Showing papers on "Maxima and minima published in 1988"

Cooling Schedules for Optimal Annealing

Abstract: A Monte Carlo optimization technique called “simulated annealing” is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima. The level of randomization is determined by a control parameter T, called temperature, which tends to zero according to a deterministic “cooling schedule.” We give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states. In the special case that the cooling schedule has parametric form Tt = c/log1 + t, the condition for convergence is that c be greater than or equal to the depth, suitably defined, of the deepest local minimum which is not a global minimum state.

Covering minima and lattice-point-free convex bodies

Abstract: The covering radius of a convex body K (with respect to a lattice L) is the least factor by which the body needs to be blown up so that its translates by lattice vectors cover the whole space. The covering radius and related quantities have been studied extensively in the geometry of numbers (mainly for convex bodies symmetric about the origin). In this paper, we define and study the "covering minima" of a general convex body. The covering radius will be one of these minima; the "lattice width" of the body will be the reciprocal of another. We derive various inequalities relating these minima. These imply bounds on the width of lattice-point-free convex bodies. We prove that every lattice-point-free body has a projection whose volume is not much larger than the determinant of the projected lattice.

Finding local maxima in a pseudo-Euclidean distance transform

Abstract: Weighted distance transforms computed by using suitable integer weights are an adequate approximation to the Euclidean distance transform. In this paper, a criterion is introduced for identifying the local maxima in a specific weighted distance transform. The reverse weighted distance transformation is also introduced to associate a disc with every labeled pixel, and the discs centered on the local maxima are proved to be maximal discs.

Exact robot navigation using cost functions: the case of distinct spherical boundaries in E/sup n/

Abstract: The utility of artificial potential functions is explored as a means of translating automatically a robot task description into a feedback control law to drive the robot actuators. A class of functions is sought which will guide a point robot amid any finite number of spherically bounded obstacles in Euclidean n-space toward an arbitrary destination point. By introducing a set of additional constraints, the subclass of navigation functions is defined. This class is dynamically sound in the sense that the actual mechanical system will inherit the essential aspects of the qualitative behavior of the gradient lines of the cost function. An existence proof is given by constructing a one parameter family of such functions; the parameter is used to guarantee the absence of local minima. >

A global search method for optimizing nonlinear systems

Abstract: The theory and implementation of a global search method of optimization in n dimensions, inspired by Kushner's method in one dimension, are presented. This method is meant to address optimization problems where the function has many extrema, where it may or may not be differentiable, and where it is important to reduce the number of evaluations of the function at the expense of increased computation. Comparisons are made to the performance of other global optimization techniques on a set of standard differentiable test functions. A new class of discrete-valued test functions is introduced, and the performance of the method is determined on a randomly generated set of these functions. Overall, this method has the power of other Bayesian/sampling techniques without the need for a separate local optimization technique for improved convergence. This makes it possible for the search to operate on unknown functions that may contain one or more discrete components. >

Optimal periodic control

Abstract: Optimization theory.- Retarded functional differential equations.- Strong local minima.- Weak local minima.- Local relaxed minima.- Tests for local properness.- A scenario for local properness.- Optimal periodic control of ordinary differential equations.

Nonlinear optimization simplified by hypersurface deformation

Abstract: A general strategy is advanced for simplifying nonlinear optimization problems, the "ant-lion" method. This approach exploits shape modifications of the costfunction hypersurface which distend basins surrounding low-lying minima (including global minima). By intertwining hypersurface deformations with steepest-descent displacements, the search is concentrated on a small relevant subset of all minima. Specific calculations demonstrating the value of this method are reported for the partitioning of two classes of irregular but nonrandom graphs, the "prime-factor" graphs and the "pi" graphs. We also indicate how this approach can be applied to the traveling salesman problem and to design layout optimization, and that it may be useful in combination with simulated annealing strategies.

Robustness in the Presence of Joint parametric Uncertainty and Unmodeled Dynamics

Abstract: It is shown that, in the case of joint real parametric and complex uncertainty, Doyle's structured singular value can be obtained as the solution of a smooth constrained optimization problem. While this problem may have local maxima, an improved computable upper bound to the structured singular value is derived, leading to a sufficient condition for robust stability and performance.

New reduction technique by step error minimization for multivariable systems

Abstract: A method of reduction of system order is introduced, in which a lower-order transfer function/transfer matrix is obtained by minimizing an error function constructed from the time responses of the system and the reduced model. The denominator of the reduced model can be selected either by retaining dominant poles or by the Routh method. The numerator of the reduced model is obtained by minimizing the step response error along with a steady-state constraint. The error function is converted into the frequency domain and the minima operation is carried out in this domain. Since the numerator polynomial selection is independent of the denominator, the method is very general and can be applied for MIMO systems as well as for SISO systems. A numerical example of a multivariable system illustrates the procedure.

