Showing papers on "Approximation algorithm published in 1973"

Stochastic approximation algorithms and applications

Abstract: This study presents the conditions of applicability of stochastic approximation algorithms that minimize a mean-square error criterion for identification of a linear discrete-time stationary system without dynamical numerator. The acceleration of the convergence is discussed. Then a tentative is outlined to overcome the previous requirement of states accessibility.

Approximation algorithms for combinatorial problems

Abstract: Simple, polynomial-time, heuristic algorithms for finding approximate solutions to various polynomial complete optimization problems are analyzed with respect to their worst case behavior, measured by the ratio of the worst solution value that can be chosen by the algorithm to the optimal value. For certain problems, such as a simple form of the knapsack problem and an optimization problem based on satisfiability testing, there are algorithms for which this ratio is bounded by a constant, independent of the problem size. For a number of set covering problems, simple algorithms yield worst case ratios which can grow with the log of the problem size. And for the problem of finding the maximum clique in a graph, no algorithm has been found for which the ratio does not grow at least as fast as 0(ne), where n is the problem size and e> 0 depends on the algorithm.

Waveform Segmentation Through Functional Approximation

Abstract: Waveform segmentation is treated as a problem of piecewise linear uniform (minmax) approximation. Various algorithms are reviewed and a new one is proposed based on discrete optimization. Examples of its applications are shown on terrain profiles, scanning electron microscope data, and electrocardiograms. The processing is sufficiently fast to allow its use on-line. The results of the segmentation can be used for pattern recognition, data compression, and nonlinear filtering not only for waveforms but also for pictures and maps. In the latter case some additional preprocessing is required and it is described in [19].

Efficient compilation of linear recursive programs

Abstract: A linear recursive program consists of a set of procedures where each procedure can make at most one recursive call. The conventional stack implementation of recursion requires time and space both proportional to n, the depth of recursion. It is shown that in order to implement linear recursion so as to execute in time n one does not need space proportional to n : ne for arbitrarily small e will do. It is also known that with constant space one can implement linear recursion in time n. We show that one can do much better : ne for arbitrarily small c. We also describe an algorithm that lies between these two: it takes time n.log(n) and space log(n). In this context one can demonstrate a speed-up theorem for linear recursion - given any constant-space program implementing linear recursion, one can effectively find another constant space program that runs faster almost everywhere.

Repro-Modeling: An Approach to Efficient Model Utilization and Interpretation

Abstract: The growing accuracy and sophistication of models in many fields often lead, paradoxically, to limitations upon their use. Computational cost, excessive input data requirements, and difficulty in interpretation of the implications of the model are typical impediments to the broad use of many complex models. Repro-modeling is an approach which, in many cases, overcomes these problems by creating an efficient input/output approximation to the model. Practical methodological approaches to this straightforward concept are proposed, including a practical algorithm for obtaining a continuous multivariate piecewise linear approximation with subregions of general form given a small randomly distributed set of input/output samples. Three examples of the successful application of repro-modeling are outlined: the impact on air quality of a traffic-restriction policy, the design of effective traffic-responsive freeway on-ramp control algorithms, and the radar return from a complex object.

Applications of a general convergence theory for outer approximation algorithms

Abstract: Eaves and Zangwill [2] have developed a very general theory of the convergence of cutting plane algorithms. This theory is applied to prove the convergence of Geoffrion's Generalized Benders Decomposition procedure (GBD) [5]. Using the insight provided by the general theory, GBD is then modified to permit the deletion of old constraints without upsetting the infinite convergence property. Finally, certain approximations of GBD are presented and the robustness of the convergence results is indicated.

New Algorithms for Network Optimization

Abstract: Two new alogrithms suitable for computer-aided optimization of networks are presented. They are both based on the nonlinear least /spl rho/th approximation approach, which has been successfully applied by the authors to microwave network design problems requiring minimax or near-minimax solutions. A basic difference here is that, instead of requiring very large values of /spl rho/, any finite value of /spl rho/, greater than 1 can be used to produce extremely accurate minimax solutions. This paper discusses a six-variable transformer example where values of /spl rho/ equal to 2, 4, 6, 10, 100, 1000, and 10 000 have all been used separately to obtain substantially the same solution. Both the adjoint network method for gradient evaluation and the Fletcher method are employed for greater efficiency. Comparisons with the razor search and grazer search methods are made. Some far-reaching observations concerning minimax design are also made.

