IA Scholar Query: Computing Convex Hulls with a Linear Solver
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 28 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440A novel hypothesis for how albatrosses optimize their flight physics in real-time: an extremum seeking model and control for dynamic soaring
https://scholar.archive.org/work/yo2hpjadgjdo7gadxsrxyixy4u
The albatross optimized flight maneuver -- known as Dynamic Soaring (DS) -- is nothing but wonder of physics, biology, and engineering. In an ideal DS cycle, this fascinating bird can travel in the desired flight direction, for free, by harvesting energy from the wind, and hence, it achieves a neutral energy cycle. This phenomenon has triggered a momentous interest among aeronautical, control and robotic engineering communities; if DS is mimicked, we have arrived at a new class of UAVs that are very energy-efficient during part (or the full) duration of their flight. However, the DS problem is highly nonlinear, under-actuated, and dependent on the wind profiles. This has resulted in decades of DS control literature that, while making progress in addressing the control problem, seem not to be aligned well with the nature of the DS phenomenon itself. The control works associated with DS in the literature rely heavily on constrained optimal control algorithms, control designs that require a mathematical expression of the objective function, and predefined wind profile models. Clearly, a functioning controller for DS that allows meaningful bio-mimicry of the albatross, needs to be autonomous, real-time, stable, and capable of tolerating the absence of the expression of the objective function (similar to what the bird does). The qualifications of such controller are the very same characteristics of ESC systems. In this paper, we show that ESC systems existing in control literature for decades are a natural characterization of the DS problem. We provide the DS problem setup, design, stability, and simulation results of the introduced ESC systems. The results, supported by comparison with optimal control solvers, emphasize that the DS phenomenon is a natural expression of ESC systems in nature and that DS can be performed autonomously and in real-time with stability guarantees.Sameer Pokhrel, Sameh A. Eisawork_yo2hpjadgjdo7gadxsrxyixy4uMon, 28 Nov 2022 00:00:00 GMTPersistently Feasible Robust Safe Control by Safety Index Synthesis and Convex Semi-Infinite Programming
https://scholar.archive.org/work/afqahmmuzbcwzm4e65lgmldm7i
Model mismatches prevail in real-world applications. Ensuring safety for systems with uncertain dynamic models is critical. However, existing robust safe controllers may not be realizable when control limits exist. And existing methods use loose over-approximation of uncertainties, leading to conservative safe controls. To address these challenges, we propose a control-limits aware robust safe control framework for bounded state-dependent uncertainties. We propose safety index synthesis to find a robust safe controller guaranteed to be realizable under control limits. And we solve for robust safe control via Convex Semi-Infinite Programming, which is the tightest formulation for convex bounded uncertainties and leads to the least conservative control. In addition, we analyze when and how safety can be preserved under unmodeled uncertainties. Experiment results show that our robust safe controller is always realizable under control limits and is much less conservative than strong baselines.Tianhao Wei, Shucheng Kang, Weiye Zhao, Changliu Liuwork_afqahmmuzbcwzm4e65lgmldm7iMon, 28 Nov 2022 00:00:00 GMTAn Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros
https://scholar.archive.org/work/wq4vgu5fmfdtrmsnbtmke6m3kq
A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which the tetrahedron's in-sphere touches those faces. This leads to a conjecture as to how the formula extends to n-dimensional simplices for all n > 3. Remarkably, for n = 3 the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios. These unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to ℤ_2^4, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subset. The paper closes by noting that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank 3 chirotope, aka affine oriented matroid.Timothy F. Havelwork_wq4vgu5fmfdtrmsnbtmke6m3kqMon, 28 Nov 2022 00:00:00 GMTPath Planning for Concentric Tube Robots: a Toolchain with Application to Stereotactic Neurosurgery
https://scholar.archive.org/work/btt7c7rfwrerrfpjyi5hrwddpy
We present a toolchain for solving path planning problems for concentric tube robots through obstacle fields. First, ellipsoidal sets representing the target area and obstacles are constructed from labelled point clouds. Then, the nonlinear and highly nonconvex optimal control problem is solved by introducing a homotopy on the obstacle positions where at one extreme of the parameter the obstacles are removed from the operating space, and at the other extreme they are located at their intended positions. We present a detailed example (with more than a thousand obstacles) from stereotactic neurosurgery with real-world data obtained from labelled MPRI scans.Matthias K. Hoffmann, Willem Esterhuizen, Karl Worthmann, Kathrin Flaßkampwork_btt7c7rfwrerrfpjyi5hrwddpyMon, 28 Nov 2022 00:00:00 GMTSafety Envelope for Orthogonal Collocation Methods in Embedded Optimal Control
