Quadratic Programming Formulation Research Articles

Turret-index optimisation with mathematical programming and metaheuristic approaches

This paper addresses the turret-index optimisation problem by proposing two mathematical programming approaches based on quadratic programming (QP) and constraint programming (CP). In addition, a metaheuristic algorithm based on weighted superposition attraction (WSA) is developed due to the combinatorial complexity of the problem. Despite attempting to linearise the QP formulation for problem resolution, both the QP and Linearized QP models prove ineffective for medium and larger-sized instances, providing solutions only for small-sized problems. In the second mathematical programming approach, a novel CP model is introduced in the literature. While CP can rapidly offer optimal solutions for small-sized problems, it only provides satisfactory solutions for medium and larger-sized problems within the predetermined computational time limits and does not guarantee optimal results. On the other hand, through computational analysis and relevant statistical tests, the proposed WSA algorithm provides the best solutions for all test problems within a reasonable computational time.

Improving Quantum Optimization Algorithms by Constraint Relaxation

Quantum optimization is a significant area of quantum computing research with anticipated near-term quantum advantages. Current quantum optimization algorithms, most of which are hybrid variational-Hamiltonian-based algorithms, struggle to present quantum devices due to noise and decoherence. Existing techniques attempt to mitigate these issues through employing different Hamiltonian encodings or Hamiltonian clause pruning, but they often rely on optimistic assumptions rather than a deep analysis of the problem structure. We demonstrate how to formulate the problem Hamiltonian for a quantum approximate optimization algorithm that satisfies all the requirements to correctly describe the considered tactical aircraft deconfliction problem, achieving higher probabilities for finding solutions compared to previous works. Our results indicate that constructing Hamiltonians from an unconventional, quantum-specific perspective with a high degree of entanglement results in a linear instead of exponential number of entanglement gates instead and superior performance compared to standard formulations. Specifically, we achieve a higher probability of finding feasible solutions: finding solutions in nine out of nine instances compared to standard Hamiltonian formulations and quadratic programming formulations known from quantum annealers, which only found solutions in seven out of nine instances. These findings suggest that there is substantial potential for further research in quantum Hamiltonian design and that gate-based approaches may offer superior optimization performance over quantum annealers in the future.

A generation and transmission expansion planning model for the electricity market with decarbonization policies

Globally, the power sector must undergo a profound transition to achieve the decarbonization development targets. Various roadmaps are implemented, but only from a macro perspective, lacking the consideration of the electricity market rules. In this paper, we develop and present a market-driven generation and transmission expansion planning (MGTEP) model considering the effectiveness of the electricity market. Specifically, generation and transmission companies incorporate hourly market trading and annual capacity investment into strategic decisions to maximize their profits, with the supply function equilibrium model to analyze bidding behaviors. An equivalent quadratic programming formulation is deployed to solve the trilevel MGTEP model. Meanwhile, the MGTEP model is coupled with decarbonization policies to support the state and federal government in assessing energy transition strategies. We implement the MGTEP model with carbon emission allowance and carbon tax policies for the southern China electricity market to achieve carbon peaking by 2030. Carbon emission allowance adopts an intensity-based cap based on generation companies' historical output. The case study results show that 50 % carbon emission allowance or 400 CNY/t carbon tax is required but with several drawbacks, including unsatisfactory decarbonization effect, excessive economic sacrifice, etc. Finally, the case study is extended to dual-track policies with different combinations of policies. An optimal combination is 70 % carbon emission allowance and 160 CNY/t carbon tax. In this case, the power sector's carbon dioxide emissions and electricity prices in the southern China electricity market would increase to 554.6 Mt and 864.34 CNY/MWh in 2030, respectively, along with a carbon price of 850 CNY/t.

Game-Theoretic Capacity Expansion Analysis Under Carbon Emission Constraints

In this paper, we develop a long-term capacity expansion analysis model that considers strategic bidding behavior by dominant generation companies together with new entry decisions, including renewables, to meet a long-term carbon reduction constraint. We deploy a centralized quadratic programming formulation of the Nash-Cournot game to analyze day-ahead electricity market prices. In our model, strategic producers simultaneously make investment and production decisions, deploying an open-loop Cournot game. We have implemented the model for Ukraine's day-ahead electricity market, which has been highly prone to market power issues. The case study results show significant inefficiencies in peak prices and explore scenarios of alternative mitigation options and their efficacies. Finally, the case study is extended to also explore implications of a low carbon future through emission-constrained runs, evaluation of the performance-based emission policy, and sensitivity on different portfolios of new renewable investors.

