Multidimensional bisection: a dual viewpoint

AbstractThis paper provides an alternative viewpoint of multidimensional bisection global optimization methods of Wood. A dual coordinate representation of convex bodies is introduced which leads to an easy implementation and eliminates the need to see the geometry of intersecting simplexes. Although developed in the context of global optimization, the techniques deal more generally with regions represented as the union of convex bodies. With this dual framework the algorithm can be implemented efficiently using any multiattribute index data structure that allows for quick range queries. A C version using a “multi-key double linked skip list” based on Pugh's skip list has been implemented

On parallel Branch and Bound frameworks for Global Optimization

Branch and Bound (B&B) algorithms are known to exhibit an irregularity of the search tree. Therefore, developing a parallel approach for this kind of algorithms is a challenge. The efficiency of a B&B algorithm depends on the chosen Branching, Bounding, Selection, Rejection, and Termination rules. The question we investigate is how the chosen platform consisting of programming language, used libraries, or skeletons influences programming effort and algorithm performance. Selection rule and data management structures are usually hidden to programmers for frameworks with a high level of abstraction, as well as the load balancing strategy, when the algorithm is run in parallel. We investigate the question by implementing a multidimensional Global Optimization B&B algorithm with the help of three frameworks with a different level of abstraction (from more to less): Bobpp, Threading Building Blocks (TBB), and a customized Pthread implementation. The following has been found. The Bobpp implementation is easy to code, but exhibits the poorest scalability. On the contrast, the TBB and Pthread implementations scale almost linearly on the used platform. The TBB approach shows a slightly better productivity

Multidimensional bisection : the performance and the context

Two aspects of the multidimensional bisection algorithms for the global optimisation of Lipschitz continuous functions are investigated. Firstly, for several test functions we examine the numerical performance of the deepest point algorithm and two acceleration procedures. Secondly, we phrase the branch and bound framework of Horst and Tuy in terms of covers, and show the algorithms to be included in this framework. A result of Basso on the convergence of localisations is extended to higher dimensions

Equivalent methods for global optimization

The envelope used by the algorithm of Breiman and Cutler [4] can be smoothed to create a better algorithm. This is equivalent to an accelerated algorithm developed by the third author and Cutler in [3] using envelopes which seemed poor ones at first sight. Explaining this anomaly lead to a general result concerning the equivalence of methods which use information from more than one point at each stage and those that only use the most recent evaluated point. Smoothing is appropriate for many algorithms, and we show it is an optimal strategy

Generating functions and the performance of backtracking adaptive search

Backtracking adaptive search is a simplified stochastic optimisation procedure which permits the acceptance of worsening objective function values. Key properties of backtracking adaptive search are defined and obtained using generating functions. Examples are given to illustrate the use of this methodology.17 page(s

Grover's quantum algorithm applied to global optimisation

Grover's quantum computational search procedure can provide the basis for implementing adaptive global optimization algorithms. A brief overview of the procedure is given and a framework called Grover adaptive search is set up. A method of Dürr and Hoyer and one introduced by the authors fit into this framework and are compared.15 page(s

Backtracking adaptive search : the distribution of the number of iterations to convergence

Backtracking adaptive search is a simplified stochastic optimiza-tion procedure which permits the acceptance of worsening objective function values. It generalizes the hesitant adaptive search, which in turn is a gener-alization of the pure adaptive search. In this paper, we use ideas from the theory of stochastic processes to determine the full distribution of the number of iterations to convergence for the backtracking adaptive search.16 page(s