Polyhedral Regions Research Articles

Analysis of Structural Parameters for a Multi-Row Inclined Side Hole Muzzle Brake in an Airplane Gun

In response to the low efficiency of a muzzle brake for a certain 30 mm airplane gun, a new type of muzzle brake with side ports tilted towards the rear of the barrel is proposed. Taking the flow state of the internal gas at the moment the projectile reaches the muzzle as the initial condition, a dynamic mesh method combining polyhedral stationary region and hexahedral structured mesh moving region is used to perform three-dimensional numerical simulation of the muzzle brake flow field considering the motion of the projectile. The influence of the number of side port rows and the angle of inclination of the side ports on the brake efficiency is analyzed. The results show that within a certain range, the brake efficiency increases with the increase in the number of side ports, but the increment gradually narrows. When the number of side port rows is 6, the brake efficiency can be increased to 42.3%. With the increase of the side hole inclination Angle, the braking efficiency increases first and then decreases. The optimal angle for this muzzle brake is between 115 degrees and 135 degrees. This study provides a reference for accurate efficiency calculation, design, and optimization of the new multi-row inclined side port muzzle brake.

Certified polyhedral decompositions of collision-free configuration space

Understanding the geometry of collision-free configuration space (C-free) in the presence of Cartesian-space obstacles is an essential ingredient for collision-free motion planning. While it is possible to check for collisions at a point using standard algorithms, to date no practical method exists for computing C-free regions with rigorous certificates due to the complexity of mapping Cartesian-space obstacles through the kinematics. In this work, we present the first to our knowledge rigorous method for approximately decomposing a rational parametrization of C-free into certified polyhedral regions. Our method, called C-Iris (C-space Iterative Regional Inflation by Semidefinite programming), generates large, convex polytopes in a rational parameterization of the configuration space which are rigorously certified to be collision-free. Such regions have been shown to be useful for both optimization-based and randomized motion planning. Based on convex optimization, our method works in arbitrary dimensions, only makes assumptions about the convexity of the obstacles in the 3D Cartesian space, and is fast enough to scale to realistic problems in manipulation. We demonstrate our algorithm’s ability to fill a non-trivial amount of collision-free C-space in several 2-DOF examples where the C-space can be visualized, as well as the scalability of our algorithm on a 7-DOF KUKA iiwa, a 6-DOF UR3e, and 12-DOF bimanual manipulators. An implementation of our algorithm is open-sourced in Drake. We furthermore provide examples of our algorithm in interactive Python notebooks.

Generalization Bounds in the Predict-Then-Optimize Framework

The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation. Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship.

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

Non-Convex Optimization: Using Preconditioning Matrices for Optimally Improving Variable Bounds in Linear Relaxations

The performance of branch-and-bound algorithms for solving non-convex optimization problems greatly depends on convex relaxation techniques. They generate convex regions which are used for improving the bounds of variable domains. In particular, convex polyhedral regions can be represented by a linear system A.x=b. Then, bounds of variable domains can be improved by minimizing and maximizing variables in the linear system. Reducing or contracting optimally variable domains in linear systems, however, is an expensive task. It requires solving up to two linear programs for each variable (one for each variable bound). Suboptimal strategies, such as preconditioning, may offer satisfactory approximations of the optimal reduction at a lower cost. In non-square linear systems, a preconditioner P can be chosen such that P.A is close to a diagonal matrix. Thus, the projection of the equivalent system P.A.x=P.b over x, by using an iterative method such as Gauss–Seidel, can significantly improve the contraction. In this paper, we show how to generate an optimal preconditioner, i.e., a preconditioner that helps the Gauss–Seidel method to optimally reduce the variable domains. Despite the cost of generating the preconditioner, it can be re-used in sub-regions of the search space without losing too much effectiveness. Experimental results show that, when used for reducing domains in non-square linear systems, the approach is significantly more effective than Gauss-based elimination techniques. Finally, the approach also shows promising results when used as a component of a solver for non-convex optimization problems.

