Geometric Optimization Problems Research Articles

Streaming Algorithms for Geometric Steiner Forest

We consider a generalization of the Steiner tree problem, the Steiner forest problem , in the Euclidean plane: the input is a multiset \(X\subseteq{\mathbb{R}}^{2}\) , partitioned into \(k\) color classes \(C_{1},\ldots,C_{k}\subseteq X\) . The goal is to find a minimum-cost Euclidean graph \(G\) such that every color class \(C_{i}\) is connected in \(G\) . We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to \(X\) . Each input point \(x\in X\) arrives with its color \(\mathsf{color}(x)\in[k]\) , and as usual for dynamic geometric streams, the input is restricted to the discrete grid \(\{1,\ldots,\Delta\}^{2}\) . We design a single-pass streaming algorithm that uses \(\operatorname{poly}(k\cdot\log\Delta)\) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio \(\alpha_{2}\) (currently \(1.1547\leq\alpha_{2}\leq 1.214\) ). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when \(k=1\) , and it is a major open question to improve the ratio to \(1+\varepsilon\) even for this special case. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and so far has not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite multiplicative approximation requires \(\Omega(k)\) bits of space.

NN-Steiner: A Mixed Neural-Algorithmic Approach for the Rectilinear Steiner Minimum Tree Problem

Recent years have witnessed rapid advances in the use of neural networks to solve combinatorial optimization problems. Nevertheless, designing the "right" neural model that can effectively handle a given optimization problem can be challenging, and often there is no theoretical understanding or justification of the resulting neural model. In this paper, we focus on the rectilinear Steiner minimum tree (RSMT) problem, which is of critical importance in IC layout design and as a result has attracted numerous heuristic approaches in the VLSI literature. Our contributions are two-fold. On the methodology front, we propose NN-Steiner which is a novel mixed neural-algorithmic framework for computing RSMTs that leverages the celebrated PTAS algorithmic framework of Arora to solve this problem (and other geometric optimization problems). Our NN-Steiner replaces key algorithmic components within Arora's PTAS by suitable neural components. In particular, NN-Steiner only needs four neural network (NN) components that are called repeatedly within an algorithmic framework. Crucially, each of the four NN components is only of bounded size independent of input size, and thus easy to train. Furthermore, as the NN component is learning a generic algorithmic step, once learned, the resulting mixed neural-algorithmic framework generalizes to much larger instances not seen in training. Our NN-Steiner, to our best knowledge, is the first neural architecture of bounded size that has capacity to approximately solve RSMT (and variants). On the empirical front, we show how NN-Steiner can be implemented and demonstrate the effectiveness of our resulting approach, especially in terms of generalization, by comparing with state-of-the-art methods (both neural and non-neural based).

Learning to Solve Hard Minimal Problems.

We present an approach to solving hard geometric optimization problems in the RANSAC framework. The hard minimal problems arise from relaxing the original geometric optimization problem into a minimal problem with many spurious solutions. Our approach avoids computing large numbers of spurious solutions. We design a learning strategy for selecting a starting problem-solution pair that can be numerically continued to the problem and the solution of interest. We demonstrate our approach by developing a RANSAC solver for the problem of computing the relative pose of three calibrated cameras, via a minimal relaxation using four points in each view. On average, we can solve a single problem in under 70 μs. We also benchmark and study our engineering choices on the very familiar problem of computing the relative pose of two calibrated cameras, via the minimal case of five points in two views.

On a numerical approach for solving some geometrical shape optimization problems in fluid mechanics

