Quadratic Programming Formulation Research Articles

A MIQP model for optimal location and sizing of dispatchable DGs in DC networks

The allocation and dimensioning of distributed generators (DGs) in direct current (DC) power grids were addressed in this study by using a mixed-integer quadratic programming (MIQP) formulation. The MIQP model corresponded to an approximation of the mixed-integer nonlinear programming (MINLP) model that represents this problem correctly. The proposed MIQP had, for its objective function, the minimization of the power losses; as constraints, it had power balance, voltage regulation, distributed generation capacity, and the number of DGs available, among others. The general algebraic modeling system (GAMS) was employed for solving the proposed MIQP as well as the MINLP formulation. Simulation results for one DC network with 21 nodes and another with 69 revealed that the proposed MIQP model obtains high-quality results regarding the locations of the generators, the objective function, and the power dispatch in comparison to the exact MINLP model and metaheuristic techniques recently reported in specialized literature.

Alternative approach for efficient OPF calculations in hybrid AC/DC power grids with VSC-HVDC systems based on shift factors

This paper introduces a novel approach for optimal power flows (OPF) in VSC-HVDC power grids based on shift factors. The optimum solution is calculated using a global equality constraint (generation minus losses equals system load) plus those related to the power set points of the VSC units. Indeed, the shift factors enable the enforcement of the VSC power flow constraints. This is the hallmark of this method which features a considerable reduction of the model complexity and the computational burden because the system nodal equations are not explicitly represented. This lies in sharp contrast with existing OPF approaches whose model complexity is inevitably increased as is the computing requirements to its solution.Three variants of the proposed OPF model are described: (i) quadratic generation cost curves with quadratic AC/DC transmission losses; (ii) quadratic generation cost curves with power losses considered by piecewise linear equations; (iii) generation cost curves and transmission losses approximated by piecewise linear functions. The effectiveness of the developed approach is assessed using three AC/DC test systems formed by three, four and six VSC stations, respectively. The results are compared against those calculated by a conventional Sequential Quadratic Programming formulation using a full, nonlinear power grid model. It is demonstrated that the optimal solutions concur very well between each other since errors inferior to 5% are obtained for the generation dispatch and nodal energy prices. Equally important is the fact that this novel approach reduces the computational time by more than 70% on average.

Comparing temporal graphs using dynamic time warping

Within many real-world networks, the links between pairs of nodes change over time. Thus, there has been a recent boom in studying temporal graphs. Recognizing patterns in temporal graphs requires a proximity measure to compare different temporal graphs. To this end, we propose to study dynamic time warping on temporal graphs. We define the dynamic temporal graph warping (dtgw) distance to determine the dissimilarity of two temporal graphs. Our novel measure is flexible and can be applied in various application domains. We show that computing the dtgw-distance is a challenging (in general) NP-hard optimization problem and identify some polynomial-time solvable special cases. Moreover, we develop a quadratic programming formulation and an efficient heuristic. In experiments on real-world data, we show that the heuristic performs very well and that our dtgw-distance performs favorably in de-anonymizing networks compared to other approaches.

Quadratically Constrained Quadratic Programming Formulation of Contingency Constrained Optimal Power Flow with Photovoltaic Generation

Contingency Constrained Optimal Power Flow (CCOPF) differs from traditional Optimal Power Flow (OPF) because its generation dispatch is planned to work with state variables between constraint limits, considering a specific contingency. When it is not desired to have changes in the power dispatch after the contingency occurs, the CCOPF is studied with a preventive perspective, whereas when the contingency occurs and the power dispatch needs to change to operate the system between limits in the post-contingency state, the problem is studied with a corrective perspective. As current power system software tools mainly focus on the traditional OPF problem, having the means to solve CCOPF will benefit power systems planning and operation. This paper presents a Quadratically Constrained Quadratic Programming (QCQP) formulation built within the matpower environment as a solution strategy to the preventive CCOPF. Moreover, an extended OPF model that forces the network to meet all constraints under contingency is proposed as a strategy to find the power dispatch solution for the corrective CCOPF. Validation is made on the IEEE 14-bus test system including photovoltaic generation in one simulation case. It was found that in the QCQP formulation, the power dispatch calculated barely differs in both pre- and post-contingency scenarios while in the OPF extended power network, node voltage values in both pre- and post-contingency scenarios are equal in spite of having different power dispatch for each scenario. This suggests that both the QCQP and the extended OPF formulations proposed, could be implemented in power system software tools in order to solve CCOPF problems from a preventive or corrective perspective.

