present, projection neural network (PNN) with bounded time delay has been widely used for solving convex quadratic programming problem (QPP). However, there is little research concerning PNN with unbounded time delay. In this paper, we propose the proportional delayed PNN to solve QPP with equality constraints. By utilizing homeomorphism mapping principle, we prove the proportional delayed PNN exists with unique equilibrium point which is the optimal solution of QPP. Simultaneously, delay-dependent criteria about global exponential stability (GES) and global polynomial stability (GPS) are also acquired by applying the method of variation of constants and inequality techniques. On the other hand, when proportional delay factor [Formula: see text] is equal to 1, the proportional delayed PNN becomes the one without time delay which still can be utilized for solving QPP. But in most situations, [Formula: see text] is not equal to 1, and time delay is unpredictable and may be unbounded in the actual neural network, which causes instability of system. Therefore, it is necessary to consider proportional delayed PNN. A numerical example demonstrates that, compared with the proportional delayed Lagrange neural network, the proportional delayed PNN is faster in terms of convergence rate. The possible reason is that appropriate parameters make the model converge to the equilibrium point along the direction of gradient descent.

Optimal Power Quality Compensation of Energy Storage System in Distribution Networks Based on Unified Multi-Phase OPF Model

Energy Storage System (ESS) is a promising solution to suppress the peak-valley difference of residential distribution networks (RDN) with high penetration of distributed photovoltaic generations. Meanwhile, it can also provide certain power quality compensations to RDN due to the flexible adjusting ability of its converter interface. To make full use of this feature, this paper investigates the optimal power quality compensation of the ESS. This is achieved by formulating and solving an optimal power flow (OPF) problem, the objectives of which are to minimize the power loss, harmonic distortion, and voltage unbalance of the network. Noticeably, the established model includes the neutral line and inter-phase loads; thus, it is unified and can be easily extended to networks with different structures. However, the coupling between different phases and the inclusion of harmonic variables significantly increase the complexity of the OPF problem. To address this issue, the optimization model is further transformed into a convex quadratic constrained quadratic program (QCQP) by replacing power flow equations with current injection equations. As a result, the problem can be solved by the primal-dual interior-point method (PDIPM) with guaranteed convergence and computational efficiency. An iterative solution method is further proposed to improve the accuracy of the solution. Finally, the proposed method is verified using a three-phase three-wire 25-node distribution network and a three-phase four-wire 162-node distribution network in North America.

Consensus models with aggregation operators for minimum quadratic cost in group decision making.

In group decision making (GDM), to facilitate an acceptable consensus among the experts from different fields, time and resources are paid for persuading experts to modify their opinions. Thus, consensus costs are important for the GDM process. Notwithstanding, the unit costs in the common linear cost functions are always fixed, yet experts will generally express more resistance if they have to make more compromises. In this study, we use the quadratic cost functions, the marginal costs of which increase with the opinion changes. Aggregation operators are also considered to expand the applications of the consensus methods. Moreover, this paper further analyzes the minimum cost consensus models under the weighted average (WA) operator and the ordered weighted average (OWA) operators, respectively. Corresponding approaches are developed based on strictly convex quadratic programming and some desirable properties are also provided. Finally, some examples and comparative analyses are furnished to illustrate the validity of the proposed models.

Strategic interactions between liquefied natural gas and domestic gas markets: A bilevel model

This paper presents a bilevel programming model to aid decision-making for two players who interact strategically in the liquefied natural gas (LNG) supply chain, possibly with conflicting interests. In the proposed model, an LNG operator is the leader and a natural gas (NG) producer is the follower. The LNG operator attempts to optimally locate LNG export terminals, purchase gas from the NG producer, and export it as LNG. The NG producer aims to optimize production, pipeline investments, and sales to domestic consumers and the LNG operator. To solve the bilevel problem, we first reformulate it as a single-level problem by exploiting the convexity of the lower-level problem. Then, we use disjunctive reformulations of complementarity constraints and piecewise linear approximations of objective function terms to convert the problem into a convex quadratic mixed-integer program (MIQP). Computational experiments confirm that the MIQP is tractable and can be solved efficiently. We apply our bilevel framework to a case study of the Gulf-Southwest region of the United States and evaluate several decision-making scenarios. The scenario results emphasize the importance of using the bilevel methodology to anticipate the effects of new LNG export facilities on NG prices across the domestic gas network. Adding these large gas-consuming facilities at specific locations puts upward pressure on domestic gas prices, although this is somewhat mitigated by the NG producer’s optimal production response to the increased demand.