This paper is devoted to the numerical study of geometrical shape optimization problems in fluid mechanics, which consist in minimizing some criterion volume cost functionals on a family of admissible doubly connected domains, constrained by steady-state Stokes boundary value problems. We establish the existence of the shape derivative of the considered cost functionals, by means of Minkowski deformation, using the shape derivative formulas, recently established in Boulkhemair and Chakib (2014). This allows us to express the shape derivative by means of the support function and to avoid the tedious computations required when one use the gradient optimization process based on the classical shape derivative, involving the vector fields, notably when one opt for the finite element discretization. So, based on the established shape derivative formulas, we propose a shape optimization numerical process for solving these problems, using the gradient descent algorithm performed by the finite element discretization, for approximating the auxiliary boundary value Stokes problems. Finally, in order to show the validity and the effectiveness of the proposed approach, we present some numerical tests obtained by solving some shape optimization problems of minimizing different cost functionals on various configurations of domains, constrained by steady-state stokes boundary value problems with different boundary conditions. These numerical simulations include some comparison results showing that the proposed approach is more efficient than the gradient approach based on the classical shape derivative, in terms of the accuracy of the solution and central processing unit (CPU) time execution.

A neurodynamic algorithm for dependent joint chance constrained geometric programs

This research studies the use of copula theory to model dependencies in joint probabilistic constrained geometric programs with dependent rows. The row vectors are assumed to follow an elliptical distribution, and their dependencies are modeled through a Gumbel–Hougaard copula. We use a log transformation to convert the chance-constrained geometric program into a deterministic optimization problem. Then we solve the resulting deterministic program using a dynamical neural network. The stability and convergence of the proposed neural network approach are demonstrated. The primary characteristic of our framework is its ability to solve the dependent joint chance constrained geometric programs without resorting to any convex approximation methods. This feature sets our approach apart from the current state-of-the-art solving techniques. The neurodynamic algorithm is finally applied to solve three geometric optimization problems.

Manifold-Constrained Geometric Optimization via Local Parameterizations.

Many geometric optimization problems contain manifold constraints that restrict the optimized vertices on some specified manifold surface. The constraints are highly nonlinear and non-convex, therefore existing methods usually suffer from a breach of condition or low optimization quality. In this article, we present a novel divide-and-conquer methodology for manifold-constrained geometric optimization problems. Central to our methodology is to use local parameterizations to decouple the optimization with hard constraints, which transforms nonlinear constraints into linear constraints. We decompose the input mesh into a set of developable or nearly-developable overlapping patches with disc topology, then flatten each patch into the planar domain with very low isometric distortion, optimize vertices with linear constraints and recover the patch. Finally, we project it onto the constrained manifold surface. We demonstrate the applicability and robustness of our methodology through a variety of geometric optimization tasks. Experimental results show that our method performs much better than existing methods.

Minimum Scan Cover and Variants: Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs , we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. To scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; and (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this article, we present polynomial-time algorithms for one-dimensional (1D) instances of MSC-TE and MSC-BE but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.

An automatic visible explainer of geometric knowledge for aeroshape design optimization based on SHAP

Aerodynamic shape design may involve complicated geometric optimization problems. When the number of geometric decision variables increases, the optimization algorithms can still search for solutions automatically, but it would work more like a black-box, for which the explanation will be of great help. This paper focuses on further mining the explanatory information of the shape optimization. A multi-objective optimization of a spacecraft is performed as a simplified case study. The LightGBM meta-model is built for the response from the geometric variables to the objectives, followed the SHAP explainer to quantify the contributions of variables. To visually interpret the deformation on shape, we propose the SHAP-Shape Map method to translate the attribution from the deformation features to the shape and make the attribution visible on the mesh surface. Much explanatory knowledge is easily obtained, for example, without seeing any flow field or pressure distribution image, the explainer successfully inferred that the significant influence of the drag coefficient occurs at the tip of the shape's nose. Furthermore, it has almost no barrier to learn that the trade-offs (conflicts) between the objectives are mainly concentrated in the nose as well. More importantly, with the proposed method, the above analysis results can be reached by anyone without specialized expertise of aerodynamics or geometric parameterization.