An inverse kinematics solution with trajectory scaling for redundant manipulators

Kinematic redundancy offers robots many useful capabilities, such as an ability to fulfill several tasks simultaneously. However, the inverse kinematics (IK) problem for redundant manipulators is not trivial. Often, pseudoinverse-based methods are used, allowing for the projection of the desired joint space velocity vector on the null space of the manipulator’s Jacobian. Unfortunately, it is not straightforward to include in the solution the joint position, velocity and acceleration bounds. On the other hand, the IK problem can be formulated as an optimization problem. A quadratic programming (QP) formulation is especially attractive for an online implementation, since it is computationally much lighter than nonlinear optimization methods. In a QP IK formulation, the joint constraints can be easily included. However, if the desired end effector trajectory becomes too demanding, then the joint space solution might not exist. To overcome this obstacle, we propose a QP IK formulation with trajectory scaling. The method is successfully tested on a model of KUKA LWR4+ 7-DOF manipulator; the joint constraints are satisfied and the motion is slowed down only when it is necessary.

An integrated path-tracking and control allocation method for autonomous racing electric vehicles

In recent years, path-tracking controllers for autonomous passenger vehicles and Control Allocation (CA) methods for handling and stability control have both received extensive discussion in the literature. However, the integration of the path-tracking control with CA methods for autonomous racing vehicles has not attracted much attention. In this study, we design an integrated path-tracking and CA method for a prototype autonomous racing electric vehicle with a particular focus on the maximising the turning speed in tight cornering. The proposed control strategy has a hierarchical structure to improve the computational efficiency: the high-level path-tracking Model Predictive Control (MPC) based on a rigid body model is designed to determine the virtual control forces according to the desired path and desired maximum velocity profile, while the low-level CA method uses a Quadratically Constrained Quadratic Programming (QCQP) formulation to distribute the individual control actuator according to the desired virtual control values. The proposed controller is validated in a high-fidelity simulation vehicle model with the computational time of the optimisation controller presented to demonstrate the real-time control performance.

Relative Transformation Estimation Based on Fusion of Odometry and UWB Ranging Data

In this article, we study the problem of estimating the four-degree-of-freedom (3-D position and heading) robot-to-robot relative frame transformation using onboard odometry and interrobot distance measurements. First, we present a theoretical analysis of the problem, namely, the derivation and interpretation of the Cramèr–Rao lower bound, the Fisher information matrix, and its determinant. Second, we propose optimization-based solutions, including a quadratically constrained quadratic programming (QCQP) formulation and its semidefinite programming (SDP) relaxation. Third, based on the theoretical results, we can detect singular configurations as well as measure the uncertainty of each individual parameter. We perform extensive simulations and real-life experiments with aerial robots to show that the proposed QCQP and SDP methods can outperform state-of-the-art approaches, especially in geometrically poor or large measurement noise conditions. In general, the QCQP method provides the best results at the expense of computational time, while the SDP method runs much faster and is sufficiently accurate in most cases.

Linear model decision trees as surrogates in optimization of engineering applications

Machine learning models are promising as surrogates in optimization when replacing difficult to solve equations or black-box type models. This work demonstrates the viability of linear model decision trees as piecewise-linear surrogates in decision-making problems. Linear model decision trees can be represented exactly in mixed-integer linear programming (MILP) and mixed-integer quadratic constrained programming (MIQCP) formulations. Furthermore, they can represent discontinuous functions, bringing advantages over neural networks in some cases. We present several formulations using transformations from Generalized Disjunctive Programming (GDP) formulations and modifications of MILP formulations for gradient boosted decision trees (GBDT). We then compare the computational performance of these different MILP and MIQCP representations in an optimization problem and illustrate their use on engineering applications. We observe faster solution times for optimization problems with linear model decision tree surrogates when compared with GBDT surrogates using the Optimization and Machine Learning Toolkit (OMLT).