Random projections for quadratic programs

Random projections map a set of points in a high dimensional space to a lower dimensional one while approximately preserving all pairwise Euclidean distances. Although random projections are usually applied to numerical data, we show in this paper that they can be successfully applied to quadratic programming formulations over a set of linear inequality constraints. Instead of solving the higher-dimensional original problem, we solve the projected problem more efficiently. This yields a feasible solution of the original problem. We prove lower and upper bounds of this feasible solution w.r.t. the optimal objective function value of the original problem. We then discuss some computational results on randomly generated instances, as well as a variant of Markowitz’ portfolio problem. It turns out that our method can find good feasible solutions of very large instances.

Computing Smooth Quasi-geodesic Distance Field (QGDF) with Quadratic Programming

Computing geodesic distances on polyhedral surfaces is an important task in digital geometry processing. Speed and accuracy are two commonly-used measurements of evaluating a discrete geodesic algorithm. In applications, such as parametrization and shape analysis, a smooth distance field is often preferred over the exact, non-smooth geodesic distance field. We use the term Quasi-geodesic Distance Field (QGDF) to denote a smooth scalar field that is as close as possible to an exact geodesic distance field. In this paper, we formulate the problem of computing QGDF into a standard quadratic programming (QP) problem which maintains a trade-off between accuracy and smoothness. The proposed QP formulation is also flexible in that it can be naturally extended to point clouds and tetrahedral meshes, and support various user-specified constraints. We demonstrate the effectiveness of QGDF in defect-tolerant distances and symmetry-constrained distances.

Algorithm 1003

Partitioning graphs is a common and useful operation in many areas, from parallel computing to VLSI design to sparse matrix algorithms. In this article, we introduce Mongoose, a multilevel hybrid graph partitioning algorithm and library. Building on previous work in multilevel partitioning frameworks and combinatoric approaches, we introduce novel stall-reducing and stall-free coarsening strategies, as well as an efficient hybrid algorithm leveraging (1) traditional combinatoric methods and (2) continuous quadratic programming formulations. We demonstrate how this new hybrid algorithm outperforms either strategy in isolation, and we also compare Mongoose to METIS and demonstrate its effectiveness on large and social networking (power law) graphs.

QMCM: Minimizing Vapnik’s bound on the VC dimension

The recently proposed Minimal Complexity Machine (MCM) learns a hyperplane classifier by minimizing a bound on the Vapnik-Chervonenkis (VC) dimension. Both the linear and kernel versions of the MCM solve a linear programming problem, in order to minimize a bound on the VC dimension. This paper proposes a new quadratic programming formulation, termed as the Quadratic MCM (QMCM), that minimizes a tighter bound on the VC dimension. We present two variants of the QMCM, that differ in the norm of the error vector being minimized. We also explore a scalable variant of the QMCM for large datasets using Stochastic Gradient Descent (SGD), and present the use of the QMCM as a viable featureselection method, in view of the the sparse nature of the models it learns. We compare the performance of the QMCM variants with LIBLinear, a linear Support Vector Machine (SVM) library; as well as against Pegasos and the linear MCM for large datasets, along with sequential feature selection methods and ReliefF. Our results validate the superiority of the QMCM in terms of statistically significant improvements on benchmark datasets from the UCI Machine Learning repository.