Delfin++: A Delaunay based algorithm for finding 3D polyhedral voids in galaxy surveys

Delfin++ (Delaunay Edge Void Finder) is a simple and time-efficient algorithm, with a single input variable, designed to find cosmological voids within a 3-dimensional distribution of galaxies by characterizing them as polyhedral regions from a Delaunay tessellation. Two density metrics are defined and used to search for density minima and construct polyhedra that represent the underdense zones. A density threshold is used to define the limit above which a galaxy will not be considered part of a void. This threshold is commonly defined as 0.2n̄, where n̄ is the mean point density of the sample, whereas the values used in this work range from 0.2n̄ to 0.33n̄. The algorithm is applied to artificial data, with different density contrasts, as well as to galaxy data from SDSS DR10. The detected voids are compared with voids that have been produced through the application of vide. Our algorithm detects roundish underdense regions in the galaxy samples, with differences and overlaps compared to the vide algorithm’s results. Closer agreement is revealed when the edge length density metric is used over large galaxy samples, allowing for the detection of up to 90% of the largest vide voids. We measure ellipticities of Delfin++ detected voids and find that their distribution is shifted towards smaller values in comparison with vide’s distribution, and with predictions from an analytical model. The voids found with Delfin++ are not intended to generate a catalog, but rather represent a proof of concept of a simpler algorithm or with fewer rules for further calibration and future production of a final catalog. Delfin++ performance tests suggest that enhanced characterization of voids can be achieved through (i) choice of other density threshold values, and (ii) early recognition and rejection of deformed polyhedra induced by missing data.

Xrd_simulator: 3D X-ray diffraction simulation software supporting 3D polycrystalline microstructure morphology descriptions.

An open source Python package named xrd_simulator, capable of simulating geometrical interactions between a monochromatic X-ray beam and a polycrystalline microstructure, is described and demonstrated. The software can simulate arbitrary intragranular lattice variations of single crystals embedded within a multiphase 3D aggregate by making use of a tetrahedral mesh representation where each element holds an independent lattice. By approximating the X-ray beam as an arbitrary convex polyhedral region in space and letting the sample be moved continuously through arbitrary rigid motions, data from standard and non-standard measurement sequences can be simulated. This implementation is made possible through analytical solutions to a modified, time-dependent version of the Laue equations. The software, which primarily targets three-dimensional X-ray diffraction microscopy (high-energy X-ray diffraction microscopy) type experiments, enables the numerical exploration of which sample quantities can and cannot be reconstructed for a given acquisition scheme. Similarly, xrd_simulator targets investigations of different measurement sequences in relation to optimizing both experimental run times and sampling.

Complexity reduction of explicit MPC based on fuzzy reshaped polyhedrons for use in industrial controllers

The explicit model predictive control (EMPC) generates the rules of control defined for a set of polyhedral regions. Online EMPC calculations consist of searching a look-up table to find the appropriate control law according to a particular state. This paper discusses the complexity of online computation and the memory required to store data in an EMPC implementation. Therefore, a new reshaping method is applied to the active regions so that the definition of the polyhedron has regular boundaries. This approach has made some improvements. First, the usable memory will be a lot less for the actual implementation compared to the traditional EMPC approach. Second, the small number of new clusters reduces search time in explicit lookup tables and speeds up overall implementation. To this end, fuzzy clustering is used to introduce a novel method of transforming polyhedrons in the context of fuzzy explicit model predictive (FEMPC) control, followed by a new fuzzy-based piece-wise affine (PWA) explicit formulation for control law calculations. The stability of the proposed method is investigated using the Lyapunov stability criteria. The proposed algorithm has been tested on a nonlinear continuous stirred tank reactor (CSTR) benchmark system and simulation tests show that the proposed approach involves a compromise between storage space requirements and online efficiency.

An efficient application of particle swarm optimization in model predictive control of constrained two-tank system

<span>Despite all the model predictive control (MPC) based solution advantages such as a guarantee of stability, the main disadvantage such as an exponential growth of the number of the polyhedral region by increasing the prediction horizon exists. This causes the increment in computation complexity of control law. In this paper, we present the efficiency of particle swarm optimization (PSO) in optimal control of a two-tank system modeled as piecewise affine. The solution of the constrained final time-optimal control problem (CFTOC) is derived, and then the PSO algorithm is used to reduce the computational complexity of the control law and set the physical parameters of the system to improve performance simultaneously. On the other hand, a new combined algorithm based on PSO is going to be used to reduce the complexity of explicit MPC-based solution CFTOC of the two-tank system; consequently, that the number of polyhedral is minimized, and system performance is more desirable simultaneously. The proposed algorithm is applied in simulation and our desired subjects are reached. The number of control law polyhedral reduces from 42 to 10 and the liquid height in both tanks reaches the desired certain value in 189 seconds. Search time and apply control law in 25 seconds.</span>

