Showing papers on "Maxima and minima published in 2011"

STOMP: Stochastic trajectory optimization for motion planning

Abstract: We present a new approach to motion planning using a stochastic trajectory optimization framework. The approach relies on generating noisy trajectories to explore the space around an initial (possibly infeasible) trajectory, which are then combined to produced an updated trajectory with lower cost. A cost function based on a combination of obstacle and smoothness cost is optimized in each iteration. No gradient information is required for the particular optimization algorithm that we use and so general costs for which derivatives may not be available (e.g. costs corresponding to constraints and motor torques) can be included in the cost function. We demonstrate the approach both in simulation and on a mobile manipulation system for unconstrained and constrained tasks. We experimentally show that the stochastic nature of STOMP allows it to overcome local minima that gradient-based methods like CHOMP can get stuck in.

Multi-target tracking by continuous energy minimization

Abstract: We propose to formulate multi-target tracking as minimization of a continuous energy function. Other than a number of recent approaches we focus on designing an energy function that represents the problem as faithfully as possible, rather than one that is amenable to elegant optimization. We then go on to construct a suitable optimization scheme to find strong local minima of the proposed energy. The scheme extends the conjugate gradient method with periodic trans-dimensional jumps. These moves allow the search to escape weak minima and explore a much larger portion of the variable-dimensional search space, while still always reducing the energy. To demonstrate the validity of this approach we present an extensive quantitative evaluation both on synthetic data and on six different real video sequences. In both cases we achieve a significant performance improvement over an extended Kalman filter baseline as well as an ILP-based state-of-the-art tracker.

Stochastic gradient descent on Riemannian manifolds

Abstract: Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically.

A Direct Least-Squares (DLS) method for PnP

Abstract: In this work, we present a Direct Least-Squares (DLS) method for computing all solutions of the perspective-n-point camera pose determination (PnP) problem in the general case (n ≥ 3). Specifically, based on the camera measurement equations, we formulate a nonlinear least-squares cost function whose optimality conditions constitute a system of three third-order polynomials. Subsequently, we employ the multiplication matrix to determine all the roots of the system analytically, and hence all minima of the LS, without requiring iterations or an initial guess of the parameters. A key advantage of our method is scalability, since the order of the polynomial system that we solve is independent of the number of points. We compare the performance of our algorithm with the leading PnP approaches, both in simulation and experimentally, and demonstrate that DLS consistently achieves accuracy close to the Maximum-Likelihood Estimator (MLE).

Discrete-continuous optimization for large-scale structure from motion

Abstract: Recent work in structure from motion (SfM) has successfully built 3D models from large unstructured collections of images downloaded from the Internet. Most approaches use incremental algorithms that solve progressively larger bundle adjustment problems. These incremental techniques scale poorly as the number of images grows, and can drift or fall into bad local minima. We present an alternative formulation for SfM based on finding a coarse initial solution using a hybrid discrete-continuous optimization, and then improving that solution using bundle adjustment. The initial optimization step uses a discrete Markov random field (MRF) formulation, coupled with a continuous Levenberg-Marquardt refinement. The formulation naturally incorporates various sources of information about both the cameras and the points, including noisy geotags and vanishing point estimates. We test our method on several large-scale photo collections, including one with measured camera positions, and show that it can produce models that are similar to or better than those produced with incremental bundle adjustment, but more robustly and in a fraction of the time.

A precise method for visualizing dispersive features in image plots

Abstract: In order to improve the advantages and the reliability of the second derivative method in tracking the position of extrema from experimental curves, we develop a novel analysis method based on the mathematical concept of curvature. We derive the formulas for the curvature in one and two dimensions and demonstrate their applicability to simulated and experimental angle-resolved photoemission spectroscopy data. As compared to the second derivative, our new method improves the localization of the extrema and reduces the peak broadness for a better visualization on intensity image plots.

