Linear Quadratic Programming Research Articles

Measurement-Noise Filtering for Automatic Discovery of Flow Splitting Ratios in ISP Networks

Network telemetry and analytics is essential for providing highly dependable services in modern computer networks. In particular, network flow analytics for ISP networks allows operators to inspect and reason about traffic patterns in their networks in order to react to anomalies. High performance network analytics systems are designed with scalability in mind, and can consequently only observe partial information about the network traffic. Still, they need to provide a holistic view of the traffic, including the distribution of different traffic flows on each link. It is impractical to monitor such fine-grained telemetry, and in large, heterogeneous networks it is often too complex and error-prone, if not impossible, to access and maintain all technical specifications and router-specific configurations needed to determine e.g. the load balancing weights used when traffic is split onto multiple paths. The ratios by which flows are split on the possible paths must be derived indirectly from the measured flow demands and link utilizations. Motivated by a case study provided by a major European ISP, we suggest an efficient method to estimate the flow splitting ratios. Our approach, based on quadratic linear programming, is scalable and achieves robustness to the measurement noise found in a typical network analytics deployment by filtering out certain constraints in the linear program. Finally, we implement an automated tool for estimating the flow splitting ratios and document its applicability on real data from the ISP.

Ultrafast Hybrid Computing Systems Enabled by Memristor‐Based Quadratic Programming Circuits

AbstractImplementing algorithms purely on digital computing platforms dramatically halts the performance of conventional computing systems. Revolutionary computing systems with extreme energy efficiency and high accuracy are demanded to handle the growing computing tasks. Here, the research on hybrid analog–digital computing platforms enabled by memristor‐based optimization solvers for achieving ultrafast computations is presented. By utilizing tunable memristors as parameters to solve linear programming (LP) and quadratic programming (QP) problems, a real‐time control algorithm for micro air vehicles (MAVs) and a support vector machine (SVM) algorithm for cancer diagnosis are implemented. These experiments demonstrate over 2000x speed‐up compared to conventional digital platforms, with negligible energy consumption, using a memristor‐based system consisting of six memristors. These findings underscore the vast potential of memristor‐based optimization solvers not only in hybrid analog–digital computing platforms but also as a transformative solution for a wide range of modern computing challenges. This approach promises significant advancements in energy efficiency and ultrafast speed, positioning it as a leading contender for next‐generation computing paradigms.

Solving a class of Thomas–Fermi equations: A new solution concept based on physics-informed machine learning

This paper presents a novel physics-informed machine learning approach, designed to approximate solutions to a specific category of Thomas–Fermi differential equations. To tackle the inherent intricacies of solving Thomas–Fermi equations, we employ the quasi-linearization technique, which transforms the original non-linear problem into a series of linear differential equations. Our approach utilizes collocation least-squares support vector regression, leveraging fractional Chebyshev functions for finite interval simulations and fractional rational Chebyshev functions for semi-infinite intervals, enabling precise solutions for linear differential equations across varied spatial domains. These selections facilitate efficient simulations across both finite and semi-infinite intervals. Key contributions of our approach encompass its versatility, demonstrated through successful approximation of solutions for diverse Thomas–Fermi problem types, including those featuring non-local integral terms, Bohr radius boundary conditions, and isolated neutral atom boundary conditions defined on semi-infinite domains. Furthermore, our method exhibits computational efficiency, surpassing classical collocation methods by solving a sequence of positive definite linear equations or quadratic programming problems. Notably, our approach showcases precision, as evidenced by experiments, including the attainment of the initial slope of the renowned Thomas–Fermi equation with an impressive 39-digit precision.

On the number of pivots of Dantzig's simplex methods for linear and convex quadratic programs

Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegatively constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency of simplex-type methods for solving these problems. Two applications of the quadratic programming results are presented.

Evaluation of Grid Support Capacity From Electrified Railways

High power level and temporal operation flexibility endow electric trains with great potential for improving grid resilience. It becomes crucial to quantify the grid support capacity (GSC) from large-scale electrified railways. This paper innovatively proposes a hierarchical train coordination and trajectory regulation method to evaluate and deliver the GSC from electrified railways. In the light of the mechanical energy consumption analysis of train running, the energy-oriented train operation model is proposed to avoid the nonlinear differential dynamics of train kinetics when coordinating large-scale trains. Then the aggregation and disaggregation of trains for grid support can be formulated as linear and convex quadratic programming problems respectively. To closely follow the reference power and enhance the feasibility of train operation, a novel iterative discretization approach is proposed to generate the precise reference trajectory. A sliding mode controller with disturbance bound adaption is devised to overcome the modeling errors and disturbances when deploying GSC in realistic environment. Comprehensive numerical examples verify the effectiveness of the proposed scheme, and demonstrate the significant grid support potential from collaborative electrified railways.