https://scholar.archive.org/work/6vnzdgimhvajjoriphpnkbnlme
Orthogonal collocation methods are direct simultaneous approaches for solving optimal control problems (OCP). A high solution accuracy is achieved with few optimization variables, making it more favorable for embedded and real-time NMPC applications. However, collocation approaches lack a guarantee about the safety of the resulting continuous trajectory as inequality constraints are only set on a finite number of collocation points. In this paper we propose a method to efficiently create a convex safety envelope containing the full trajectory such that the solution fully satisfies the OCP constraints. We make use of the Bernstein approximations of a polynomial's extrema and span the solution over an orthogonal basis using the Legendre polynomials. The tightness of the safety envelope estimation, high spectral accuracy of the method in solving the underlying differential equations, fast rate of convergence and little conservatism are properties of the presented approach making it a suitable method for safe real-time NMPC deployment. We show that our method has comparable computational performance to the pseudospectral approaches and can approximate the original OCP more accurately and up to 9 times more quickly than standard multiple-shooting methods in autonomous driving applications, without adding complexity to the formulation.Jean Pierre Allamaa, Panagiotis Patrinos, Herman Van der Auweraer, Tong Duy Sonwork_6vnzdgimhvajjoriphpnkbnlmeSun, 27 Nov 2022 00:00:00 GMTOn the convex hull of convex quadratic optimization problems with indicators
https://scholar.archive.org/work/hgunudy6tberdgjy3jym6fuubq
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are "finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.Linchuan Wei, Alper Atamtürk, Andrés Gómez, Simge Küçükyavuzwork_hgunudy6tberdgjy3jym6fuubqSun, 27 Nov 2022 00:00:00 GMTEfficient Joint Object Matching via Linear Programming
https://scholar.archive.org/work/55wmebaczvdsdmdzzz36smlmpm
Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear programming (LP) relaxations with theoretical performance guarantees for joint object matching. We start by proposing a new characterization of consistent partial maps; this in turn enables us to formulate joint object matching as an integer linear programming (ILP) problem. To construct strong LP relaxations, we study the facial structure of the convex hull of the feasible region of this ILP, which we refer to as the joint matching polytope. We present an exponential family of facet-defining inequalities that can be separated in strongly polynomial time, hence obtaining a partial characterization of the joint matching polytope that is both tight and cheap to compute. To analyze the theoretical performance of the proposed LP relaxations, we focus on permutation group synchronization, an important special case of joint object matching. We show that under the random corruption model for the input maps, a simple LP relaxation, that is, an LP containing only a very small fraction of the proposed facet-defining inequalities, recovers the ground truth with high probability if the corruption level is below 40%. Finally, via a preliminary computational study on synthetic data, we show that the proposed LP relaxations outperform a popular SDP relaxation both in terms of recovery and tightness.Antonio De Rosa, Aida Khajaviradwork_55wmebaczvdsdmdzzz36smlmpmSat, 26 Nov 2022 00:00:00 GMTNon-Euclidean Contraction Analysis of Continuous-Time Neural Networks
https://scholar.archive.org/work/tqkvo3sgkja6bgamf7c76yvcrq
Critical questions in dynamical neuroscience and machine learning are related to the study of continuous-time neural networks and their stability, robustness, and computational efficiency. These properties can be simultaneously established via a contraction analysis. This paper develops a comprehensive non-Euclidean contraction theory for continuous-time neural networks. First, for non-Euclidean ℓ_1/ℓ_∞ logarithmic norms, we establish quasiconvexity with respect to positive diagonal weights and closed-form worst-case expressions over certain matrix polytopes. Second, for locally Lipschitz maps (e.g., arising as activation functions), we show that their one-sided Lipschitz constant equals the essential supremum of the logarithmic norm of their Jacobian. Third and final, we apply these general results to classes of continuous-time neural networks, including Hopfield, firing rate, Persidskii, Lur'e and other models. For each model, we compute the optimal contraction rate and corresponding weighted non-Euclidean norm via a linear program or, in some special cases, via a Hurwitz condition on the Metzler majorant of the synaptic matrix. Our non-Euclidean analysis establishes also absolute, connective, and total contraction properties.Alexander Davydov, Anton V. Proskurnikov, Francesco Bullowork_tqkvo3sgkja6bgamf7c76yvcrqSat, 26 Nov 2022 00:00:00 GMTGlobal weight optimization of frame structures with polynomial programming