Nonlinear Model Predictive Control applied to Concentrated Solar Power Plants

This papers presents a nonlinear model predictive controller (NMPC) for temperature control in solar collector fields. The proposed NMPC uses feedback linearization for handling the systems nonlinearities in a mixed integer quadratic programming (MIQP) formulation that makes the constraints convex in the new coordinates, making the controller solution an optimal one. Several simulations are shown with real data and a validated model of solar field to illustrate the advantages of the proposed strategy over other general-purpose NMPC methods, showing great improvement in constraint satisfaction.

Tail Risks in Large Portfolio Selection: Penalized Quantile and Expectile Minimum Deviation Models

Accurate estimation and optimal control of tail risk is important for building portfolios with desirable properties, especially when dealing with a large set of assets. In this work, we consider optimal asset allocations strategies based on the minimization of two asymmetric deviation measures, related to quantile and expectile regression, respectively. Their properties are discussed in relation with the `risk quadrangle' framework introduced by Rockafellar et al (2013) and compared to traditional strategies, such as the mean-variance portfolio. In order to control estimation error and improve the out-of-sample performances of the proposed models, we include ridge and elastic-net regularization penalties. Finally, we propose quadratic programming formulations for the optimization problems. Simulations and real-world analyses on multiple datasets, allow to discuss pros and cons of the different methods. The results show that the ridge and elastic-net allocations are effective in improving the out-of-sample performances, especially in large portfolios, compared to the un-penalized ones.

Data maximum dispersion classifier in projection space for high-dimension low-sample-size problems

Various applications in different fields, such as gene expression analysis or computer vision, suffer from data sets with high-dimensional low-sample-size (HDLSS), which has posed significant challenges for standard statistical and modern machine learning methods. In this paper, we propose a novel linear binary classifier, denoted by Data Maximum Dispersion classifier (DMDC), which is applicable to general data, especially HDLSS such as genetic microarrays, image and so on. DMDC maximize data dispersion in projection space to find the hyperplane for classification. Our proposed model has several advantages compared to those widely used approaches. First, it works well on HDLSS. Second, it is not sensitive to the intercept term b. Third, the implement of DMDC is easy and holds low computational complexity owing to solve the similar Convex Quadratic Programming formulation as in SVM, rather than the SOCP in the DWD related methods. Fourth, it is robust to the model specification for various real applications. We conduct the evaluations on one simulated and seven real-world benchmark data sets, including face classification, radiomics and mRNA classification. The experiment results manifested the superiority of DMDC compared to the state-of-art algorithms in most cases, or at least obtains comparable results.

Reverse Blending: An economically efficient approach to the challenge of fertilizer mass customization

Reasoned fertilization, which is a major concern for sustainable and efficient agriculture, consists of applying customized fertilizers which requires a very large increase in the number of fertilizer formulae, involving increasing costs due to the multiplication of production batch, of storage areas and of transportation constraints. An alternative solution is given by adopting a Reverse Blending approach, which is a new Blending Problem where inputs are non-pre-existing composite materials that need to be defined in both number and composition, simultaneously with the quantities to be used in the blending process, such as to meet the specifications of a wide variety of outputs, while keeping their number as small as possible. This would replace the production of a large variety of small batches of fertilizers by few large batches of new composite materials whose blending may be performed close to end-users (delayed differentiation), delivering substantial production and logistics cost savings, well in excess of remote blending costs. Reverse Blending presents some analogies with the Pooling Problem which is a two-stage Blending Problem where primary inputs are existing raw materials. An adapted version of this problem may be used to facilitate the design of new composite materials used by Reverse Blending. This paper presents the Reverse Blending approach, whose modelling is based on a quadratic programming formulation, and a large case study to demonstrate its feasibility. Reverse Blending, therefore, may be a disruptive approach to successfully reengineer not only the fertilizer supply chain but any other industry operating in blending contexts to meet a great diversity.