DMPC-based load frequency control of multi-area power systems with heterogeneous energy storage system considering SoC consensus

The energy storage system (ESS) has been widely used for the load frequency control (LFC) of power systems. The heterogeneous ESS (HESS) consisting of various types of energy storage units (ESUs) with different regulation characteristics creates many difficulties in designing a high-performance LFC strategy for a multi-area power system. To obtain the satisfied dynamic response of the LFC scheme against load disturbances and keep the state of charge (SoC) balance of the HESS, this paper proposes a distributed model predictive control (DMPC) method to design the LFC strategy. Firstly, the dynamic model of the power system with HESS is established. Then, the distributed predictive model for LFC design is formulated by considering the LFC objective, the SoC balance requirement, and the operational constraints of the generators and the HESS. Subsequently, the optimal LFC strategy is obtained by solving the distributed predictive model based on the linear quadratic programming (LQP) method. Finally, we make case studies to validate the effectiveness of the proposed method. Compared with the PI-based LFC, the proposed DMPC-based LFC performs better in stabilizing the system frequency, narrowing the area control error (ACE), and keeping the SoC of all the ESUs consistent.

Research on Stability Control Algorithm of Distributed Drive Bus under High-Speed Conditions

Aiming at the instability problem of a four-wheel independent drive electric bus under high-speed conditions, this paper first designs a vehicle yaw stability controller based on a linear two-degree-of-freedom model and a linear quadratic programming (LQR) algorithm. A vehicle roll stability controller is then designed based on a linear three-degree-of-freedom model and a model predictive control algorithm (MPC). Moreover, a coordinated control rule based on the lateral load transfer rate (LTR) is designed for the coupled problem of yaw and roll dynamics. Finally, the effectiveness of the proposed control algorithm is verified by simulation. The obtained results show that when the vehicle is running at a high speed of 90 km/h, the stability control algorithm can control the yaw rate tracking error within 0.05 rad/s. In addition, the control algorithm can reduce the maximum amplitude of the side slip angle, the maximum value of the roll angle, the maximum value of the roll angular velocity, and the amplitude of the lateral acceleration by more than 96%, 81.1%, 65.0%, and 11.1%, respectively.

Safety considerations of static sensitization emulsification technology for Kings-type emulsion explosives based on quadratic linear programming algorithm

Abstract This paper collects data on the static sensitization emulsification process flow of Kings-type emulsion explosives, analyzes the collected data by combining the indicator factors and probability density functions in the time-domain analysis, and designs a safety monitor for the static sensitization process flow. In order to pursue more rigorous production safety, the static sensitization process optimization objective function was determined, and the function was constrained as well as quadratically linearized to construct a static sensitization process safety optimization model based on quadratic linear programming. After analysis, it is concluded that the safety factors affecting the static sensitization process are temperature, sensitizer, number of sensitizer stages and pressure. Other conditions are the same, the foaming density can finally reach 1.23g/cm 3 when low-temperature static sensitization, but the time spent is longer 14h, and the foaming density decreases from 1.41g/cm 3 to 1.09g/cm 3 as the amount of sensitizer increases.

Constrained distributed consensus control of homogeneous vehicle platoons with bidirectional communication

Vehicle platooning is deemed a promising solution to improve traffic safety, reduce fuel consumption and increase traffic throughput and road capacity. Road throughput can be increased by driving at small inter-vehicle distances. Adjusting the spacing with both the preceding and the succeeding vehicles using bidirectional communication can potentially improve the platoon cohesiveness and is crucial when considering the synchronized merging scenario. In this paper, a distributed consensus controller is presented which relies on bidirectional platoon communication. To take into account the constraints of the system state and input constraints are implemented using control barrier functions via linear quadratic programming. Along with the consensus-based controller synthesis, platoon stability proof and string stability analysis, effectiveness of the controller is verified in a simulation study.