https://scholar.archive.org/work/2txqknvbtnbixgvc4u5uzvffma
Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic structure and adopt the Lasserre hierarchy of relaxations to compute the global minimizers. While this hierarchy generates a natural sequence of lower bounds, we show, under mild assumptions, how to project the relaxed solutions onto the feasible set of the original problem and thus construct feasible upper bounds. Based on these bounds, we develop a simple sufficient condition of global ε-optimality. Finally, we prove that the optimality gap converges to zero in the limit if the set of global minimizers is convex. We demonstrate these results by means of two academic illustrations.Marek Tyburec, Michal Kočvara, Martin Kružíkwork_2txqknvbtnbixgvc4u5uzvffmaFri, 25 Nov 2022 00:00:00 GMTGeodesic Models with Convexity Shape Prior
https://scholar.archive.org/work/tj3amvw57ngmtbc3gw6zvdqdee
The minimal geodesic models based on the Eikonal equations are capable of finding suitable solutions in various image segmentation scenarios. Existing geodesic-based segmentation approaches usually exploit image features in conjunction with geometric regularization terms, such as Euclidean curve length or curvature-penalized length, for computing geodesic curves. In this paper, we take into account a more complicated problem: finding curvature-penalized geodesic paths with a convexity shape prior. We establish new geodesic models relying on the strategy of orientation-lifting, by which a planar curve can be mapped to an high-dimensional orientation-dependent space. The convexity shape prior serves as a constraint for the construction of local geodesic metrics encoding a particular curvature constraint. Then the geodesic distances and the corresponding closed geodesic paths in the orientation-lifted space can be efficiently computed through state-of-the-art Hamiltonian fast marching method. In addition, we apply the proposed geodesic models to the active contours, leading to efficient interactive image segmentation algorithms that preserve the advantages of convexity shape prior and curvature penalization.Da Chen and Jean-Marie Mirebeau and Minglei Shu and Xuecheng Tai and Laurent D. Cohenwork_tj3amvw57ngmtbc3gw6zvdqdeeFri, 25 Nov 2022 00:00:00 GMTConvergence estimates for the Magnus expansion II. C^*-algebras
https://scholar.archive.org/work/vsvlik4mgrakdd7acmaknpxltu
We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker–Campbell–Hausdorff expansion are also made. In this Part II, we consider the case of C^*-algebras, i. e. essentially the case of operators on Hilbert spaces. We present the spectral approach to the Magnus expansion in the context of the conformal range (which is a projection of the Davis–Wielandt shell), allowing a more effective approach. This makes possible to clarify certain convergence properties of the BCH expansion related to the critical cumulative norm π. In particular, we prove that for finite dimensional matrices A,B, the norm condition A_2+B_2≤π implies that the BCH expansion of A and B is convergent. Several counterexamples regarding convergence of the Magnus and BCH expansions are presented. In the rest, we prove growth estimates for the Magnus expansion in the setting of Hilbert space operators, both in terms of the overall sum and the individuals terms.Gyula Lakoswork_vsvlik4mgrakdd7acmaknpxltuThu, 24 Nov 2022 00:00:00 GMTMultidimensional rank-one convexification of incremental damage models at finite strains
https://scholar.archive.org/work/2yo5cq2a6vdidpc7lpsthc3mai
This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, the relaxation techniques circumvent the problem of non-existence of minimizers and prevent mesh dependency of the solutions of discretized boundary value problems using finite elements. By the combination, modification and parallelization of the underlying convexification algorithms the approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step size control strategies prevents stability issues related to local minima in the energy landscape and the computation of derivatives. Special techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations.Daniel Balzani, Maximilian Köhler, Timo Neumeier, Malte A. Peter, Daniel Peterseimwork_2yo5cq2a6vdidpc7lpsthc3maiThu, 24 Nov 2022 00:00:00 GMTConvex Fairness Measures: Theory and Optimization
https://scholar.archive.org/work/7kubg64luvc3xpy3dwor23xt3u