Covering and packing of triangles intersecting a straight line

We study four geometric optimization problems: set cover , hitting set , piercing set , and independent set with right-triangles (a triangle is a right-triangle whose base is parallel to the x -axis, perpendicular is parallel to the y -axis, and the slope of the hypotenuse is − 1 ). The input triangles are constrained to be intersecting a straight line . The straight line can either be a horizontal or an inclined line (a line whose slope is − 1 ). A right-triangle is said to be a λ -right-triangle , if the length of both its base and perpendicular is λ . For 1-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set problems are NP -hard, whereas the piercing set and independent set problems are in P . The same results hold for 1-right-triangles where the triangles are intersecting a horizontal line instead of an inclined line. We prove that the piercing set and independent set problems with right-triangles intersecting an inclined line are NP -hard. Finally, we give an n O ( ⌈ log c ⌉ + 1 ) time exact algorithm for the independent set problem with λ -right-triangles intersecting a straight line such that λ takes more than one value from [ 1 , c ] , for some integer c . We also present O ( n 2 ) -time dynamic programming algorithms for the independent set problem with 1-right-triangles where the triangles intersect a horizontal line and an inclined line.

Geometric Optimization of Distributed MIMO Radar System for Accurate Localization in Multiple Key Subareas

In this paper, we study a geometric optimization problem of distributed multi-input multi-output (MIMO) radar system. We aim to enhance both the system surveillance and localization performance by adjusting the node positions. Different from existing researches, from a practical perspective, we consider the importance differences of the performance in different subareas and the coupled relationship between different radar performance. To optimize the node placement scheme, we first establish evaluation metrics respectively for surveillance and localization performance. Then, we formulate a multi-objective geometric optimization problem with complex coupled constraints. Considering that the final optimization problem is difficult to tackle due to its high dimensionality, non-convexity, and especially the complex coupled constraint, we propose a novel self-constrained multi-objective particle swarm optimization (SC-MOPSO) algorithm for fast computation. The SC-MOPSO algorithm can be efficiently applied to the established optimization problem with the complex coupled constraint satisfied. Moreover, the obtained Pareto-optimal solution set is more complete in comparison with the state-of-the-art algorithms. Finally, various numerical results show that the proposed method can effectively enhance both the system surveillance and localization performance while the complex coupled constraints are satisfied.

Generation of quasi‐developable Q‐Bézier strip via PSO‐based shape parameters optimization

The generation of quasi‐developable from two boundary curves finds extensive application in manufacturing system. In this paper, we present a new method to generate a quasi‐developable from two Bézier boundary curves via particle swarm optimization (PSO)‐based shape parameters optimization. Owing to the powerful and flexible shape adjustment ability of Q‐Bézier curves, we introduce the multiple shape parameters of Q‐Bézier model into original boundary curves for developability optimization. We formulate the generation of quasi‐developable as a geometric optimization problem in which the objective function is defined as the developability degree of the strip and search within the value range of the shape parameters for the set of optimal shape parameters by PSO to achieve a higher‐accuracy developability. Modeling examples demonstrate that the proposed method is effective, easy to implement, which provides users with the overall shape control of the output strip. With the control points keeping unchanged, the fairness of the surface and its approximation to the original surface can be guaranteed naturally, and the number of optimization variables is greatly reduced compared with previous methods.

Linear-size universal discretization of geometric center-based problems in fixed dimensions

Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are $k$-Means, Geometric $k$-Median/Center, Continuous Facility Location, $m$-Variance, etc. We prove that, for any fixed $\varepsilon>0$, every set of $n$ input points in fixed-dimensional space with the metric induced by any vector norm admits a set of $O(n)$ candidate centers which can be computed in almost linear time and which contains a $(1+\varepsilon)$-approximation of each point of space with respect to the distances to all the input points. It gives a universal approximation-preserving reduction of geometric center-based problems with arbitrary continuity-type objective functions to their discrete versions where the centers are selected from a fairly small set of candidates. The existence of such a linear-size set of candidates is also shown for any metric space of fixed doubling dimension.

State of the Art on Computational Design of Assemblies with Rigid Parts

AbstractAn assembly refers to a collection of parts joined together to achieve a specific form and/or functionality. Designing assemblies is a non‐trivial task as a slight local modification on a part's geometry or its joining method could have a global impact on the structural and/or functional performance of the whole assembly. Assemblies can be classified as structures that transmit force to carry loads and mechanisms that transfer motion and force to perform mechanical work. In this state‐of‐the‐art report, we focus on computational design of structures with rigid parts, which generally can be formulated as a geometric modeling and optimization problem. We broadly classify existing computational design approaches, mainly from the computer graphics community, according to high‐level design objectives, including fabricability, structural stability, reconfigurability, and tileability. Computational analysis of various aspects of assemblies is an integral component in these design approaches. We review different classes of computational analysis and design methods, discuss their strengths and limitations, make connections among them, and propose possible directions for future research.