Optimal multi-period crop procurement and distribution policy with minimum support prices

Governments of many developing countries offer minimum earnings to farmers through crop Minimum support price (MSP), which guarantees minimum revenue to farmers who cultivate the crop. The crop procured by a government through MSP is later distributed among poor consumers at subsidized prices through public distribution systems (PDS). Thus, governments combine MSP with the PDS scheme to simultaneously ensure farmers’ welfare and tackle poor consumers’ hunger and nutrition problems.However, improper implementation of the schemes causes burgeoning stockpiles in government warehouses, which eventually causes a substantial increase in operational costs and food wastage. In this paper, we characterize the optimal joint MSP and PDS policy over a finite horizon for two objectives: (i) minimize the total cost and (ii) maximize the net value after accounting for farmers’ benefit.Using the data over 15 years, we show that the optimal policy can substantially reduce operational costs (by about 85%) and improve the net value (by about 1.8 trillion INR). We use a linearly constrained quadratic stochastic dynamic program formulation to model the problem. We derive a simple and efficient joint MSP and PDS operational policy that provides the optimal quantity of crop that should be disbursed through PDS in the current period and the optimal MSP that should be announced for the subsequent period.Our policy is easy to compute and implement and can help policymakers efficiently implement MSP and PDS programs jointly. Our model provides a basic framework to solve the joint MSP and PDS problem and can be modified to incorporate other aspects through additional constraints.

Quadratic programming based partitioning for Block Cimmino with correct value representation

The block Cimmino method is successfully used for the parallel solution of large linear systems of equations due to its amenability to parallel processing. Since the convergence rate of block Cimmino depends on the orthogonality between the row blocks, advanced partitioning methods are used for faster convergence. In this work, we propose a new partitioning method that is superior to the state-of-the-art partitioning method, GRIP, in several ways. Firstly, our proposed method exploits the Mongoose partitioning library which can outperform the state-of-the-art methods by combining the advantages of classical combinatoric methods and continuous quadratic programming formulations. Secondly, the proposed method works on the numerical values in a floating-point format directly without converting them to integer format as in GRIP. This brings an additional advantage of obtaining higher quality partitionings via better representation of numerical values. Furthermore, the preprocessing time is also improved since there is no overhead in converting numerical values to integer format. Finally, we extend the Mongoose library, which originally partitions graphs into only two parts, by using the recursive bisection paradigm to partition graphs into more than two parts. Extensive experiments conducted on both shared and distributed memory architectures demonstrate the effectiveness of the proposed method for solving different types of real-world problems.

Constrained multi‐location assortment optimization under the multinomial logit model

AbstractWe study the assortment optimization problem in an online setting where a retailer uses multiple distribution centers (DC) to fulfill orders from multiple regions. Customer choice in each region follows a multinomial logit model. Each DC can carry up to a pre‐specified number of products. Outbound shipping cost to a region depends on the DC that ships the order. The problem is to determine which products to carry in each DC and which products to offer for sale in each region to maximize the expected profit. We first show that the problem is NP‐complete. We develop a conic quadratic mixed integer programming formulation and suggest a family of valid inequalities. We also show that a special case with identical choice models can be solved as a linear program. This LP solution approach can be used to develop heuristics for the general case. Numerical experiments show that our conic approach outperforms the mixed integer linear programming formulation and enables us to solve moderately sized instances optimally. The experiments also show that not allowing cross‐shipments or not considering them in assortment decisions may lead to substantial losses and LP‐based heuristics can be effective in practice.

A Hierarchical Framework of Decision Making and Trajectory Tracking Control for Autonomous Vehicles

Most of the existing research in the field of autonomous vehicles (AVs) addresses decision making, planning and control as separate factors which may affect AV performance in complex driving environments. A hierarchical framework is proposed in this paper to address the problem mentioned above in environments with multiple lanes and surrounding vehicles. Firstly, high-level decision making is implemented by a finite-state machine (FSM), according to the relative relationship between the ego vehicle (EV) and the surrounding vehicles. After the decision is made, a cluster of quintic polynomial equations is established to generate the path connecting the initial position to the candidate target positions, according to the traffic states of the EV and the target vehicle. The optimal path is selected from the cluster, based on the quadratic programming (QP) framework. Then, the speed profile is generated, based on the longitudinal displacement–time graph (S–T graph), considering the vehicle motion state constraints and collision avoidance. The smoothed speed profile is created through another QP formulation, in the space created by the dynamic-programming (DP) method. Finally, the planned path and speed profile are combined and sent to the lower level control module, which consists of the linear quadratic regulator (LQR) for lateral trajectory tracking control, and a double PID controller for longitudinal control. The performance of the proposed framework was validated in a co-simulation environment using PreScan, MATLAB/Simulink and CarSim, and the results demonstrate that the proposed framework is capable of addressing most ordinary scenarios on a structured road, with reasonable decisions and controlling abilities.