Theory of optimal algorithms

Abstract: Motivations for studying computational complexity are discus* recent results in algebraic and analytic computational complexity This report is based on an invited paper presented at a Conference for Numerical Mathematics held at Loughborough, England. ed. Some re surveyed, on Software DD ,Nr651473 ( P A G E n S / N 0 1 0 1 8 0 7 6 8 0 1 Unlimited S e c u r i t y C l a s i f i c a t i o n

Optimal Trajectories for Multidimensional Nonlinear Processes by Iterated Dynamic Programming

Abstract: A trajectory optimization technique for multidimensional nonlinear processes is presented. Problems which are cast in a discrete-time mold are considered. The method is based on dynamic programming and employs a combination of the technique of functional approximation and the method of region-limiting strategies. The cost function at each stage is approximated by a quadratic polynomial in a region which is restricted to be of a size judged appropriate to reduce the error in the approximation. Minimal costs are evaluated at a set of points, called base points. A new control trajectory and an improved state trajectory are then generated within an extrapolation region. The iterative application of this procedure yields an optimal trajectory. Contained in the algorithm is a simple procedure which eliminates matrix inversion to determine the coefficients of the approximating polynomial. The present algorithm is applicable to problems with one bounded control action. It accounts for inequality constraints on state variables in a straightforward manner. The algorithm is applied to solve a number of trajectory optimization problems.

Near-optimal state estimation for interconnected systems

Abstract: A new algorithm is proposed for estimating the state of a nonlinear stochastic system when only noisy observations of the state are available. The state estimation problem is formulated as a modaltrajectory maximum-likelihood estimation problem. The resulting minimization problem is analogous to the nonlinear tracking problem in optimal control theory. By viewing the system as an interconnection of lower-dimensional subsystems and applying the so-called ?-coupling technique, which originated in the study of the sensitivity of control systems to parameter variations, a near-optimal state estimation algorithm is derived which has the properties that all computations can be performed in parallel at the subsystem level and only linear equations need be solved. The principal attraction of the method is that significant reductions in the computational requirements relative to other approximate algorithms can be achieved when the system is large-dimensional.

New Algorithms for Network Optimization

Abstract: Two new algorithms suitable for optimization of networks are presented. Least pth approximation is used in such a way that any finite value of p > 1 can be used to produce minimax results very efficiently.

Quasi-Newton algorithms: Approaches and motivations

Abstract: This paper surveys some of the unifying approaches used to derive formulae for updating the inverse Hessian approximations in quasi-Newton algorithms and presents a new approach of this kind based on geometric considerations. The paper discusses the intuitive motivations for these approaches and their potential in providing explanations for observed behavior of such algorithms.

A segmented algorithm for solving a class of state constrained discrete optimal control problems

Abstract: A segmented algorithm for solving state constrained optimal control problems is presented. Its main feature is that it may be implemented on an array of digital computers. Such an implementation exhibits two attractive properties. The first one is that the computational time is reduced due to the parallelism of computations achieved. The second property is that a graceful degradation of performance is realized, i.e., the array produces approximate solutions in the presence of hardware faults.

A mixed stochastic optimization algorithm and its applications in pattern recognition

Abstract: The problem of the minimization of a functional of several parameters is treated. It is supposed that the gradient of the functional is known but the stochastic approximation method cannot be applied since its convergence is not guaranteed due to the form of the functional. A mixed random search-stochastic approximation method insuring this convergence is developed. Several applications in the field of pattern recognition are highlighted.

Generalized linear minimax approximation of system functions with constraints

Abstract: Analysis and an algorithm are providedfor the linear minimax approximation of system functions when equality constraints are imposed upon the approximant and/or the basis functions do not form a Chebyshev set in the region of approximation. The latter situation is restricted to the case of common zeros shared by all basis functions. The theory and algorithm are especially useful in the approximation of arbitrary system functions by functions with restricted Fourier transforms.

On the realization of an optimal decision rule by means of a hierarchial system

Abstract: This paper deals with the realization of a best approximation of an unknown optimal decision rule by means of a self-learning hierarchical system. This optimization implies the simultaneous use of adaptive estimation and classification techniques. An unsupervised learning discrete algorithm based on stochastic approximation theory is proposed as a solution to this problem. Some applications are suggested, particularly, in the field of modelling of static and dynamic non-linear relationships. Numerical results are given.

Maximum cost trajectories in optimally controlled stochastic systems

Abstract: An optimal control in a stochastic system is often one which minimizes the mean value of a specified performance index. Exclusive concern with this single parameter of dynamic response may mask anomalous behavior which occurs with small probability. Before utilizing an optimal controller further study of its properties is warranted. This paper provides an algorithm for finding an approximation to the least favorable trajectory for the system.