We describe an algorithm for computing the value function for "all source, single destination " discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path problem on a finite graph. The method is illustrated by two numerical examples, namely a single pendulum on a cart and a parametrically driven inverted double pendulum.

The Matlab toolbox OPERA TB is a set of Matlab m- les, which solves basic linear and discrete optimization problems in operations research and mathematical programming. Included are routines for linear programming (LP), network programming (NP), integer programming (IP) and dynamic programming (DP). OPERA TB, like the nonlinear programming toolbox NLPLIB TB, is a part of TOMLAB � an environment in Matlab for research and teaching in optimization. Linear programs are solved either by direct call to a solver routine or to a multisolver driver routine, or interactively, using the Graphical User Interface (GUI) or a menu system. From OPERA TB it is possible to call solvers in the Math Works Optimization Toolbox and, using a MEX- le interface, general-purpose solvers implemented in Fortran or C. The focus is on dense problems, but sparse linear programs may be solved using the commercial solver MINOS. Presently, OPERA TB implements about thirty algorithms and includes a set of test examples and demonstration les. This paper gives an overview of OPERA TB and presents test results for medium size LP problems. The tests show that the OPERA TB solver converges as fast as commercial Fortran solvers and is at least ve times faster than the simplex LP solver in the Optimization Toolbox 2.0andtwice as fast as the primal-dual interior-pointLP solver in the same toolbox. Running the commercial Fortran solvers using MEX- le interfaces gives a speed-up factor of ve to thirty- ve.

The paper presents the toolbox NLPLIB TB 1.0 (NonLinear Programming LIBrary); a set of Matlab solvers, test problems, graphical and computational utilities for unconstrained and constrained optimization, quadratic programming, unconstrained and constrained nonlinear least squares, box-bounded global optimization, global mixed-integer nonlinear programming, and exponential sum model fitting.

This paper proposes a nonparametric estimator for general regression functions with multiple regressors. The method used here is motivated by a remarkable result derived by Kolmogorov (1957) and later tightened by Lorentz (1966). In short, any continuous function f(x 1 ; : : : ; x d ) has the representation P 2d+1 k=1 ~ g( 1 ~ OE k (x 1 ) + \Delta \Delta \Delta + d ~ OE k (x d )), where ~ g(\Delta) is a continuous function, OE k (\Delta), k = 1; : : : ; 2d + 1, is Lipschitz of order one and strictly increasing, and j , j = 1; : : : ; d, is some constant. Extending this result to the case of smoother functions, we restrict f(\Delta) to be of the form k=1 g k ( k;1 OE k (x 1 ) + \Delta \Delta \Delta + k;d OE k (x d )), 1 t ! 1, where both g k (\Delta) and OE k (\Delta) are three times continuously differentiable and OE k (\Delta) is nondecreasing. These functions are estimated using regression cubic B-splines, which have excellent numerical and approximating properties. One of the main contributions of this paper is that we develop a method for imposing monotonicity on the cubic B-splines, a priori, such that the estimator is dense in the set of all monotonic cubic B-splines. The method requires only 2(r+1)+1 restrictions per each OE k (\Delta), where r is the number of interior knots. Rates of convergence in L 2 are the same as the optimal rate for the one-dimensional case. A simulation experiment shows that the estimator works well when t is small, f(\Delta) does not belong to this class of linear superpositions, and optimization is performed using the back-fitting algorithm. The monotonic restriction has many other applications besides the one presented here, such as estimating a demand function. With only r + 2 more constraints, it is also possible to impose ...

. Much progress has been made in constrained nonlinear optimization in the past ten years, but most large-scale problems still represent a considerable obstacle. In this survey paper we will attempt to give an overview of the current approaches, including interior and exterior methods and algorithms based upon trust regions and line searches. In addition, the importance of software, numerical linear algebra and testing will be addressed. We will try to explain why the difficulties arise, how attempts are being made to overcome them and some of the problems that still remain. Although there will be some emphasis on the LANCELOT and CUTE projects, the intention is to give a broad picture of the state-of-the-art. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA 2 Parallel Algorithms Team, CERFACS, 42 Ave. G. Coriolis, 31057 Toulouse Cedex, France 3 Central Computing Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England ...

We discuss the advantages and problems associated with fitting geometric data of the human torso obtained from magnetic resonance imaging, with high order (bicubic Hermite) surface elements. These elements preserve derivative (C 1 ) continuity across element boundaries and permit smooth anatomically accurate surfaces to be obtained with relatively few elements. These elements are fitted to the data with a new non-linear fitting procedure that minimises the error in the fit whilst maintaining C 1 continuity with non-linear constraints. Non-linear Sobelov smoothing is also incorporated into this fitting scheme. The structures fitted along with their corresponding Root Mean Squared (RMS) error, number of elements used and number of degrees-of-freedom (dof) per variable are: epicardium (0.91 mm, 40 elements, 142 dof), left lung (1.66 mm, 80 elements, 309 dof), right lung (1.69 mm, 80 elements, 309 dof), skeletal muscle surface (1.67 mm, 264 elements, 1010 dof), fat layer (1.79 mm, 264 e...

this article, we consider the set of functions G such that G(X; U) = H

Computer aided molecular design is a strategy in which a set of structural groups are systematically combined to form molecules with desired properties. In this paper, a mathematical programming based approach to computer aided molecular design is presented. Using a set of structural groups, the problem is formulated as a mixed integer nonlinear program in which discrete variables represent the number of each type of structural groups present in the candidate compound. The augmentedpenalty /outer-approximation algorithm is used to solve the MINLP to obtain compound(s) with an optimum value of an appropriate performance index such that molecular structural constraints, physical property constraints and process design limitations are met. With the current renewed interest in the environment, the suggested approach is applied to refrigerant design with an environmental constraint. The results indicate the viability of this approach. INTRODUCTION The chemical industry is constantly explo...

TOMLAB is a general purpose, open and integrated Matlab development environment for research and teaching in optimization on Unix and PC systems. One motivation for TOMLAB is to simplify research on practical optimization problems, giving easy access to all types of solvers; at the same time having full access to the power of Matlab. The design principle is: define your problem once, optimize using any suitable solver. In this paper we discuss the design and contents of TOMLAB, as well as some applications where TOMLAB has been successfully applied. TOMLAB is based on NLPLIB TB, a Matlab toolbox for nonlinear programming and parameter estimation, and OPERA TB 1.0, a Matlab toolbox for linear and discrete optimization. More than 65 different algorithms and graphical utilities are implemented. It is possible to call solvers in the Matlab Optimization Toolbox and general-purpose solvers implemented in Fortran or C using a MEX-file interface. Currently, MEX-file interfaces have been developed for

. Quasi-Newton methods are reliable and ecient on a wide range of problems, but they can require many iterations if the problem is ill-conditioned or if a poor initial estimate of the Hessian is used. In this paper, we discuss methods designed to be more ecient in these situations. All the methods to be considered exploit the fact that quasi-Newton methods accumulate approximate second-derivative information in a sequence of expanding subspaces. Associated with each of these subspaces is a certain reduced approximate Hessian that provides a complete but compact representation of the second derivative information approximated up to that point. Algorithms that compute an explicit reduced Hessian approximation have two important advantages over conventional quasi-Newton methods. First, the amount of computation for each iteration is signicantly less during the early stages. This advantage is increased by forcing the iterates to linger on a manifold whose dimension can be signicantly sma...