Hybrid trilinear and bilinear programming for aligning partially overlapping point sets

In many applications, we need algorithms which can align partially overlapping point sets and are invariant to the corresponding transformations. In this work, a method possessing such properties is realized by minimizing the objective of the robust point matching (RPM) algorithm. We first show that the RPM objective is a cubic polynomial. We then utilize the convex envelopes of trilinear and bilinear monomials to derive its lower bound function. The resulting lower bound problem has the merit that it can be efficiently solved via linear assignment and low dimensional convex quadratic programming. We next develop a branch-and-bound (BnB) algorithm which only branches over the transformation variables and runs efficiently. Experimental results demonstrated better robustness of the proposed method against non-rigid deformation, positional noise and outliers in case that outliers are not mixed with inliers when compared with the state-of-the-art approaches. They also showed that it has competitive efficiency and scales well with problem size.

Proximal Stabilized Interior Point Methods and Low-Frequency-Update Preconditioning Techniques

In this work, in the context of Linear and convex Quadratic Programming, we consider Primal Dual Regularized Interior Point Methods (PDR-IPMs) in the framework of the Proximal Point Method. The resulting Proximal Stabilized IPM (PS-IPM) is strongly supported by theoretical results concerning convergence and the rate of convergence, and can handle degenerate problems. Moreover, in the second part of this work, we analyse the interactions between the regularization parameters and the computational footprint of the linear algebra routines used to solve the Newton linear systems. In particular, when these systems are solved using an iterative Krylov method, we are able to show—using a new rearrangement of the Schur complement which exploits regularization—that general purposes preconditioners remain attractive for a series of subsequent IPM iterations. Indeed, if on the one hand a series of theoretical results underpin the fact that the approach here presented allows a better re-use of such computed preconditioners, on the other, we show experimentally that such (re)computations are needed only in a fraction of the total IPM iterations. The resulting regularized second order methods, for which low-frequency-update of the preconditioners are allowed, pave the path for an alternative class of second order methods characterized by reduced computational effort.

Linear pencils and quadratic programming problems with a quadratic constraint

Given bounded selfadjoint operators A and B acting on a Hilbert space H, consider the linear pencil P(λ)=A+λB, λ∈R. The set of parameters λ such that P(λ) is a positive (semi)definite operator is characterized. These results are applied to solving a quadratic programming problem with an equality quadratic constraint (or a QP1EQC problem).

Core-selecting auction-based mechanisms for service function chain provisioning and pricing in NFV markets

With the rapid development of network function virtualization (NFV), an ever-increasing number of enterprises, businesses and operators are resorting to network service provisioning by means of the service function chains (SFCs). Toward this trend, NFV markets are emerging, where users rent SFCs from the network service provider (NSP) to execute their tasks, and the NSP uses the computation resources across geo-distributed clouds to construct various SFCs. Such NFV markets need well-designed pricing mechanisms for SFCs since the users have motivations to strategize around their payments. The Vickrey–Clarke–Groves (VCG) mechanism is a natural fit here, which can ensure that truthful bidding is a dominant strategy. However, VCG mechanism is susceptible to the problems of collusion and shill bidding. To address these problems, in this paper, we design a core-selecting auction-based mechanism (CSAM) for SFC provisioning and pricing in the NFV market. This mechanism considers dynamic provisioning, flexible deployment and reservation prices across geo-distributed clouds. Specifically, we first formulate the core of NFV auction, and prove that an in-core outcome can prevent collusion and shill bidding. Next, we use a couple of correlated linear program (LP) and quadratic program (QP) to design an in-core VCG-nearest payment algorithm, aiming to acquire an in-core outcome and minimize users’ incentives to bid untruthfully. Simulation results verify the strict theoretical analysis, and validate the effectiveness and efficiency of our proposed CSAM.

Continuous-Time Receding-Horizon Estimation via Primal-Dual Dynamics and Stability Analysis

Receding-Horizon Estimation (RHE) is an optimal filterling approach which uses past series of the plant's measured output and input and finds estimated states based on linear programming or quadratic programming. It is known that RHE can estimate the plant state to which the Kalman filter cannot be applied due to modeling errors. This paper considers the new computational form of RHE based on the principal dual gradient algorithm. The proposed form is expressed by the dynamical system, so we can consider the computational stability based on the dynamical system theory. This paper discusses the continuous-time representation of the RHE algorithm (continuous-time RHE) and filter characteristics to improve the convergence performance of the estimation. On the basis of the small gain theorem, the dynamics of RHE is analyzed. Also, the characteristics of continuous-time RHE is demonstrated via a vehicle path tracking control problem.