Reinforcement Learning Enabled Safety-Critical Tracking of Automated Vehicles with Uncertainties via Integrated Control-Dependent, Time-Varying Barrier Function, and Control Lyapunov Function

Model uncertainties are considered in a learning-based control framework that combines control dependent barrier function (CDBF), time-varying control barrier function (TCBF), and control Lyapunov function (CLF). Tracking control is achieved by CLF, while safety-critical constraints during tracking are guaranteed by CDBF and TCBF. A reinforcement learning (RL) method is applied to jointly learn model uncertainties that related to CDBF, TCBF, and CLF. The learning-based framework eventually formulates a quadratic programming (QP) with different constraints of CDBF, TCBF and CLF involving model uncertainties. It is the first time to apply the proposed learning-based framework for safety-guaranteed tracking control of automated vehicles with uncertainties. The control performances are validated for two different single-lane change maneuvers via Simulink/CarSim® co-simulation and compared for the cases with and without learning. Moreover, the learning effects are discussed through explainable constraints in the QP formulation.

Multi-agent Distributed Model Predictive Control with Connectivity Constraint

In cooperative multi-agent robotic systems, coordination is necessary in order to complete a given task. Important examples include search and rescue, operations in hazardous environments, and environmental monitoring. Coordination, in turn, requires simultaneous satisfaction of safety critical constraints, in the form of state and input constraints, and a connectivity constraint, in order to ensure that at every time instant there exists a communication path between every pair of agents in the network. In this work, we present a model predictive controller that tackles the problem of performing multi-agent coordination while simultaneously satisfying safety critical and connectivity constraints. The former is formulated in the form of state and input constraints and the latter as a constraint on the second smallest eigenvalue of the associated communication graph Laplacian matrix, also known as Fiedler eigenvalue, which enforces the connectivity of the communication network. We propose a sequential quadratic programming formulation to solve the associated optimization problem that is amenable to distributed optimization, making the proposed solution suitable for control of multi-agent robotics systems relying on local computation. Finally, the effectiveness of the algorithm is highlighted with a numerical simulation.

Mixed integer programming and quadratic programming formulations for the interval count problem

A graph is an interval graph if its vertex set corresponds to a family of intervals on the real line, called a model, such that two distinct vertices are adjacent in the graph if and only if their corresponding intervals intersect each other. The minimum number of interval lengths that suffices to represent a model of a given interval graph is its interval count. The use of mathematical optimization techniques for solving interval count problems was first explored by Joos et al.[1]. In more detail, given a bipartition of vertices into classes of lengths, the authors propose an efficient linear programming based algorithm for solving the interval count two problem. However, so far, no mathematical formulation exists in the literature for general interval count. As a contribution in that direction, we introduce a mixed integer programming formulation for the exact value of interval count, parameterized by the largest interval length. Additionally, we also propose a quadratic formulation for a valid upper bound on interval count. Solution algorithms for these formulations were tested on interval count instances found in the literature. As an outcome of these experiments, the algorithm for the upper bound formulation was shown to run much faster than its exact solution counterpart. Furthermore, the upper bounds thus obtained were frequently certified as optimal by the exact algorithm.

An exact solution method and a genetic algorithm-based approach for the unit commitment problem in conventional power generation systems

The unit commitment problem (UCP) is one of the fundamental problems in power systems planning and operations that comprises two decisions: commitment and dispatching of conventional generating units. The objective is to minimize total operating costs -fuel and start-up costs- while satisfying several operational and technical constraints. The UCP is characterized as a highly constrained mixed-integer nonlinear NP-hard problem, which makes it difficult to develop a rigorous optimization method for real-size systems. Hence, we devise an efficient mixed-integer quadratic programming formulation as an exact method with brand-new linear representations for each of the three crucial constraint sets, namely minimum uptime/downtime, start-up and ramp-up/down constraints. Furthermore, to be able to solve a large-scale UCP and to deal with its complexities, we propose a Genetic Algorithm-based matheuristic approach that can provide optimal/near-optimal solutions quickly, thanks to its unique binary-integer coding scheme and several problem-specific operators. During the genetic evolution, commitment and dispatching schedules are determined by combining genetic operations and the Improved Lambda Iteration Method reinforced by the incorporation of average fuel cost optimization and ramp rate limits. The final dispatching schedule is then determined via a start-up adjustment procedure and an efficient quadratic programming model. The computational experiments show that both proposed exact approach and GA-based matheuristic can provide satisfactorily good schedules even for large-scale conventional power systems in quite a reasonable computation time.