Optimal Algorithm for Shared Network Communication Bandwidth in IoT Applications

In recent years, a variety of wired and wireless network communication protocols in the field of industrial control have become increasingly mature. The purpose of this paper is to provide a Shared network communication bandwidth optimization management algorithm for large-scale industrial networked control systems in Internet of things applications. This algorithm is based on the generalized geometric convex optimization method and can realize the optimal allocation of Shared network communication bandwidth resources. L2 networked control systems is used in this paper for the establishment of various numerical relations between the control performance and the communication network parameters. Based on the generalized geometric convex optimization method for the numerical relationship between convex analysis and fitting, convexity, and with the convex analysis and the numerical relationship between convexity fitting as constraint conditions, the results of integrity for networked control systems with large-scale resource allocation target will share the optimal management of network resources as a generalized geometric convex optimization problem. Using convex optimization software package for optimizing the optimal global solution of management problem, i. e. the optimal allocation of resources, the algorithm realizes the stability of each networked control system and achieve optimal L2 control performance. It is concluded that the predetermined transmission rate between the network node one and network node two, the data flow information sent by the network node two to the network node one is read, the delay time and packet loss rate between the two nodes are determined, the delay time is reduced by about 8 seconds, and the packet loss rate is greatly reduced by 78%.

From Forecast to Decisions in Graphical Models: A Natural Gradient Optimization Approach

Graphical models and in particular Hidden Markov Models or their continuous space equivalent, the so called Kalman filter model, are a powerful tool to make some inference that can be used in decision making contexts. The estimation of their parameters is usually based on the Expectation Maximization approach as this is a natural statistical way to train them. When used for decision making, it may be more relevant to find parameters that are relevant to our decisions rather than just try to fit the model from a statistical point of view. Hence, we can reformulate the determination of graphical model as an inference problem where the true concern is the quality of the decisions from the forecast given by the model. We show that the resulting optimization problem can be reformulated as an information geometric optimization problem and introduce a natural gradient descent strategy that incorporates additional meta parameters. We show that our approach is a strong alternative to the celebrated EM approach for learning in graphical models. Actually, our natural gradient based strategy leads to learning optimal parameters for the final objective function (which is our decision) without artificially trying to fit a distribution that may not correspond to the real one. We support our theoretical findings with the question of decision in financial markets and show that the learned model performs better than traditional practitioner methods and is less prone to overfitting.

Systematic Beam Management in mmWave Networks: Tradeoff Among Beam Coverage, Link Budget, and Interference Control

In this paper, we study the beam management problem in mmWave networks from a systematic perspective. Instead of maximizing the network performance in terms of user sum-rate, we improve the beam coverage, which could potentially reduce beam tracking overhead for mobile users. Moreover, to compensate the severe signal attenuation in mmWave bands, we configure the beam to meet the user link budget requirement. To reduce the interference among users, we take the inter-beam interference from both main-lobe and side-lobe into consideration. Overall, we formulate our beam management scheme as a non-linear integer optimization problem. By deliberately transforming it to a geometric optimization problem and designing a rounding method, we develop a suboptimal solution, with low computational complexity. Based on this, we further design a scheduling user set refining method, which can assist user scheduling in the MAC layer, to make the beam configuration request within the antenna capability. Extensive simulation results demonstrate that our proposed algorithm could manage the beam on-line, while satisfying beam configuration requests for beam coverage, link budget, and inter-beam interference control.

