This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well-known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen-Segupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

High performance applications involving large data sets require the efficient and flexible use of multiple disks. In an external memory machine with D parallel, independent disks, only one block can be accessed on each disk in one I/O step. This restriction leads to a load balancing problem that is perhaps the main inhibitor for adapting single-disk external memory algorithms to multiple disks. This paper shows that this problem can be solved efficiently using a combination of randomized placement, redundancy and an optimal scheduling algorithm. A buffer of O(D) blocks suffices to support efficient writing of arbitrary blocks if blocks are distributed uniformly at random to the disks (e.g., by hashing). If two randomly allocated copies of each block exist, N arbitrary blocks can be read within dN=De + 1 I/O steps with high probability. In addition, the redundancy can be reduced from 2 to 1 + 1=r for any integer r. These results can be used to emulate the simple and powerful "single-disk multi-head" model of external computing [1] on the physically more realistic independent disk model [33] with small constant overhead. This is faster than a lower bound for deterministic emulation [3].

This survey covers the state of the art of techniques for solving general purpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably nd all solutions (if there are nitely many). The core of the material is presented in sucient detail that the survey may serve as a text for teaching constrained global optimization.

A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, convert the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of ffl-finite convergence and ffl-global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.

For the design and analysis of algorithms that process huge data sets, a machine model is needed that handles parallel disks. There seems to be a dilemma between simple and flexible use of such a model and accurate modelling of details of the hardware. This paper explains how many aspects of this problem can be resolved. The programming model implements one large logical disk allowing concurrent access to arbitrary sets of variable size blocks. This model can be implemented efficienctly on multiple independent disks even if zones with different speed, communication bottlenecks and failed disks are allowed. These results not only provide useful algorithmic tools but also imply a theoretical justification for studying external memory algorithms using simple abstract models.

. We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multidimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension. 1. Introduction and Basic Concepts Our underlying problem is: (1) find all global minimizers to f(x) subject to x 2 X; where X ae R m is a compact right parallelepiped with face...

present s a series of new algorit hms for reliably graphingt wo-dimensional implicit equat ions and inequalit ies. A clear st andard for int erpret ingt he graphs generat ed byt wo-dimensional graphing soft ware is int roduced and used t o evaluat et he present ed algorit hms. The first approach present ed uses a st andard int erval arit hmet ic library. This approach is shownt o be fault y; an analysis oft he failure reveals a limit at ion of st andard int erval arit hmet ic. Subsequent algorit hms are developed in parallel wit h improvement s and ext#E sions t# t# e int erval ari t#met# c used byt he graphing algorit hms. Graphs exhibit ing a variet y of mat hemat ical and art ist ic phenomena are shownt o be graphed correct ly byt he present ed algorit hms. A brief comparison of t he final algorit hm present edt o ot her graphing algorit hms is included.

An important branch in scientific computing is parameter estimation. Given a mathematical model and observation data, parameters are sought to explain physical properties as well as possible. In order to find these parameters an optimization problem is often formed, frequently a nonlinear least squares problem. This thesis mainly contributes to the development of tools, techniques, and theories for nonlinear least squares problems that lack a well-defined solution. Specifically, the intention is to generalize regularization methods for linear inverse problems to also handle nonlinear inverse problems. The investigation started by considering an exactly rank-deficient problem, i.e., a problem with a dependency among the parameters. It turns out that such a problem can be formulated as a nonlinear minimum norm problem. To solve this optimization problem two regularization methods are proposed: A Gauss-Newton Tikhonov regularized method and a minimum norm GaussNewton method. It is shown t...

Many problems in control system analysis and design can be posed in a setting where a system with a fixed model structure and nominal parameter values is affected by parameter variations. An example is parametric robustness analysis, where the parameters might represent physical quantities that are known only to within a certain accuracy, or vary depending on operating conditions etc. Frequently asked questions here deal with performance issues: "How bad can a certain performance measure of the system be over all possible values of the parameters?" Another example is parametric controller design, where the parameters represent degrees of freedom available to the control system designer. A typical question here would be: "What is the best choice of parameters, one that optimizes a certain design objective?" Many of the questions above may be directly restated as optimization problems: If q denotes the vector of parameters, Q

Interval analysis is a powerful tool which allows to design branch-and-bound algorithms able to solve many global optimization problems. In this paper we present new adaptive multisection rules which enable the algorithm to choose the proper multisection type depending on simple heuristic decision rules. Moreover, for the selection of the next box to be subdivided, we investigate new criteria. Both the adaptive multisection and the subinterval selection rules seem to be specially suitable for being used in inequality constrained global optimization problems. The usefulness of these new techniques is shown by computational studies.