In computer graphics and computational geometry, a bounding volume for a set of objects is a closed volume that completely contains the union of the objects in the set. The average behavior of the ellipsoid algorithm is too slow, making it not competitive with the simplex algorithm. The optimal bounding ellipsoid for parameter identification is proposed in. A note on approximate minimum volume enclosing ellipsoid of. Recent developments in optimal bounding ellipsoidal parameter. Minimum volume enclosing ellipsoid file exchange matlab. Do the optimal sequential outer bounding sets e n, o n and p n provided by those three algorithms converge to a singleton if the membership set. Pdf comparative study of blind equalizers based on.

In 16 ellipsoid type of activation function is proposed for feedforward neural networks. A summary of the routines in ell lib is provided in sec. Pdf nonlinear system identification with a feedforward. The notion of psum appears in the brunnminkowskifirey. The extremal volume ellipsoids of convex bodies, their. The obe algorithms have been developed using a settheoretic formulation and are applicable to autoregressive with exogenous input arx models with bounded noise. Efficiently approximating the minimumvolume bounding box of a. In 15 unsupervised and supervised learning laws in the form of ellipsoids are used to find and tune the fuzzy function rules. It is also possible to approximate the minimum bounding box volume, to within any constant factor greater than one, in linear time. Nearoptimal deterministic algorithms for volume computation via. Nov 26, 20 we give a deterministic algorithm for computing an m ellipsoid of a convex body, matching a known lower bound. Optimal bounding ellipsoid obe algorithms offer an attractive alternative to traditional least squares methods for identifying linearinparameters signal and system models due to their low. This paper investigates a lagrangian dual problem for solving the optimal power. Optimal bounding ellipsoid algorithms compute ellipsoidal approximations to regions in the parameter space that are consistent with the observed data and the model assumptions.

Ellipsoidal state bounding, journal of optimization theory and. It is a problem of convex optimization to determine the minimumvolume. This algorithm has a low computational burden which allows to use it easily. Optimal bounding ellipsoid obe algorithms comprise a class of novel recursive identification methods for affineinparameters system and signal models.

Tracking characteristics of an obe parameterestimation. The first ellipsoidal set method for setmembership state estimation was established in and developed to optimal control problem with parametric families of ellipsoids in. Unifying the landmark developments in optimal bounding ellipsoid identification unifying the landmark developments in optimal bounding ellipsoid identification deller, j. This paper analyzes the performance of an optimal bounding ellipsoid obe algorithm for tracking timevarying parameters with incrementally bounded time variations. This formulation leads to convex optimization problems that can be essentially.

Extended ellipsoidal outerbounding setmembership estimation. It is a problem of convex optimization to determine the minimum volume. The ellipsoid algorithm for linear programming is a specific application of the ellipsoid. Nonlinear system identification with a feedforward neural network and an optimal bounded ellipsoid algorithm. The algorithms in this class 2,57 are temporally recursive and yield ellipsoids which are optimal, in a sense to be defined later. Therefore, given just the bounding ellipsoid you cannot determine the bounded ellipsoid. On equivalence and computational efficiency of the major. The ellipsoid algorithm is the rst polynomialtime algorithm discovered for linear programming. Multiweight optimization in bounding ellipsoid algorithms. The next theorem provides a characterization of a bounding ellipsoid for.

Among the set membership identification methods, the chosen algorithm is an optimal bounding ellipsoid type algorithm. Section 3 presents a simple initial volume approximation scheme, which constitutes the initialization stage of our algorithm. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a finite number of steps. For simplicity we generally refer to e for all n 1 as an ellipsoid. Do interpretable optimal bounding ellipsoid algorithms converge. However, the theoretical implications of the algorithm are very important, in particular, providing the. The algorithm for doing this involves finding an approximation to the diameter of the point set, and using a box oriented towards this diameter as an initial approximation to the minimum volume bounding box. We describe a number of algorithms to perform basic geometric operations. Phan department of business analytics and mathematical sciences, ibm t.

Sqd, which forces the probability density function pdf at the equalizer. Multiweight optimization in optimal bounding ellipsoid algorithms. Multiweight optimization in optimal bounding ellipsoid. Compared to normal learning algorithms, for example backpropagation, the optimal bounded ellipsoid obe algorithm has some better properties, such as faster convergence, since it has a similar. Rao, member, ieee, and yihfang huang, ieee abstmctrecently there seems to have been a resurgence of interest in recursive parameterbounding algorithms. Pdf in this paper, we provide two algorithms for blind equalization in a bounded. Under the minimumvolume and minimumtrace criteria, optimal ellipsoidal state bounding algorithms were introduced in. Neural network training with optimal bounded ellipsoid algorithm article in neural computing and applications 186. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Bounding volumes are used to improve the efficiency of geometrical operations by using simple volumes to contain more complex o. One of the main features of these temporally recursive algorithms is a discerning up.