Disjoint Bilinear Optimization: A Two-Stage Robust Optimization Perspective

In this paper, we focus on a subclass of quadratic optimization problems, that is, disjoint bilinear optimization problems. We first show that disjoint bilinear optimization problems can be cast as two-stage robust linear optimization problems with fixed-recourse and right-hand-side uncertainty, which enables us to apply robust optimization techniques to solve the resulting problems. To this end, a solution scheme based on a blending of three popular robust optimization techniques is proposed. For disjoint bilinear optimization problems with a polyhedral feasible region and a general convex feasible region, we show that, under mild regularity conditions, the convex relaxations of the original bilinear formulation and its two-stage robust reformulation obtained from a reformulation-linearization-based technique and linear decision rules, respectively, are equivalent. For generic bilinear optimization problems, the convex relaxations from the reformulation-linearization-based technique are generally tighter than the one from linear decision rules. Numerical experiments on bimatrix games, synthetic disjoint bilinear problem instances, and convex maximization problems demonstrate the efficiency and effectiveness of the proposed solution scheme. Summary of Contribution: Computing solutions for disjoint bilinear optimization problems are of much interest in real-life applications, yet they are, in general, computationally intractable. This paper proposes a computationally tractable approximation as well as a convergent algorithm to the optimal values of such problems. Extensive computational experiments on (i) (constrained) bimatrix games, (ii) synthetic disjoint bilinear problems, and (iii) convex maximization problems are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

Fast computation of binary search tree for PWA functions representation using intersection classification

Explicit Model Predictive Control (eMPC) aims to overcome the runtime computational effort from Model Predictive Control (MPC) by computing offline the overall optimization procedure. The result from eMPC is a Piecewise Affine (PWA) function that defines a relationship between the system state and the optimal control action. To improve the PWA evaluation during runtime, Binary Search Trees (BST) are employed to represent PWA functions, which may be time prohibitive considering the total time to obtain the tree. This note presents a new classification strategy of polyhedral regions with respect to hyperplanes to build BST for PWA functions. The proposed solution is based on the verification of intersection between regions and hyperplanes, resulting in a decrease by almost half in the number of optimization problems, and consequently, in the total time to build the BST, which is the main time-consuming task. The computational time improvement is verified on numerical examples of different sizes, where the same BST is obtained for the original and the proposed classification methods.

A posteriori error analysis of a momentum conservative Banach spaces based mixed-FEM for the Navier–Stokes problem

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

A new deterministic global computing algorithm for solving a kind of linear fractional programming

This paper investigates a class of linear fractional programming (LFP) problem, which minimizes the sum of a finite number of linear fractional functions over a polyhedral region. Firstly, the equivalence problem (EP) of the LFP problem is given by a new two-stage transformation method. Secondly, considering the characteristics that the branch-and-bound algorithm can guarantee the global optimality of the solution to an optimization problem, and then based on the EP, we discuss the bounding operation, branching operation, pruning operation and rectangle-region reduction technique of this algorithm. After that, the convergence of the algorithm is proved and its computational complexity is deduced from the worst case. Finally, some experiments are reported to verify the effectiveness, feasibility and other performance of the proposed algorithm.

Super-k: A piecewise linear classifier based on Voronoi tessellations

Voronoi tessellations are used to partition the Euclidean space into polyhedral regions, which are called Voronoi cells. Labeling the Voronoi cells with the class information, we can map any classification problem into a Voronoi tessellation. In this way, the classification problem changes into a query of just finding the enclosing Voronoi cell. In order to accomplish this task, we have developed a new algorithm which generates a labeled Voronoi tessellation that partitions the training data into polyhedral regions and obtains interclass boundaries as an indirect result. It is called Supervised k-Voxels or in short Super-k. We are introducing Super-k as a foundational new algorithm and opening the possibility of a new family of algorithms. In this paper, it is shown via comparisons on certain datasets that the Super-k algorithm has the potential of providing comparable performance of the well-known SVM family of algorithms.