Position and Orientation Tunnel-Following NMPC of Robot Manipulators Based on Symbolic Linearization in Sequential Convex Quadratic Programming

The tunnel-following nonlinear model predictive control (NMPC) scheme allows to exploit acceptable deviations around a path reference. This is done by using convex-over-nonlinear functions as objective and constraints in the underlying optimal control problem (OCP). The convex-over-nonlinear structure is exploited by algorithms such as the generalized Gauss-Newton (GGN) method or the sequential convex quadratic programming (SCQP) method to reduce the computational complexity of the OCP solution. However, the modeling effort and engineering time required to implement these methods is high. We address the problem of reducing the modeling effort in the implementation of SCQP, focusing on a standard sequential quadratic programming (SQP) implementation where symbolic linearization is applied to the nonlinear part of the convex-over-nonlinear functions in the objective and constraints. The novelty of this letter is twofold. It introduces a novel operator that applies symbolic linearization in a transparent and easy way to solve nonconvex OCPs with the SCQP method, and introduces a meaningful representation of an orientation-tunnel for robotic applications by means of a convex-over-nonlinear constraint, which preserves the convexity exploitation by the SCQP method. The proposed technique is demonstrated in a tunnel-following task for a 7-degrees-of-freedom manipulator.

A Support Vector Neural Network for P300 EEG Signal Classification

Brain computer interface (BCI) P300 speller can help severely disabled patients communicate and control with external machines or robots, so that the classification methods of P300 Electroencephalogram (EEG) signal play an important role in the development of BCI system and technology. In this paper, a novel support vector neural network (SVNN) is proposed and developed to obtain more accurate and effective EEG classification results. It is the first time to combine linear variational inequality (LVI) based primal-dual neural network with convex quadratic programming problem based on support vector machine to solve the classification problem. It has been proved that the support vector neural network globally converges to the optimal solution of convex optimization problem and iterates the parameters in the form of matrix, which means that the method has global convergence and parallelism. The proposed SVNN method is used to solve the classification problem of P300 EEG signals. Experimental results on data set IIb from BCI competition II and data set II from BCI competition III show that the accuracy of the proposed SVNN method is 100% and 98%, respectively. Compared with most of the state-of-theart algorithms, SVNN has the highest recognition accuracy and information transfer rate.

QPALM: a proximal augmented lagrangian method for nonconvex quadratic programs

We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an \(R\)-linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.

An efficient arc-search interior-point algorithm for convex quadratic programming with box constraints

Although the classical LQR design method has been very successful in real world engineering designs, in some cases, the classical design method needs modifications because of the saturation in actuators. This modified problem is sometimes called the constrained LQR design. For discrete systems, the constrained LQR design problem is equivalent to a convex quadratic programming problem with box constraints. We will show that the interior-point method is very efficient for this problem because an initial interior point is available, a condition which is not true for general convex quadratic programming problem. We will devise an effective and efficient algorithm for the constrained LQR design problem using the special structure of the box constraints and a recently introduced arc-search technique for the interior-point algorithm. We will prove that the algorithm is polynomial and has the best-known complexity bound for the convex quadratic programming. The proposed algorithm is implemented in MATLAB. An example for the constrained LQR design is provided to show the effectiveness and efficiency of the design method. The proposed algorithm can easily be used for model predictive control.

Robustness verification of ReLU networks via quadratic programming

Neural networks are known to be sensitive to adversarial perturbations. To investigate this undesired behavior we consider the problem of computing the distance to the decision boundary (DtDB) from a given sample for a deep neural net classifier. In this work we present a procedure where we solve a convex quadratic programming (QP) task to obtain a lower bound on the DtDB. This bound is used as a robustness certificate of the classifier around a given sample. We show that our approach provides better or competitive results in comparison with a wide range of existing techniques.

A new projection neural network for linear and convex quadratic second-order cone programming

A new projection neural network approach is presented for the linear and convex quadratic second-order cone programming. In the method, the optimal conditions of the linear and convex second-order cone programming are equivalent to the cone projection equations. A Lyapunov function is given based on the G-norm distance function. Based on the cone projection function, the descent direction of Lyapunov function is used to design the new projection neural network. For the proposed neural network, we give the Lyapunov stability analysis and prove the global convergence. Finally, some numerical examples and two kinds of grasping force optimization problems are used to test the efficiency of the proposed neural network. The simulation results show that the proposed neural network is efficient for solving some linear and convex quadratic second-order cone programming problems. Especially, the proposed neural network can overcome the oscillating trajectory of the exist projection neural network for some linear second-order cone programming examples and the min-max grasping force optimization problem.