Adaptive training of multilayer neural networks using a least squares estimation technique

Abstract: A technique is developed for the training of artificial neural networks, using a modification of the Marquardt-Levenberg optimization technique. An adaptive choice of the convergence rate factor mu , based on the contribution of each neuron in the minimization of the error function, is presented that can be very useful in handling the problem of local minima of the error function. The proposed algorithm is more powerful but also more elaborate than backpropagation. Moreover, it can be shown that in some applications its computational complexity can be made similar to that of backpropagation by using fast implementations of the least-squares method. >

Automatic test generation using neural networks

Abstract: An automatic test pattern generation (ATPG) methodology that has the potential to exploit fine-grain parallel computing and relaxation techniques is described. The approach is radically different from the conventional methods used to generate tests for circuits from their gate-level descriptions. A digital circuit is represented as a bidirectional network of neurons. The circuit function is coded in the firing thresholds of neurons and the weights of interconnection links. This neural network is suitably reconfigured for solving the ATPG problem. A fault is injected into the neural network and an energy function is constructed with global minima at test vectors. Global minima are determined by a probabilistic relaxation technique augmented by a directed search. Preliminary results on combinational circuits confirm the feasibility of the technique. >

An algorithm for compressing digital contour data

Abstract: When contour lines are digitised in a stream mode, excessive data is generated. Some algorithms are needed to remove this excess. In this paper, these algorithms are reviewed and compared, and finally a new algorithm is proposed based on selecting local maxima and minima.

Dynamics of Alopex Process: Application to Optimization Problems

Abstract: The Alopex process has been applied to a variety of problems in which maxima or minima of functions of many parameters are to be determined. We present here different forms of the algorithm and a simple theory of the dynamics of the process. Computer simulations of several applications are presented to illustrate the versatility of the method. An artificial visual system is described in which partial stimuli are completed and patterns are created out of noise with only a scalar cost function to guide the process.

An investigation on local minima of a Hopfield network for optimization circuits

Abstract: The Hopfield architecture can be utilized in the VLSI implementation of several important optimization functions. A description is given of the properties of local minima in the energy function of Hopfield networks. A novel design technique to eliminate such local minima has been developed. The neural-based analog-to-digital converter is used as an example to demonstrate this design technique. Experimental results agree well with theoretical calculations on the output characteristics of the analog-to-digital converter. >

Gradient descent fails to separate

Abstract: In the context of neural network procedures, it is proved that gradient descent on a surface defined by a sum of squared errors can fail to separate families of vectors. Each output is assumed to be a differentiable monotone transformation (typically the logistic) of a linear combination of inputs. Several examples are given of two families of vectors for which a linear combination exists that will serve to separate the two families. However, the minimum cost solution does not yield the desired combination. The examples include several cases where there are no local minima, as well as a one-layer system showing local minima with a large basin of attraction. In contrast to the perceptron convergence theorem, which proves that the perceptron architecture, there is no convergence theorem for gradient descent which would allow correct classification. The theorem disproves the presumption made in recent years, that barring local minima, gradient descent will find the best set of weights for a given problem. >

Searching for optimal configurations by simulated tunneling

Abstract: Optimization theory deals with algorithms finding the lowest cost (energy) configuration in a minimal number of steps. When the cost function has many local minima, the deterministic algorithms become easily trapped in suboptimal solutions. The simulated annealing method tries to overcome this difficulty by introducting thermal noise in the problem. Here we explore the possibility of implementing search procedures analogous to the quantum tunneling effect. The suggested dynamics is a guided diffusion process of an interactingpopulation of configurations. Different dynamical aspects of the search process are formulated first in a simple one-dimensional tight-binding model with a hierarchical potential. The new algorithm is then applied to the Traveling Salesman Problem. It is demonstrated that the use of interacting, evolving populations of tours representing our “wave packet” leads to systematic improvements and possibly, to the optimal tour. In addition, the structure of the cost function landscape for a given instance becomes locally accessible. The performance of the algorithm and its implications for parallel computing and “genetic” programming are briefly discussed.

Analysis of alternative realizations of adaptive IIR filters

Abstract: A general theory, based on an analysis of stationary points, presented which shows that whenever a direct-form IIR (infinite-impulse-response) filter with unimodal MSE (mean-squared-error) surface is transformed into an alternate realization, the MSE surface associated with the new structure may have additional stationary points, which are either new equivalent minima (and hence indistinguishable at the filter output), or saddle points, which are unstable solutions in the parameter space. The general theory is specialized to parallel and cascade forms. It is also shown that, for both the parallel and cascade forms, a gradient algorithm will find a global minimum as long as there is some noise present to jitter the solution away from the reduced-order manifolds which may contain saddle points. Experimental examples were presented to illustrate that the predicted behavior is indeed observed in practice. >