LMI conditions for evaluating and increasing stability margins of uncertain continuous-time linear systems

This article investigates the robust stability of uncertain continuous-time linear systems with parameters belonging to intervals. Differently from the techniques available in the literature, the proposed method, formulated in terms of an iterative algorithm based on LMIs, treats the interval bounds as optimization variables and can be used, for instance, to search for the maximum variation around nominal values. Experiments involving the fragility analysis of controllers and volume maximization of the polyhedral region centered on nominal parameters are presented to illustrate possible applications of the proposed technique.

Polyhedral Regions of Stability for Aperiodic Sampled-Data Linear Control Systems With Saturating Inputs

This work proposes a method to assess the local asymptotic stability and to provide polyhedral estimates of the region of attraction of the origin (RAO) of linear systems under aperiodic sampled-data control and saturating inputs. The approach is based on a discrete-time model that describes the behavior of the system state between consecutive sampling instants. It corresponds to a difference inclusion defined from a partition of the intersampling interval and from the saturated and nonsaturated (SNS) embedding of saturation functions. A method to construct a contractive polyhedral set for this model is proposed. It is shown that this set induces a local Lyapunov function strictly decreasing at the sampling instants and that it is an estimate of the RAO of the continuous-time closed-loop system.

Learning the Optimal Strategy of Power System Operation With Varying Renewable Generations

Optimal dispatch of modern power systems often entails efficiently solving large-scale optimization problems, especially when generators have to respond to the fast fluctuation of renewable generation. This paper develops a method to learn the optimal strategy from a mixed-integer quadratic program with time-varying parameters, which can model many power system operation problems such as unit commitment and optimal power flow. Different from existing machine learning methods that learn a map from the parameter to the optimal action, the proposed method learns the map from the parameter to the optimal integer solution and the optimal basis, forming a discrete pattern. Such a framework naturally gives rise to a classification problem: the parameter set is partitioned into polyhedral regions; in each region, the optimal 0-1 variable and the set of active constraints remain unchanged, and the optimal continuous variables are affine functions in the parameter. The outcome of classification is compared with analytical results derived from multi-parametric programming theory, showing interesting connections between traditional mathematical programming theory and the interpretability of the learning-based method. Tests on a small-scale problem demonstrate the partition of the parameter set learned from data meets the theoretical outcome. More tests on the IEEE 57-bus system and a real-world 1881-bus system validate the performance of the proposed method with a high-dimensional parameter for which the analytical method is intractable.

Online Identification of Piecewise Affine Systems Using Integral Concurrent Learning

Piecewise affine (PWA) systems are attractive models that can represent various hybrid systems with local affine subsystems and polyhedral regions due to their universal approximation properties. The identification problem of PWA systems amounts to estimating the number of subsystems, parameters of each subsystem, and the corresponding polyhedral partitions via state-input vectors. In this paper, we propose a novel approach to address the online identification problem of continuous-time PWA systems in state-space form. Specifically, an online active mode recognition algorithm and a generalized integral concurrent learning identifier are presented to acquire the number of subsystems, the switching sequence, and the parameter of each subsystem. In addition, we develop the optimization problem for the polyhedral partition estimation, which is solved by using the estimated switching sequence and subsystem parameters. The effectiveness of the proposed identification approach is demonstrated via simulation results.

On the Quasiconcave Multilevel Programming Problems

Multilevel programming appears in many decision-making situations. Investigation of the main properties of quasiconcave multilevel programming (QCMP) problems, to date, is limited to bilevel programming (only two levels). In this paper, first, we present an extension of the properties of quasiconcave bilevel programming (QCBP) problems for the case when three levels exist. Then, by induction on n (the number of levels), we prove the existence of an extreme point of the polyhedral constraint region that solves the QCMP problem under given conditions. Ultimately, a number of numerical examples are illustrated to verify the results.

Using geometric mean to compute robust mixture designs

AbstractMixture experiments involve developing a dedicated formulation for specific applications. We propose the weighted optimality criterion using the geometric mean as the objective function for the genetic algorithms. We generate a robust mixture design using genetic algorithms (GAs) of which the region of interest is an irregularly shaped polyhedral region formed by constraints on proportions of the mixture component. When specific terms in the initial model display unimportant effects, it is assumed that they are removed. The design generation objective requires model robustness across the set of the reduced models of the design. Proposing an alternative way to tackle the problem, we find that the proposed GA designs based on G‐ or/and IV‐efficiency are robust to model misspecification.