QpSWIFT: A Real-Time Sparse Quadratic Program Solver for Robotic Applications

In this letter, we present qpSWIFT, a real-time quadratic program (QP) solver. Motivated by the need for a robust embedded QP solver in robotic applications, qpSWIFT employs standard primal-dual interior-point method, along with Mehrotra predictor–corrector steps and Nesterov–Todd scaling. The sparse structure of the resulting Karush–Kuhn–Tucker linear system in the QP formulation is exploited, and sparse direct methods are utilized to solve the linear system of equations. To further accelerate the factorization process, we only modify the corresponding rows of the matrix factors that change during iterations and cache the nonzero Cholesky pattern. qpSWIFT is library free, written in ANSI-C and its performance is benchmarked through standard problems that could be cast as QP. Numerical results show that qpSWIFT outperforms state-of-the-art solvers for small scale problems. To evaluate the performance of the solver, a real-time implementation of the solver in the model predictive control framework through experiments on a quadrupedal robot are presented.

Interference Exploitation Precoding for Multi-Level Modulations: Closed-Form Solutions

We study closed-form interference-exploitation precoding for multi-level modulations in the downlink of multi-user multiple-input single-output (MU-MISO) systems. We consider two distinct cases: first, when the number of served users is not larger than the number of transmit antennas at the base station (BS), we mathematically derive the optimal precoding structure based on the Karush-Kuhn-Tucker (KKT) conditions. By formulating the dual problem, the precoding problem is transformed into a pre-scaling operation using quadratic programming (QP) optimization. We further consider the case where the number of served users is larger than the number of transmit antennas at the BS. By employing the pseudo inverse, we show that the optimal solution of the pre-scaling vector is equivalent to a linear combination of the right singular vectors corresponding to zero singular values, and derive the equivalent QP formulation. We also present the condition under which multiplexing more streams than the number of transmit antennas is achievable. For both considered scenarios, we propose a modified iterative algorithm to obtain the optimal precoding matrix, as well as a sub-optimal closed-form precoder. Numerical results validate our derivations on the optimal precoding structures for multi-level modulations, and demonstrate the superiority of interference-exploitation precoding for both scenarios.

Safe Trajectory Generation for Complex Urban Environments Using Spatio-Temporal Semantic Corridor

Planning safe trajectories for autonomous vehicles in complex urban environments is challenging since there are numerous semantic elements (such as dynamic agents, traffic lights, and speed limits) to consider. These semantic elements may have different mathematical descriptions, such as obstacle, constraint, and cost. It is non-trivial to tune the effects from different combinations of semantic elements for a stable and generalizable behavior. In this letter, we propose a novel unified spatio-temporal semantic corridor (SSC) structure, which provides a level of abstraction for different types of semantic elements. The SSC consists of a series of mutually connected collision-free cubes with dynamical constraints posed by the semantic elements in the spatio-temporal domain. The trajectory generation problem then boils down to a general quadratic programming formulation. Thanks to the unified SSC representation, our framework can generalize to any combination of semantic elements. Moreover, our formulation provides a theoretical guarantee that the entire trajectory is safe and constraint-satisfied, by using the convex hull and hodograph properties of piecewise Bezier curve parameterization. We also release the code of our method to accommodate benchmarking.

Path-Based Distribution Feeder Reconfiguration for Optimization of Losses and Reliability

This paper presents a path-based modeling framework for the distribution feeder reconfiguration (DFR) problem. The framework maps the decision variables— on/off status indicators for the paths—into linear expressions for the network flows and reliability indices. These linear expressions are suitably deployed for DFR optimization, where reliability can feature either as an objective or as a constraint. Additionally, the usual objective of loss minimization can also be incorporated, leading to a multi-objective DFR optimization framework. In its most general form, the proposed framework is a mixed-integer quadratic programming formulation, and can be solved using existing solvers. The paper makes three other important contributions: first, a probabilistic approach is adopted to validate the approximation of load point failure rates as the sum of failure rates of upstream components; second, a graph algorithm is proposed for identifying topologies with best reliability; and third, simulation studies with representative loading patterns are used to identify the appropriate choice of loading conditions for optimizing overall active power losses. Application of the proposed framework to standard test systems demonstrates its ability to optimize network losses or reliability or both. Numerical results also show the effect of topology and loading conditions on the performance of optimized topologies.