An approach based on constrained nonnegative matrix factorization to unmix hyperspectral data

Abstract: Nonnegative matrix factorization (NMF) has been recently applied to solve the hyperspectral unmixing problem because it ensures nonnegativity and needs no assumption for the presence of pure pixels. However, the algorithm has a large amount of local minima due to the obvious nonconvexity of the objective function. In order to improve its performance, auxiliary constraints can be introduced into the algorithm. In this paper, we propose a new approach named abundance separation and smoothness constrained NMF by introducing two constraints, namely, abundance separation and smoothness, into the NMF algorithm. These constraints are based on two properties of hyperspectral imagery. First, usually, every ground object presents dominance in a specific region of the entire image scene and the correlation is weak between different endmembers. Second, moving through various regions, ground objects usually vary slowly and abrupt changes rarely appear. We also propose a learning algorithm to further improve the performance of our method, from which the auxiliary constraints are removed at an appropriate time. The proposed algorithm retains all the advantages of NMF and effectively overcomes the shortcoming of local minima at the same time. Experimental results based on synthetic and real hyperspectral data show the superiority of the proposed algorithm with respect to other state-of-the-art approaches.

Software for Generation of Classes of Test Functions with Known Local and Global Minima for Global Optimization

Abstract: A procedure for generating non-differentiable, continuously differentiable, and twice continuously differentiable classes of test functions for multiextremal multidimensional box-constrained global optimization and a corresponding package of C subroutines are presented. Each test class consists of 100 functions. Test functions are generated by defining a convex quadratic function systematically distorted by polynomials in order to introduce local minima. To determine a class, the user defines the following parameters: (i) problem dimension, (ii) number of local minima, (iii) value of the global minimum, (iv) radius of the attraction region of the global minimizer, (v) distance from the global minimizer to the vertex of the quadratic function. Then, all other necessary parameters are generated randomly for all 100 functions of the class. Full information about each test function including locations and values of all local minima is supplied to the user. Partial derivatives are also generated where possible.

Finite Element Model Updating Using Evolutionary Strategy for Damage Detection

Abstract: : Structural health monitoring through the use of finite element model updating techniques for dispersed civil infrastructures usually deals with minimizing a complex, nonlinear, nonconvex, high-dimensional cost function with several local minima. Hence, stochastic optimization algorithms with promising performance in solving global optimization problems have received considerable attention for finite element model updating purposes in recent years. In this study, the performance of an evolutionary strategy in the finite element model updating approach was investigated for damage detection in a quarter-scale two-span reinforced concrete bridge system which was tested experimentally at the University of Nevada, Reno. The damage sequence in the structure was induced by a range of progressively increasing excitations in the transverse direction of the specimen. Intermediate nondestructive white noise excitations and response measurements were used for system identification and damage detection purposes. It is shown that, when evaluated together with the strain gauge measurements and visual inspection results, the applied finite element model updating algorithm of this article could accurately detect, localize, and quantify the damage in the tested bridge columns throughout the different phases of the experiment.

An inflationary differential evolution algorithm for space trajectory optimization

Abstract: In this paper we define a discrete dynamical system that governs the evolution of a population of agents. From the dynamical system, a variant of Differential Evolution is derived. It is then demonstrated that, under some assumptions on the differential mutation strategy and on the local structure of the objective function, the proposed dynamical system has fixed points towards which it converges with probability one for an infinite number of generations. This property is used to derive an algorithm that performs better than standard Differential Evolution on some space trajectory optimization problems. The novel algorithm is then extended with a guided restart procedure that further increases the performance, reducing the probability of stagnation in deceptive local minima.

An Inflationary Differential Evolution Algorithm for Space Trajectory Optimization

Abstract: In this paper, we define a discrete dynamical system that governs the evolution of a population of agents. From the dynamical system, a variant of differential evolution (DE) is derived. It is then demonstrated that, under some assumptions on the differential mutation strategy and on the local structure of the objective function, the proposed dynamical system has fixed points toward which it converges with probability one for an infinite number of generations. This property is used to derive an algorithm that performs better than standard DE on some space trajectory optimization problems. The novel algorithm is then extended with a guided restart procedure that further increases the performance, reducing the probability of stagnation in deceptive local minima.

Weak Sharp Minima on Riemannian Manifolds

Abstract: This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and Hadamard manifolds. A number of the results obtained in this paper are also new for the case of conventional problems in finite-dimensional Euclidean spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper.