Fast trajectory planning and control of a lab-scale 3D gantry crane for a moving target in an environment with obstacles

In this work, the real-time optimal trajectory planning, together with a cascaded tracking controller, is presented for a three-dimensional (3D) gantry crane in an environment with static obstacles and a dynamically moving target considering dynamic constraints and control input limits. State-of-the-art trajectory optimization-based approaches require long computation times and cannot quickly respond to changes in the target state. The focus of this paper lies on a novel trajectory planning algorithm, which consists of two steps. First, an offline trajectory planner is implemented to compute a time-optimal, collision-free, and dynamically feasible trajectory database that connects all possible initial states of the gantry crane from a predefined starting subspace to the target states in a target subspace. Second, based on linear constrained quadratic programming, the online trajectory replanner makes use of this trajectory database to generate an optimal trajectory in real time that accounts for all changes in the target state. Additionally, a trajectory tracking controller is developed to take into account the dynamic constraints of the gantry crane and to compensate for possible model inaccuracies, disturbances, and other non-modeled effects. Both simulation and experimental results are presented to demonstrate the performance of the proposed trajectory (re)planning algorithm and the control concept.

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.

Scalable Optimal Control Allocation: Linear and Quadratic Programming Methods Applied to Active Capacitor Balancing in Modular Multilevel Converters

The paper introduces a novel, scalable control allocation (CA) formulation for active capacitor balancing of the Modular Multilevel Converter (MMC). The formulation is obtained by introducing a converter arm model that is also scalable and accommodates an MMC with any number of phases and any number of submodules (SM). The CA algorithm involves the fast real-time minimization of an error cost function formalized as linear programming (LP) and quadratic programming (QP) problems. The real-time coding is implemented in both cases using an interior-point method. A hardware-in-the-loop (HIL) test procedure is used to demonstrate the ability of the CA to perform active capacitor voltage balancing. The influence of QP vs LP on the system behavior is also evaluated, and tracking errors using QP are found to be more evenly distributed than with LP.

Guest Editorial

Evolutionary Algorithms (EAs) are nature-inspired population-based search methods which work on Darwinian principles of natural selection. Due to their strong search capability and simplicity of implementation, EAs have been successfully applied to solve many complex optimization problems, which cannot be easily solved by traditional mathematical programming approaches, such as linear programming, quadratic programming, and convex optimization. Despite the great success enjoyed by EAs, it is worth noting that existing EA solvers usually conduct the search process from scratch, regardless of how similar the new problem encountered is to those already solved in the past. Therefore, conventional EAs do not learn from previous problems and the search capabilities of the EA solvers do not automatically grow with problem-solving experiences. However, in reality, since problems seldom exist in isolation, solving one problem may thus yield useful information for solving other related problems. In the literature, there is a growing interest in conducting research on evolutionary transfer optimization (ETO) in recent years: a paradigm that integrates EA solvers with knowledge learning and transfer across related domains to achieve better optimization efficiency and performance.

Inference on estimators defined by mathematical programming

We propose an inference procedure for a class of estimators defined as the solutions to linear and convex quadratic programming problems in which the coefficients in both the objective function and the constraints of the problem are estimated from data and hence involve sampling error. We argue that the Karush–Kuhn–Tucker conditions that characterize the solutions to these programming problems can be treated as moment conditions; by doing so, we transform the problem of inference on the solution to a constrained optimization problem (which is non-standard) into one involving inference on inequalities with pre-estimated coefficients, which is better understood. Our approach is valid regardless of whether the problem has a unique solution or multiple solutions. We apply our method to various portfolio selection models, in which the confidence sets can be non-convex, lower-dimensional manifolds.

An evolutionary algorithm for large margin classification

Classification is an essential task in the field of machine learning, where finding a maximum margin classifier is one of its central problems. In this work, an evolutionary algorithm is constructed, relying on the convexity properties of the version space, to evolve a population of perceptron classifiers in order to find a solution that approximates the maximum margin. Unlike other methods whose solutions explore the problem’s dual formulation, usually requiring the solution of a linear constraint quadratic programming problem, the proposed method requires only the evaluation of the margin values. Hyperspherical coordinates are used to guarantee feasibility when generating new individuals and for the population to be uniformly distributed through the search space. To control the number of generations, we developed a stop criteria based on a lower bound function which asymptotically approximates the margin curves providing a stop margin that satisfies a $$\beta $$ -approximation of the optimal margin. Experiments were performed on artificial and real datasets, and the obtained results indicate the potential to adopt the proposed algorithm for solving practical problems.