Lecture notes on the ellipsoid algorithm the simplex algorithm was the. Convergence of optimal sequential outer bounding sets in. E2 being given ellipsoids, and p a given point, the tasks considered include. Ellipsoidal and gaussian kalman filter model for discrete. A linear statespace model is used, with the timevarying parameters represented by the state vector. The computational complexity of the optimal bounding ellipsoid obe algorithms is much lower than that of the exact polytope bounding algorithms 8 and nonrecursive linear programming based algorithms 9. Nearoptimal deterministic algorithms for volume computation. The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases osman guler. Jan 20, 2009 the solver is based on khachiyan algorithm, and the final solution is different from the optimal value by the prespecified amount of tolerance. The benefits including low computational efficiency, superior tracking ability, and selective updating that permits processor multitasking are enhanced by multiweight mw optimization in which the data history is considered in determining update times and optimal weights on the observations.

Smallest ellipsoid containing psum of ellipsoids with. Optimal bounding ellipsoid obe algorithms with interpretable optimization volume or trace criteria converge under the condition that the disturbance process. A lower bound for the objective function value overan ellipse is obtained by writing f as the sum of a convex and a concave function and replacing the concave part by an a. Tracking of timevarying parameters using optimal bounding. On the parameterized computation of minimum volume outer. Stochastic convergence of optimal bounding ellipsoid. We show in section 4 that there is a version of the ellipsoid algorithm that can be applied to find an e optimal solution of cpd in at most o n2 in cddl iterations of the ellipsoid algorithm, where each iteration must either perform a separation cut on one of the cones cx or cy, or must perform a related optimality cut. This leads to a nearly optimal deterministic algorithm for estimating the volume of a convex body and improved deterministic algorithms for fundamental lattice problems under general norms. Optimal bounding ellipsoid obe algorithms offer an attractive alternative to traditional leastsquares methods for identification and filtering problems involving affineinparameters signal and. Tracking characteristics of an obe parameter estimation algorithm ashok k. This matrix contains all the information regarding the shape of the ellipsoid. An optimal volume ellipsoid algorithm for parameter set estimation. Arma parameter estimation using a novel recursive estimation. Recursive blind equalization with an optimal bounding.

In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. The optimal bounding ellipsoid obe algorithms are viable alternatives to conventional adaptive filtering algorithms in situations where the noise does not. Uobe is based on generalized weighted recursive least squares in which very broad. Comparisons are made to the optimal bounding ellipsoid obe algorithm of fogel and. Lagrangian duality and branchandbound algorithms for optimal power flow dzung t. In fact, it is a nonconvex optimization problem for determining the minimum ellipsoid containing the. Compared to normal learning algorithms, for example backpropagation, the optimal bounded ellipsoid obe algorithm has some better properties, such as faster convergence, since it has a similar structure as kalman filter. Two families of algorithms were developed in order to determine the optimal outer bounding ellipsoid, one in estimation theory, known as optimal bounding ellipsoid algorithm obe and the other in linear programming, known as ellipsoid with parallel cuts algorithms epc 3, 4. Minimum enclosing ellipsoid to maximal enclosed ellipsoid. Nonlinear system identification with a feedforward neural. Consider the yellow and orange convex figures, which have the same bounding ellipsoid but different bounded ellipsoids. Comparative study of blind equalizers based on optimal bounding ellipsoid algorithms under awgn and fading channels.

Neural network training with optimal bounded ellipsoid algorithm. Solve the optimization problem 4445 of determining the maximum volume ellipsoid for the optimal solutions e. Conditionbased complexity of convex optimization in conic. Ddimensional vector containing the center of the ellipsoid. Ellipsoidal bounds on state trajectories for discretetime. If the address matches an existing account you will receive an email with instructions to reset your password.

Optimal bounding ellipsoid obe algorithms offer an. It also coincides with a central shallowcut ellipsoid method for the polar of s, and is the frankwolfe 1956 method specialized for d. Efficiently approximating the minimumvolume bounding box of. This paper describes some fundamental contributions to the theory and applicability of optimal bounding ellipsoid obe algorithms for signal processing. Research article extended ellipsoidal outerbounding set. On khachiyans algorithm for the computation of minimum. Unifying the landmark developments in optimal bounding. A quite general class of optimal bounding ellipsoid obe algorithms including all methods published to date, can be unified into a single framework called the unified obe uobe algorithm.

Recent developments in optimal bounding ellipsoidal. In particular, what happens to the outer bounding sets if the noise is indeed a sequence of random variables. Optimal bounding ellipsoid algorithms for adaptive blind. Lecture notes on the ellipsoid algorithm the simplex algorithm was the rst algorithm proposed for linear programming, and although the algorithm is quite fast in practice, no variant of it is known to be polynomial time. Smallest ellipsoid containing psum of ellipsoids with application to reachability analysis abhishek halder abstractwe study the problem of ellipsoidal bounding of convex setvalued data, where the convex set is obtained by the psum of.

1137 531 1524 1449 1108 1170 345 1439 41 606 747 153 1249 1493 1363 294 343 1334 1347 924 794 1201 920 1234 20 726 274 1407 372 625 104 99 399 1465 190 994 1335