Choosing where to go: Complete 3D exploration with stereo

Abstract: This paper is about the autonomous acquisition of detailed 3D maps of a-priori unknown environments using a stereo camera - it is about choosing where to go. Our approach hinges upon a boundary value constrained partial differential equation (PDE) - the solution of which provides a scalar field guaranteed to have no local minima. This scalar field is trivially transformed into a vector field in which following lines of max flow causes provably complete exploration of the environment in full 6 degrees of freedom (6-DOF). We use a SLAM system to infer the position of a stereo pair in real time and fused stereo depth maps to generate the boundary conditions which drive exploration. Our exploration algorithm is parameter free, is as applicable to 3D laser data as it is to stereo, is real time and is guaranteed to deliver complete exploration. We show empirically that it performs better than oft-used frontier based approaches and demonstrate our system working with real and simulated data.

Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation method.

Abstract: The stationary points (SPs) of a potential-energy landscape play a crucial role in understanding many of the physical or chemical properties of a given system. However, unless they are found analytically, no efficient method is available to obtain all the SPs of a given potential. We present a method, called the numerical polynomial-homotopy-continuation method, which numerically finds all the SPs, and is embarrassingly parallelizable. The method requires the nonlinearity of the potential to be polynomial-like, which is the case for almost all of the potentials arising in physical and chemical systems. We also certify the numerically obtained SPs so that they are independent of the numerical tolerance used during the computation. It is then straightforward to separate out the local and global minima. As a first application, we take the XY model with power-law interaction, which is shown to have a polynomial-like nonlinearity, and we apply the method.

Infinite-Horizon Model Predictive Control for Periodic Tasks with Contacts

Abstract: We present a method that combines offline trajectory optimization and online Model Predictive Control (MPC), generating robust controllers for complex periodic behavior in domains with unilateral constraints (e.g., contact with the environment). MPC offers robust and adaptive control even in high-dimensional domains; however, the online optimization gets stuck in local minima when the domains has discontinuous dynamics. Some methods of trajectory optimization that are immune to such problems, but these are often too slow to be applied online. In this paper, we use offline optimization to find the limit-cycle solution of an infinite-horizon average-cost optimal-control task. We then compute a local quadratic approximation of the Value function around this limit cycle. Finally, we use this quadratic approximation as the terminal cost of an online MPC. This combination of an offline solution of the infinite-horizon problem with an online MPC controller is known as Infinite Horizon Model Predictive Control (IHMPC), and has previously been applied only to simple stabilization objectives. Here we extend IHMPC to tackle periodic tasks, and demonstrate the power of our approach by synthesizing hopping behavior in a simulated robot. IHMPC involves a limited computational load, and can be executed online on a standard laptop computer. The resulting behavior is extremely robust, allowing the hopper to recover from virtually any perturbation. In real robotic domains, modeling errors are inevitable. We show how IHMPC is robust to modeling errors by altering the morphology of the robot; the same controller remains effective, even when the underlying infinite-horizon solution is no longer accurate.

Two-Dimensional Time-Dependent Vortex Regions Based on the Acceleration Magnitude

Abstract: Acceleration is a fundamental quantity of flow fields that captures Galilean invariant properties of particle motion. Considering the magnitude of this field, minima represent characteristic structures of the flow that can be classified as saddle- or vortex-like. We made the interesting observation that vortex-like minima are enclosed by particularly pronounced ridges. This makes it possible to define boundaries of vortex regions in a parameter-free way. Utilizing scalar field topology, a robust algorithm can be designed to extract such boundaries. They can be arbitrarily shaped. An efficient tracking algorithm allows us to display the temporal evolution of vortices. Various vortex models are used to evaluate the method. We apply our method to two-dimensional model systems from computational fluid dynamics and compare the results to those arising from existing definitions.