Formula omitted]-Kernel-free soft quadratic surface support vector regression

In this paper, we propose a new regression method called the ∊-kernel-free soft quadratic surface support vector regression (∊-SQSSVR). After converting the n-dimensional regression problem into the (n+1)-dimensional classification problem, the principle of maximizing the sum of relative geometrical margin of each training point is used to construct our optimization problem, where the quadratic surface is restricted to be a hyperparaboloid by setting both the (n+1)-th row and (n+1)-th column of the corresponding matrix to be zero. The existence and uniqueness of the optimal solution to both primal and dual problems are also addressed. It should be pointed out that our model is nonlinear and kernel-free, so it does not need to select kernel function and corresponding parameters. At the same time, it is highly interpretable. In addition, our model is still a quadratic convex programming problem similar to the standard SVR. To visualize the effectiveness of our ∊-SQSSVR, 6 artificial datasets and 15 benchmark datasets are implemented in numerical experiments. The results show that our method is less time-consuming and as good as the nonlinear standard SVR with kernel function in comprehensive performances.

Does ESG Impact Really Enhance Portfolio Profitability?

Over the last few decades, growing attention to the topic of social responsibility has affected financial markets and institutional authorities. Indeed, recent environmental, social, and financial crises have inevitably led regulators and investors to take into account the sustainable investing issue; however, the question of how Environmental, Social, and Governance (ESG) criteria impact financial portfolio performances is still open. In this work, we examine a multi-objective optimization model for portfolio selection, where we add to the classical Mean-Variance analysis a third non-financial goal represented by the ESG scores. The resulting optimization problem, formulated as a convex quadratic programming, consists of minimizing the portfolio variance with parametric lower bounds on the levels of the portfolio expected return and ESG. We provide here an extensive empirical analysis on five datasets involving real-world capital market indexes from major stock markets. Our empirical findings typically reveal the presence of two behavioral patterns for the 16 Mean-Variance-ESG portfolios analyzed. Indeed, over the last fifteen years we can distinguish two non-overlapping time windows on which the inclusion of portfolio ESG targets leads to different regimes in terms of portfolio profitability. Furthermore, on the most recent time window, we observe that, for the US markets, imposing a high ESG target tends to select portfolios that show better financial performances than other strategies, whereas for the European markets the ESG constraint does not seem to improve the portfolio profitability.

Multi-category intuitionistic fuzzy twin support vector machines with an application to plant leaf recognition

The intuitionistic fuzzy twin support vector machine for multi-categorization is developed in this study, which incorporates both structural and empirical risk concepts. In this method, each training pattern is first aggregated with the appropriate membership and non-membership degrees, which describe the position of a pattern in relation to its class centre and surrounding circumstances in input or feature space, and then the separating hyperplane is constructed using the kernel function and convex quadratic programming. Empirical findings on an artificial and thirteen UCI standard datasets show that it outperforms well-known existing methods including improved support vector machines, K-nearest neighbour, logistic regression, decision trees, random forests, and multilayer perceptrons. Furthermore, the suggested classifier with linear, polynomial, and Gaussian kernels has been used to identify the leaves of various plants, where the shape, texture, and margin data are extracted from the leaf in order to categorize the plant species. The method’s generalization capacity is demonstrated by the classification results on two leaf datasets of thirty and one hundred species, respectively. To compare the suggested method’s prediction capacity with others, statistical analysis is performed using two non-parametric tests, Friedman and Wilcoxon, with a 5% threshold of significance. The results show that the proposed method yields better performance for both linear and non-linear kernels.

A Convex Dynamic Approach for Globally Optimal Profit in Supply Chains

Supply chain finance aims to coordinate multiple stakeholders to maximize the flow of cash and internal and external funding along the supply chain, as shown in prior research. From a regulatory standpoint, the goal of this paper is to maximize the profitability of an entire supply chain. As a result, a constrained finite time Linear Quadratic Regulation (LQR) approach is provided for determining an entity’s optimal profit state in a supply chain. The framework is represented by discrete-time linear dynamical equations for each entity in the supply chain network, taking state and input variables into account. The problem is formulated in terms of a convex quadratic programming optimization for which several numerically efficient algorithms are readily available. In order to validate the approach, it was tested on two topologies. The first topology is a fully connected supply chain with six nodes; the second is a simple topology based on the Iranian pharmaceutical supply chain. The results indicate that the proposed approach successfully planned production and financing decisions within the simulated supply chain and obtained globally optimal profit for all supply chain stakeholders.

Error propagation model and optimal control method for the quality of remanufacturing assembly

In order to improve the quality of remanufacturing assembly with uncertainty for the sustainability of remanufacturing industry, an error propagation model of the remanufacturing assembly process and its optimal control method are established. First, the state space model of error propagation is established by taking the work-in-process parameter errors of each process as the initial state of the procedure and the parameters of remanufactured parts and operation quantities as the input. Then, the quality control issue of remanufacturing assembly is transformed into a convex quadratic programming with constraints based on this model. Finally, the proposed method is used to control the remanufactured-crankshaft assembly quality. The experimental results show that the axial-clearance consistency and the crankshaft torque are improved, and the one-time assembly success rate of a remanufactured crankshaft is increased from 96.97%to 99.24%. This study provides a theoretical model and method support for the quality control of remanufacturing assembly and has a practical effect on improving the quality of remanufactured products.