A branch and price algorithm for single-machine completion time variance

This paper studies a single machine scheduling problem to minimize the completion time variance (CTV) of all jobs. A new time-indexed quadratic programming formulation is proposed, in which the quadratic terms are further linearized into a mixed integer programming (MIP) with non-negative continuous variables. To solve the problem to optimality, a branch and price (BP) algorithm is designed, which combines the Lagrangian relaxation with clusters. Extensive computational experiments on three sets of benchmark problem instances and randomly generated ones are conducted and the results show that the proposed BP algorithm outperforms state-of-the-art approaches.

On the Optimal Design of a Bipartite Matching Queueing System

We consider a multi-class multi-server queueing system and study the problem of designing an optimal matching topology (or service compatibility structure) between customer classes and servers under a FCFS-ALIS service discipline. Specifically, we are interested in finding matching topologies that optimize --in a Pareto efficiency-- sense the trade-off between two competing objectives: (i) minimizing customers' waiting time delays and (ii) maximizing matching rewards generated by pairing customers and servers. Our analysis of the problem is divided in three main parts. First, under heavy-traffic conditions, we show that any bipartite matching system can be partitioned into a collection of complete resource pooling (CRP) subsystems, which are interconnected by means of a direct acyclic graph (DAG). We show that this DAG together with the aggregate service capacity on each CRP component fully determine the vector of steady-state waiting times. In particular, we show that the average (scaled) steady-state delay across all customer classes is asymptotically equal to the number of CRP components divided by the total system capacity. Second, since computing matching rewards under a FCFS-ALIS service discipline is computationally infeasible as the number of customer classes and servers grow large, we propose a quadratic programming (QP) formulation to approximate matching rewards. We show that the QP formulation is exact for a number of instances of the problem and provides a very good approximation in general. Extensive numerical experiments show that in over 98% of problem instances the relative error between the exact rewards and the QP approximate rewards is less than 2%. Lastly, combining our characterization of average delays in terms of the number of CRP components and the quadratic programming formulation to compute matching rewards, we propose a mixed-integer linear program (MILP) that can be used to find the set of matching topologies that define the Pareto frontier of reward-delay pairs.

Model-Predictive Control Strategy for an Array of Wave-Energy Converters

To facilitate the commercialization of wave energy in an array or farm environment, effective control strategies for improving energy extraction efficiency of the system are important. In this paper, we develop and apply model-predictive control (MPC) to a heaving point-absorber array, where the optimization problem is cast into a convex quadratic programming (QP) formulation, which can be efficiently solved by a standard QP solver. We introduced a term for penalizing large slew rates in the cost function to ensure the convexity of this function. Constraints on both range of the states and the input capacity can be accommodated. The convex formulation reduces the computational hurdles imposed on conventional nonlinear MPC. For illustration of the control principles, a point-absorber approximation is adopted to simplify the representation of the hydrodynamic coefficients among the array by exploiting the small devices to wavelength assumption. The energy-capturing capabilities of a two-cylinder array in regular and irregular waves are investigated. The performance of the MPC for this two-WEC array is compared to that for a single WEC, and the behavior of the individual devices in head or beam wave configuration is explained. Also shown is the reactive power required by the power takeoff system to achieve the performance.

Mixed-integer programming formulation of a data-driven solver in computational elasticity

This paper presents a mixed-integer quadratic programming formulation of an existing data-driven approach to computational elasticity. This formulation is suitable for application of a standard mixed-integer programming solver, which finds a global optimal solution. Therefore, the results obtained by the presented method can be used as benchmark instances for any other algorithm. Preliminary numerical experiments are performed to compare quality of solutions obtained by the proposed method and a heuristic used in the data-driven computational mechanics.