Hierarchical Line Integration

Abstract: This paper presents an acceleration scheme for the numerical computation of sets of trajectories in vector fields or iterated solutions in maps, possibly with simultaneous evaluation of quantities along the curves such as integrals or extrema. It addresses cases with a dense evaluation on the domain, where straightforward approaches are subject to redundant calculations. These are avoided by first calculating short solutions for the whole domain. From these, longer solutions are then constructed in a hierarchical manner until the designated length is achieved. While the computational complexity of the straightforward approach depends linearly on the length of the solutions, the computational cost with the proposed scheme grows only logarithmically with increasing length. Due to independence of subtasks and memory locality, our algorithm is suitable for parallel execution on many-core architectures like GPUs. The trade-offs of the method - lower accuracy and increased memory consumption - are analyzed, including error order as well as numerical error for discrete computation grids. The usefulness and flexibility of the scheme are demonstrated with two example applications: line integral convolution and the computation of the finite-time Lyapunov exponent. Finally, results and performance measurements of our GPU implementation are presented for both synthetic and simulated vector fields from computational fluid dynamics.

Software for Generation of Classes of Test Functions with Known Local and Global Minima for Global Optimization

Abstract: A procedure for generating non-differentiable, continuously differentiable, and twice continuously differentiable classes of test functions for multiextremal multidimensional box-constrained global optimization and a corresponding package of C subroutines are presented. Each test class consists of 100 functions. Test functions are generated by defining a convex quadratic function systematically distorted by polynomials in order to introduce local minima. To determine a class, the user defines the following parameters: (i) problem dimension, (ii) number of local minima, (iii) value of the global minimum, (iv) radius of the attraction region of the global minimizer, (v) distance from the global minimizer to the vertex of the quadratic function. Then, all other necessary parameters are generated randomly for all 100 functions of the class. Full information about each test function including locations and values of all local minima is supplied to the user. Partial derivatives are also generated where possible.

Global convergence of radial basis function trust region derivative-free algorithms *

Abstract: We analyze globally convergent derivative-free trust region algorithms relying on radial basis function interpolation models. Our results extend the recent work of Conn, Scheinberg, and Vicente [SIAM J. Optim., 20 (2009), pp. 387–415] to fully linear models that have a nonlinear term. We characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first-order critical points for the ORBIT algorithm of Wild, Regis, and Shoemaker [SIAM J. Sci. Comput., 30 (2008), pp. 3197–3219]. Using ORBIT, we present numerical results for different types of radial basis functions on a series of test problems. We also demonstrate the use of ORBIT in finding local minima on a computationally expensive environmental engineering problem.

Efficient moves for global geometry optimization methods and their application to binary systems.

Abstract: We show that molecular dynamics based moves in the minima hopping method are more efficient than saddle point crossing moves. For binary systems we incorporate identity exchange moves in a way that allows one to avoid the generation of high energy configurations. Using this modified minima hopping method, we re-examine the binary Lennard-Jones benchmark system with up to 100 atoms and we find a large number of new putative global minima.

Optimum operation of wells in coastal aquifers

Abstract: The combination of two powerful analysis techniques, simulation and optimisation, produces an engineering design tool that can aid in the formulation of design criteria and assist decision makers in assessing the impacts of design trade-offs. This paper presents a management model with the economic objective of maximising the net benefit derived from a pumped freshwater volume while eliminating the utility cost in coastal aquifers threatened by saltwater intrusion. Because the search space is both discontinuous and extremely wide, and also contains numerous local minima, evolutionary algorithms can be helpful in the search for near-optimal solutions. In this paper, a simulation model of the sharp saltwater–freshwater interface is considered to be time-independent. Moreover, the honey-bee mating optimisation (HBMO) algorithm is applied to a test case and the results are compared with those of a previously developed genetic algorithm (GA) for the same case. The comparison indicates that the HBMO algorithm c...

Analyzing search topology without running any search: on the connection between causal graphs and h +

Abstract: The ignoring delete lists relaxation is of paramount importance for both satisficing and optimal planning. In earlier work, it was observed that the optimal relaxation heuristic h+ has amazing qualities in many classical planning benchmarks, in particular pertaining to the complete absence of local minima. The proofs of this are hand-made, raising the question whether such proofs can be lead automatically by domain analysis techniques. In contrast to earlier disappointing results - the analysis method has exponential runtime and succeeds only in two extremely simple benchmark domains - we herein answer this question in the afirmative. We establish connections between causal graph structure and h+ topology. This results in low-order polynomial time analysis methods, implemented in a tool we call TorchLight. Of the 12 domains where the absence of local minima has been proved, TorchLight gives strong success guarantees in 8 domains. Empirically, its analysis exhibits strong performance in a further 2 of these domains, plus in 4 more domains where local minima may exist but are rare. In this way, TorchLight can distinguish "easy" domains from "hard" ones. By summarizing structural reasons for analysis failure, TorchLight also provides diagnostic output indicating domain aspects that may cause local minima.