We propose a new parameterized class of fairness measures, convex fairness measures, suitable for optimization contexts. This class includes our new proposed order-based fairness measure and several popular measures (e.g., deviation-based measures, Gini deviation). We provide theoretical analyses and derive a dual representation of these measures. Importantly, this dual representation renders a unified mathematical expression and a geometric characterization for convex fairness measures through their dual sets. Moreover, we propose a generic framework for optimization problems with a convex fairness measure objective, including reformulations and solution methods. Finally, we provide a stability analysis on the choice of convex fairness measures in the objective of optimization models.Man Yiu Tsang, Karmel S. Shehadehwork_7kubg64luvc3xpy3dwor23xt3uThu, 24 Nov 2022 00:00:00 GMTEfficient Separation of RLT Cuts for Implicit and Explicit Bilinear Products
https://scholar.archive.org/work/j6zohyydhjeplakbva2vgtj7de
The reformulation-linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs is developed based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. A detailed computational study based on implementations in two solvers evaluates the performance impact of the proposed methods.Ksenia Bestuzheva, Ambros Gleixner, Tobias Achterbergwork_j6zohyydhjeplakbva2vgtj7deThu, 24 Nov 2022 00:00:00 GMTStable-Set and Coloring bounds based on 0-1 quadratic optimization
https://scholar.archive.org/work/6ty45n6ukzaqpfu5wq2awqreiy
We consider semidefinite relaxations of Stable-Set and Coloring, which are based on quadratic 0-1 optimization. Information about the stability number and the chromatic number is hidden in the objective function. This leads to simplified relaxations which depend mostly on the number of vertices of the graph. We also propose tightenings of the relaxations which are based on the maximal cliques of the underlying graph. Computational results on graphs from the literature show the strong potential of this new approach.Dunja Pucher, Franz Rendlwork_6ty45n6ukzaqpfu5wq2awqreiyWed, 23 Nov 2022 00:00:00 GMTApproximate streamsurfaces for flow visualization
https://scholar.archive.org/work/ioq4pmkibbcxpixxi5xq4452tq
Instantaneous features of three-dimensional velocity fields are most directly visualized via streamsurfaces. It is generally unclear, however, which streamsurfaces one should pick for this purpose, given that infinitely many such surfaces pass through each point of the flow domain. Exceptions to this rule are vector fields with a nondegenerate first integral whose level surfaces globally define a continuous, one-parameter family of streamsurfaces. While generic vector fields have no first integrals, their vortical regions may admit local first integrals over a discrete set of streamtubes, as Hamiltonian systems are known to do over Cantor sets of invariant tori. Here we introduce a method to construct such first integrals approximately from velocity data, and show that their level sets indeed frame vortical features of the velocity field in examples in which those features are known from Lagrangian analysis. Moreover, we test our method in numerical data sets, including a flow inside a V-junction and a turbulent channel flow. For the latter, we propound an algorithm to pin down the most salient barriers to momentum transport up to a given scale providing a way out of the occlusion conundrum that typically accompanies other vortex visualization methods.Stergios Katsanoulis, Florian Kogelbauer, Roshan Kaundinya, Jesse Ault, George Hallerwork_ioq4pmkibbcxpixxi5xq4452tqWed, 23 Nov 2022 00:00:00 GMTCornering Large-N_c QCD with Positivity Bounds
https://scholar.archive.org/work/4ontlqy7uve7hjz7s4ip63cln4
The simple analytic structure of meson scattering amplitudes in the large-N_c limit, combined with positivity of the spectral density, provides precise predictions on low-energy observables. Building upon previous studies, we explore the allowed regions of chiral Lagrangian parameters and meson couplings to pions. We reveal a structure of kinks at all orders in the chiral expansion and develop analytical tools to show that kinks always correspond to amplitudes with a single light pole. We build (scalar- and vector-less) deformations of the Lovelace-Shapiro and Coon UV-complete amplitudes, and show that they lie close to the boundaries. Moreover, constraints from crossing-symmetry imply that meson couplings to pions become smaller as their spin increases, providing an explanation for the success of Vector Meson Dominance and holographic QCD. We study how these conclusions depend on assumptions about the high-energy behavior of amplitudes. Finally, we emphasize the complementarity between our results and Lattice computations in the exploration of large-N_c QCD.Clara Fernandez, Alex Pomarol, Francesco Riva, Francesco Sciottiwork_4ontlqy7uve7hjz7s4ip63cln4Tue, 22 Nov 2022 00:00:00 GMTGeneralized and Scalable Optimal Sparse Decision Trees
https://scholar.archive.org/work/ml5kozqn7vfnljayfckbiuynoe