FuncPipe: A Pipelined Serverless Framework for Fast and Cost-Efficient Training of Deep Learning Models

Training deep learning (DL) models in the cloud has become a norm. With the emergence of serverless computing and its benefits of true pay-as-you-go pricing and scalability, systems researchers have recently started to provide support for serverless-based training. However, the ability to train DL models on serverless platforms is hindered by the resource limitations of today's serverless infrastructure and DL models' explosive requirement for memory and bandwidth. This paper describes FuncPipe, a novel pipelined training framework specifically designed for serverless platforms that enable fast and low-cost training of DL models. FuncPipe is designed with the key insight that model partitioning can be leveraged to bridge both memory and bandwidth gaps between the capacity of serverless functions and the requirement of DL training. Conceptually simple, we have to answer several design questions, including how to partition the model, configure each serverless function, and exploit each function's uplink/downlink bandwidth. In particular, we tailor a micro-batch scheduling policy for the serverless environment, which serves as the basis for the subsequent optimization. Our Mixed-Integer Quadratic Programming formulation automatically and simultaneously configures serverless resources and partitions models to fit within the resource constraints. Lastly, we improve the bandwidth efficiency of storage-based synchronization with a novel pipelined scatter-reduce algorithm. We implement FuncPipe on two popular cloud serverless platforms and show that it achieves 7%-77% cost savings and 1.3X-2.2X speedup compared to state-of-the-art serverless-based frameworks.

Wind Power Prediction Interval Based on Predictive Density Estimation Within a New Hybrid Structure

Wind power prediction interval (WPPI) is the most common technique to represent wind power (WP) uncertainty. This article proposes a novel WPPI approach developed based on predictive density estimation (DE). Unlike most WPPI models in the literature, the proposed model does not need to solve a high-dimensional optimization problem for model training. It optimizes the WPPIs using a single control variable—the bandwidth (BW) of DE—and trains the model directly and noniteratively using the quantiles extracted from the WP predictive density. For predictive DE, a novel application-specific method has been developed based on generalized cross-entropy (GCE). A precise but straightforward technique is designed to determine the optimal BW that results in the optimal WPPIs. The original GCE-based DE problem is also transformed into a convex quadratic programming formulation that can be solved quickly and uniquely. The WPPI model is employed in a new hybrid deterministic/probabilistic WPP (HDPWP) framework. Different from the conventional HDPWP approach that constructs WPPIs based on the point prediction error, the proposed framework incorporates WP point prediction among the predictor variables in the WPPI model, thereby improving performance. The effectiveness of the proposed methods is confirmed through extensive simulations and comparisons using real-world WP generation datasets.

Maximizing Convergence Time in Network Averaging Dynamics Subject to Edge Removal

We consider the consensus interdiction problem (CIP), in which the goal is to maximize the convergence time of consensus averaging dynamics subject to removing a limited number of network edges. We first show that CIP can be cast as an effective resistance interdiction problem (ERIP), in which the goal is to remove a limited number of network edges to maximize the effective resistance between a source node and a sink node. We show that ERIP is strongly NP-hard, even for bipartite graphs of diameter three with fixed source/sink edges, and establish the same hardness result for the CIP. We then show that both ERIP and CIP cannot be approximated up to a (nearly) polynomial factor assuming the exponential time hypothesis. Subsequently, we devise a polynomial-time $mn$-approximation algorithm for the ERIP that only depends on the number of nodes $n$ and the number of edges $m$ but is independent of the size of edge resistances. Finally, using a quadratic program formulation for the CIP, we devise an iterative approximation algorithm to find a first-order stationary solution for the CIP and evaluate its good performance through numerical experiments.

Linear approximations to improve lower bounds of a physician scheduling problem in emergency rooms

The physician assignment process consists of coverage of shifts and duties allocated to physicians in a planning period, taking into account work regulations, individual preferences, and organizational rules, which mostly conflict with each other. In this work, we propose a reformulated mixed-integer programming model based on the literature to tackle fairness in physician scheduling in Emergency Rooms (ERs). In particular, we propose two mixed-integer quadratic programming formulations that consider quadratic costs and two models with linear costs. Our approaches provide balanced schedules concerning target hours and weekends in terms of fairness. Our models also provide a high degree of demand coverage, providing decision-makers a significant advantage.