A universal optimisation network

Abstract: An optimization algorithm was developed that causes the simultaneous convergence of a large number of parameters determining the value of a scaler cost function. The procedure is iterative and stochastic, and tends to avoid local extrema. It is shown that the number of iterations required for convergence of the cost function increases linearly with the number of parameters. The procedure is universal in that it can be applied without modification to a large variety of optimization problems. Several examples are discussed, and results of computer simulations are presented. >

On the uniqueness of local minima for general abstract nonlinear least-squares problems

Abstract: The effectiveness of the inversion of a mapping phi defined on a set C by nonlinear least-squares techniques relies on, among other things, the uniqueness of local minima of the least-squares criterion, which ensures that the numerical optimisation algorithm (if they do) converges towards the global minimum of the least-squares functional. The author defines a number y depending only on C and phi which, if the size of phi (C) is not too large with respect to its curvature, is strictly positive, thus yielding the uniqueness of all local minima having a value smaller than y. The condition y>0 requires neither convexity of C nor any monotonic property of phi , but involves the computation of an infimum over delta C* delta C of first and second derivatives of phi . Numerical applications to the estimation of two parameters in a parabolic equation are given.

Non-Abelian self-interaction of a test loop.

Abstract: The classical reparametrization-invariant SU(2) self-interactions of a test string are analyzed. Dynamically noncollapsing loop configurations in real space and on the intrinsic S/sub 2/ manifold are shown to exist, reflecting a combined effect of non-Abelian charges and currents. The dynamics leads naturally to a double-well ''Weierstrass'' potential whose minima specify two discrete configurations in the internal space.

A survey of large time asymptotics of simulated annealing algorithms

Abstract: Simulated annealing is a probabilistic algorithm for minimizing a general cost function which may have multiple local minima The amount of randomness in this algorithm is controlled by the “temperature”, a scalar parameter which is decreased to zero as the algorithm progresses We consider the case where the minimization is carried out over a finite domain and we present a survey of several results and analytical tools for studying the asymptotic behavior of the simulated annealing algorithm, as time goes to infinity and temperature approaches zero

A discrete search technique for global optimization

Abstract: We have developed a new algorithm which uses the trajectories of a discrete dynamical system to sample the domain of an objective function in search of global minima. We demonstrate the effectiveness of this algorithm by applying it to a model geometry optimization problem. Significant improvements in optimization efficiency are demonstrated, in that our algorithm returns lower minima than conventional line minimization in 79% of the optimization runs we made. The method is orders of magnitude less computationally intensive than simulated annealing, while returning good minima for functions possessing many local minima. The method is simple to program, and requires only two-point gradient calculations for its implementation.

On the computation of stationary points on potential energy hypersurfaces

Abstract: In this article, a modification of a procedure proposed by Zirrilli et al. for solving nonlinear equations is presented. The method permits the computation of minima and saddle points of energy functionals. The Muller-Brown test potential and the quantum chemical description of some proton transfer reactions are given as examples.

Synthesis of arbitrary and conformal arrays using non-linear optimization techniques

Abstract: Two related synthesis problems associated with coherent sensor array processing are discussed. The first problem concerns the determination of the geometric arrangement of a fixed number of array elements to achieve an optimal response; the other is to find a set of shading weights for an array of predetermined geometry. The case of an array of sensors with arbitrary 3-D geometry or conformity to a curved sheet in space, which in general do not have analytic solutions, is treated. These synthesis problems are solved by the application of nonlinear optimization techniques. Methods are presented that are suitable for configuring arrays of arbitrary geometry, applicable for most coherent signal-processing algorithms, along with a method for determining weights useful for conventional beamforming. As with most nonlinear optimization problems, the solutions represent only local optimality and not global optimality. The usual technique of reoptimization with random or perturbed initial values should be used to insure that suitable local minima are found. >

Error bounds for the nonlinear filtering of signals with small diffusion coefficients

Abstract: New upper and lower bounds for the nonlinear filtering problem are presented. The lower bounds are especially useful in the region of small diffusion coefficients where previously known bounds are inefficient. The upper and lower bounds are shown to be tight. An example demonstrating the tightness of the bounds is presented. >

Parametric optimization: Pathfollowing with jumps

Abstract: We consider finite dimensional optimization problems depending on one real parameter t Recently, Jongen/Jonker/Twilt [9] studied the generic behaviour of such problems Based on this investigation, we propose a partial concept for finding a suitably fine discretization 0=to<…

A continuous set covering problem as a quasidifferentiable optimization problem

Abstract: In this paper we present an algorithm to solve a family of finite covering problems in . Given a compact, finitely convex decomposable set and an integer we are looking for the centers and the minimal radius of m balls with the property . It will be shown that this problem can be reduced to the computation of Dirichtlet tessellations (Voronoi sets) and the computation of minima of quasidifferentiable optimization problems.