Decision tree optimization is notoriously difficult from a computational perspective but essential for the field of interpretable machine learning. Despite efforts over the past 40 years, only recently have optimization breakthroughs been made that have allowed practical algorithms to find optimal decision trees. These new techniques have the potential to trigger a paradigm shift where it is possible to construct sparse decision trees to efficiently optimize a variety of objective functions without relying on greedy splitting and pruning heuristics that often lead to suboptimal solutions. The contribution in this work is to provide a general framework for decision tree optimization that addresses the two significant open problems in the area: treatment of imbalanced data and fully optimizing over continuous variables. We present techniques that produce optimal decision trees over a variety of objectives including F-score, AUC, and partial area under the ROC convex hull. We also introduce a scalable algorithm that produces provably optimal results in the presence of continuous variables and speeds up decision tree construction by several orders of magnitude relative to the state-of-the art.Jimmy Lin, Chudi Zhong, Diane Hu, Cynthia Rudin, Margo Seltzerwork_ml5kozqn7vfnljayfckbiuynoeTue, 22 Nov 2022 00:00:00 GMTMixed Integer Linear Program model for optimized scheduling of a vanadium redox flow battery with variable efficiencies, capacity fade, and electrolyte maintenance
https://scholar.archive.org/work/sngrtngjl5anfedr4yvoj6jtzu
Redox Flow Batteries are a promising option for large-scale stationary energy storage. The vanadium redox flow battery is the most widely commercialized system thanks to its chemical stability and performance. This work aims to optimize the scheduling of a vanadium flow battery that stores energy produced by a renewable power plant, keeping into account a thorough characterization of the battery performance, with variable efficiencies and capacity fade effects. A detailed characterization of the battery performance improves the calculation of the optimal number of cycles and revenue associated with the battery use if compared to the results obtained using simpler models, which take into account constant efficiencies and no capacity fade effects. The presented problem is nonlinear due to the functions of the battery efficiency, which depend upon charging and discharging powers and state of charge with nonlinear, non-convex correlations. The problem is linearized using convex hulls. The optimization program also calculates the progressive battery capacity fade due to undesired secondary electrochemical reactions and the economic impact of capacity restoration through periodic maintenance. The final problem is solved as a Mixed-Integer Linear Program (MILP) to guarantee the global optimality of the linearized problem. The proposed optimization model has been applied to two different case studies: a case of energy arbitrage and a case of load-shifting. The optimization results have been compared to those obtained with constant battery efficiency models, which do not consider the capacity fade effects. Results show that simpler models overestimate the optimal number of cycles of the battery and the revenue by up to 15% if they do not take into account the degradation model of the battery, and respectively up to 32% and 42% if they also assume constant efficiency for the battery.Diana Cremoncini and Guido Francesco Frate and Aldo Bischi and Lorenzo Ferrariwork_sngrtngjl5anfedr4yvoj6jtzuTue, 22 Nov 2022 00:00:00 GMTMaximum Likelihood Estimation of Log-Concave Densities on Tree Space
https://scholar.archive.org/work/d5k45azeibgjbcwjnnuwfmvtlm
Phylogenetic trees are key data objects in biology, and the method of phylogenetic reconstruction has been highly developed. The space of phylogenetic trees is a nonpositively curved metric space. Recently, statistical methods to analyze the set of trees on this space are being developed utilizing this property. Meanwhile, in Euclidean space, the log-concave maximum likelihood method has emerged as a new nonparametric method for probability density estimation. In this paper, we derive a sufficient condition for the existence and uniqueness of the log-concave maximum likelihood estimator on tree space. We also propose an estimation algorithm for one and two dimensions. Since various factors affect the inferred trees, it is difficult to specify the distribution of sample trees. The class of log-concave densities is nonparametric, and yet the estimation can be conducted by the maximum likelihood method without selecting hyperparameters. We compare the estimation performance with a previously developed kernel density estimator numerically. In our examples where the true density is log-concave, we demonstrate that our estimator has a smaller integrated squared error when the sample size is large. We also conduct numerical experiments of clustering using the Expectation-Maximization (EM) algorithm and compare the results with k-means++ clustering using Fr\'echet mean.Yuki Takazawa, Tomonari Seiwork_d5k45azeibgjbcwjnnuwfmvtlmTue, 22 Nov 2022 00:00:00 GMT