Chern numbers hiding in time-of-flight images

Abstract: We present a technique for detecting topological invariants---Chern numbers---from time-of-flight images of ultracold atoms. We show that the Chern numbers of integer quantum Hall states of lattice fermions leave their fingerprints in the atoms' momentum distribution. We analytically demonstrate that the number of local maxima in the momentum distribution is equal to the Chern number in two limiting cases, for large hopping anisotropy and in the continuum limit. In addition, our numerical simulations beyond these two limits show that these local maxima persist for a range of parameters. Thus, an everyday observable in cold atom experiments can serve as a useful tool to characterize and visualize quantum states with nontrivial topology.

Maximum Power Point Tracking for PV system under partial shading condition via particle swarm optimization

Abstract: Performance of Photovoltaic (PV) system is greatly dependent on the solar irradiation and operating temperature. Due to partial shading condition, the characteristics of a PV system considerably change and often exhibit several local maxima with one global maxima. Conventional Maximum Power Point Tracking (MPPT) techniques can easily be trapped at local maxima under partial shading. This significantly reduced the energy yield of the PV systems. In order to solve this problem, this paper proposes a Maximum Power Point tracking algorithm based on particle swarm optimization (PSO) that is capable of tracking global MPP under partial shaded conditions. The performance of proposed algorithm is evaluated by means of simulation in MATLAB Simulink. The proposed algorithm is applied to a grid connected PV system, in which a Boost (step up) DC-DC converter satisfactorily tracks the global peak.

Algorithm for Optimal Mode Scheduling in Switched Systems

Abstract: This paper considers the problem of computing the schedule of modes in a switched dynamical system, that minimizes a cost functional defined on the trajectory of the system's continuous state variable. A recent approach to such optimal control problems consists of algorithms that alternate between computing the optimal switching times between modes in a given sequence, and updating the mode-sequence by inserting to it a finite number of new modes. These algorithms have an inherent inefficiency due to their sparse update of the mode-sequences, while spending most of the computing times on optimizing with respect to the switching times for a given mode-sequence. This paper proposes an algorithm that operates directly in the schedule space without resorting to the timing optimization problem. It is based on the Armijo step size along certain Gateaux derivatives of the performance functional, thereby avoiding some of the computational difficulties associated with discrete scheduling parameters. Its convergence to local minima as well as its rate of convergence are proved, and a simulation example on a nonlinear system exhibits quite a fast convergence.

Multiple testing of local maxima for detection of peaks in 1d.

Abstract: A topological multiple testing scheme for one-dimensional domains is proposed where, rather than testing every spatial or temporal location for the presence of a signal, tests are performed only at the local maxima of the smoothed observed sequence. Assuming unimodal true peaks with finite support and Gaussian stationary ergodic noise, it is shown that the algorithm with Bonferroni or Benjamini–Hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate, and is power consistent, as the search space and the signal strength get large, where the search space may grow exponentially faster than the signal strength. Simulations show that error levels are maintained for nonasymptotic conditions, and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks, akin to the matched filter theorem in signal processing. The methods are illustrated in an analysis of electrical recordings of neuronal cell activity.

Parameter identification in dynamic systems using the homotopy optimization approach

Abstract: Identifying the parameters in a mathematical model governed by a system of ordinary differential equations is considered in this work. It is assumed that only partial state measurement is available from experiments, and that the parameters appear nonlinearly in the system equations. The problem of parameter identification is often posed as an optimization problem, and when deterministic methods are used for optimization, one often converges to a local minimum rather than the global minimum. To mitigate the problem of converging to local minima, a new approach is proposed for applying the homotopy technique to the problem of parameter identification. Several examples are used to demonstrate the effectiveness of the homotopy method for obtaining global minima, thereby successfully identifying the system parameters.