On the minimum-area rectangular and square annulus problem

In this paper, we address the minimum-area rectangular and square annulus problem, which asks a rectangular or square annulus of minimum area, either in a fixed orientation or over all orientations, that encloses a set P of n input points in the plane. To our best knowledge, no nontrivial results on the problem have been discussed in the literature, while its minimum-width variants have been intensively studied. For a fixed orientation, we show reductions to well-studied problems: the minimum-width square annulus problem and the largest empty rectangle problem, yielding algorithms of time complexity O(nlog2⁡n) and O(nlog⁡n) for the rectangular and square cases, respectively. In arbitrary orientation, we present O(n3)-time algorithms for the rectangular and square annulus problem by enumerating all maximal empty rectangles over all orientations. The same approach is shown to apply also to other geometric optimization problems with similar flavor, resulting in O(n3)-time algorithms. As a result, we improve the previously best algorithm for the minimum-width square annulus problem by a factor of logarithm. We also consider bicriteria optimization variants, computing a minimum-width minimum-area or minimum-area minimum-width annulus.

A new geometrically nonlinear topology optimization formulation for controlling maximum displacement

This article presents a novel formulation for geometric nonlinear topology optimization problems. In practical engineering, maximum deflection is frequently used to quantify the stiffness of continuum structures, yet not applied generally as the optimization constraint in geometrically nonlinear topology optimization problems. In this study, the maximum nodal displacement is formulated as a sole constraint. The p-mean aggregation function is adopted to efficiently treat a mass of local displacement constraints imposed on nodes in the user-specified region. The sensitivities of the objective and constraint functions with respect to relative densities are derived. The effect of the aggregate parameter on the final design is further investigated through numerical examples. By comparison with final designs from the traditional formulation, i.e. minimization end compliance with the volume fraction constraint, or minimization of total volume subject to multiple nodal displacement constraints, the optimized results clearly demonstrate the necessity for and efficiency of the present approach.

Approximating Global Optimum for Probabilistic Truth Discovery

The problem of truth discovery arises in many areas such as database, data mining, data crowdsourcing and machine learning. It seeks trustworthy information from possibly conflicting data provided by multiple sources. Due to its practical importance, the problem has been studied extensively in recent years. Two competing models were proposed for truth discovery, weight-based model and probabilistic model. While $$(1+\epsilon )$$ -approximations have already been obtained for the weight-based model, no quality guaranteed solution has been discovered yet for the probabilistic model. In this paper, we focus on the probabilistic model and formulate it as a geometric optimization problem. Based on a sampling technique and a few other ideas, we achieve the first $$(1 + \epsilon )$$ -approximation solution. Our techniques can also be used to solve the more general multi-truth discovery problem. We validate our method by conducting experiments on both synthetic and real-world datasets (teaching evaluation data) and comparing its performance to some existing approaches. Our solutions are closer to the truth as well as global optimum based on the experimental result. The general technique we developed has the potential to be used to solve other geometric optimization problems.

Geometric optimization using nonlinear rotation-invariant coordinates

Geometric optimization problems are at the core of many applications in geometry processing. The choice of a representation fitting an optimization problem can considerably simplify solving the problem. We consider the Nonlinear Rotation-Invariant Coordinates (NRIC) that represent the nodal positions of a discrete triangular surface with fixed combinatorics as a vector that stacks all edge lengths and dihedral angles of the mesh. It is known that this representation associates a unique vector to an equivalence class of nodal positions that differ by a rigid body motion. Moreover, integrability conditions that ensure the existence of nodal positions that match a given vector of edge lengths and dihedral angles have been established. The goal of this paper is to develop the machinery needed to use the NRIC for solving geometric optimization problems. First, we use the integrability conditions to derive an implicit description of the space of discrete surfaces as a submanifold of an Euclidean space and a corresponding description of its tangent spaces. Secondly, we reformulate the integrability conditions using quaternions and provide explicit formulas for their first and second derivatives facilitating the use of Hessians in NRIC-based optimization problems. Lastly, we introduce a fast and robust algorithm that reconstructs nodal positions from almost integrable NRIC. We demonstrate the benefits of this approach on a collection of geometric optimization problems. Comparisons to alternative approaches indicate that NRIC-based optimization is particularly effective for problems involving near-isometric deformations.