Complexity analysis of interior point methods for convex quadratic programming based on a parameterized Kernel function

The kernel functions play an important role in the amelioration of the computational complexity of algorithms. In this paper, we present a primal-dual interior-point algorithm for solving convex quadratic programming based on a new parametric kernel function. The proposed kernel function is not logarithmic and not self-regular. We analysis a large and small-update versions which are based on a new kernel function. We obtain the best known iteration bound for large-update methods, which improves signicantly the so far obtained complexity results. Thisresult is the rst to reach this goal.

Nonlinear burn control in ITER using adaptive allocation of actuators with uncertain dynamics

ITER will be the first tokamak to sustain a fusion-producing, or burning, plasma. If the plasma temperature were to inadvertently rise in this burning regime, the positive correlation between temperature and the fusion reaction rate would establish a destabilizing positive feedback loop. Careful regulation of the plasma’s temperature and density, or burn control, is required to prevent these potentially reactor-damaging thermal excursions, neutralize disturbances and improve performance. In this work, a Lyapunov-based burn controller is designed using a full zero-dimensional nonlinear model. An adaptive estimator manages destabilizing uncertainties in the plasma confinement properties and the particle recycling conditions (caused by plasma–wall interactions). The controller regulates the plasma density with requests for deuterium and tritium particle injections. In ITER-like plasmas, the fusion-born alpha particles will primarily heat the plasma electrons, resulting in different electron and ion temperatures in the core. By considering separate response models for the electron and ion energies, the proposed controller can independently regulate the electron and ion temperatures by requesting that different amounts of auxiliary power be delivered to the electrons and ions. These two commands for a specific control effort (electron and ion heating) are sent to an actuator allocation module that optimally maps them to the heating actuators available to ITER: an electron cyclotron heating system (20 MW), an ion cyclotron heating system (20 MW), and two neutral beam injectors (16.5 MW each). Two different actuator allocators are presented in this work. The first actuator allocator finds the optimal mapping by solving a convex quadratic program that includes actuator saturation and rate limits. It is nonadaptive and assumes that the mapping between the commanded control efforts and the allocated actuators (i.e. the effector model) contains no uncertainties. The second actuator allocation module has an adaptive estimator to handle uncertainties in the effector model. This uncertainty includes actuator efficiencies, the fractions of neutral beam heating that are deposited into the plasma electrons and ions, and the tritium concentration of the fueling pellets. Furthermore, the adaptive allocator considers actuator dynamics (actuation lag) that contain uncertainty. This adaptive allocation algorithm is more computationally efficient than the aforementioned nonadaptive allocator because it is computed using dynamic update laws so that finding the solution to a static optimization problem is not required at every time step. A simulation study assesses the performance of the proposed adaptive burn controller augmented with each of the actuator allocation modules.

On a primal-dual Newton proximal method for convex quadratic programs

This paper introduces QPDO, a primal-dual method for convex quadratic programs which builds upon and weaves together the proximal point algorithm and a damped semismooth Newton method. The outer proximal regularization yields a numerically stable method, and we interpret the proximal operator as the unconstrained minimization of the primal-dual proximal augmented Lagrangian function. This allows the inner Newton scheme to exploit sparse symmetric linear solvers and multi-rank factorization updates. Moreover, the linear systems are always solvable independently from the problem data and exact linesearch can be performed. The proposed method can handle degenerate problems, provides a mechanism for infeasibility detection, and can exploit warm starting, while requiring only convexity. We present details of our open-source C implementation and report on numerical results against state-of-the-art solvers. QPDO proves to be a simple, robust, and efficient numerical method for convex quadratic programming.

Синтез нелинейного прогнозирующего управления вентильным электроприводом оптико-механических комплексов

The paper proposes a method to synthesize the nonlinear discrete predictive position controller in tracking electric drive control system of optical-mechanical complexes based on permanent-magnet synchronous motor. The technique for constructing mathematical discrete nonlinear vector-matrix model of permanent-magnet synchronous motor in rotor rotating coordinate for vector control is given. The permanentmagnet synchronous motor is operated under conditions of two-mass elastic load. The steps to synthesize nonlinear predictive control algorithms by using extended linearization technique and optimization objective function with convex quadratic programming method are analyzed. Computer simulation results are carried out on MATLAB / Simulink environment. In carried out simulation, the performance of discrete predictive controller in the position loop of optical-mechanical complexes tracking control system is assessed with different kind of control signals and under the influence of external disturbance.