Leocadio G. Casado

New interval methods for constrained global optimization

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.

A new multisection technique in interval methods for global optimization

Abstract A new multisection technique in interval methods for global optimization is investigated, and numerical tests demonstrate that the efficiency of the underlying global optimization method can be improved substantially. The heuristic rule is based on experiences that suggest the subdivision of the current subinterval into a larger number of pieces only if it is located in the neighbourhood of a minimizer point. An estimator of the proximity of a subinterval to the region of attraction to a minimizer point is utilized. According to the numerical study made, the new multisection strategies seem to be indispensable, and can improve both the computational and the memory complexity substantially.

New Interval Analysis Support Functions Using Gradient Information in a Global Minimization Algorithm

The performance of interval analysis branch-and-bound global optimization algorithms strongly depends on the efficiency of selection, bounding, elimination, division, and termination rules used in their implementation. All the information obtained during the search process has to be taken into account in order to increase algorithm efficiency, mainly when this information can be obtained and elaborated without additional cost (in comparison with traditional approaches). In this paper a new way to calculate interval analysis support functions for multiextremal univariate functions is presented. The new support functions are based on obtaining the same kind of information used in interval analysis global optimization algorithms. The new support functions enable us to develop more powerful bounding, selection, and rejection criteria and, as a consequence, to significantly accelerate the search. Numerical comparisons made on a wide set of multiextremal test functions have shown that on average the new algorithm works almost two times faster than a traditional interval analysis global optimization method.

Experiments with a new selection criterion in a fast interval optimization algorithm

Usually, interval global optimization algorithms use local search methods to obtain a good upper (lower) bound of the solution. These local methods are based on point evaluations. This paper investigates a new local search method based on interval analysis information and on a new selection criterion to direct the search. When this new method is used alone, the guarantee to obtain a global solution is lost. To maintain this guarantee, the new local search method can be incorporated to a standard interval GO algorithm, not only to find a good upper bound of the solution, but also to simultaneously carry out part of the work of the interval B&B algorithm. Moreover, the new method permits improvement of the guaranteed upper bound of the solution with the memory requirements established by the user. Thus, the user can avoid the possible memory problems arising in interval GO algorithms, mainly when derivative information is not used. The chance of reaching the global solution with this algorithm may depend on the established memory limitations. The algorithm has been evaluated numerically using a wide set of test functions which includes easy and hard problems. The numerical results show that it is possible to obtain accurate solutions for all the easy functions and also for the investigated hard problems.

A suite of algorithms for key distribution and authentication in centralized secure multicast environments

The Extended Euclidean algorithm provides a fast solution to the problem of finding the greatest common divisor of two numbers. In this paper, we present three applications of the algorithm to the security and privacy field. The first one allows one to privately distribute a secret to a set of recipients with only one multicast communication. It can be used for rekeying purposes in a Secure Multicast scenario. The second one is an authentication mechanism to be used in environments in which a public-key infrastructure is not available. Finally, the third application of the Extended Euclidean algorithm is a zero-knowledge proof that reduces the number of messages between the two parts involved, with the aid of a central server.

Branch-and-Bound interval global optimization on shared memory multiprocessors

The focus of this paper is on the analysis and evaluation of a type of parallel strategies applied to the algorithm Advanced Multidimensional Interval analysis Global Optimization (AMIGO). We investigate two parallel versions of AMIGO, called Parallel AMIGO (PAMIGO) algorithm, Global-PAMIGO and Local-PAMIGO. The idea behind our study is that in order to exploit the potential parallelism of algorithms, researchers need to adapt them to the target computer architectures. Our PAMIGO algorithms have been designed for shared memory architectures and are based on a threaded programming model, which is suitable to be run on current personal computers with multicore processors. Our first experimental results show a promising speed-up up to four process units. We analyse the loss of efficiency when the number of process units is greater than four by obtaining a profile of the algorithm executions. Secondly we experiment with the use of a local memory allocator per thread. This increases the efficiency by reducing the number of lock conflicts given by the standard system memory allocator. Our experimental results for both PAMIGO versions, using up to 15 process units, obtain a good performance for hard to solve problems on unicore and multicore processors. It is noteworthy that both versions of PAMIGO obtain a similar performance. Our experiments may be useful for researchers who use parallel B&B algorithms.

The semi-continuous quadratic mixture design problem: Description and branch-and-bound approach

The semi-continuous quadratic mixture design problem (SCQMDP) is described as a problem with linear, quadratic and semi-continuity constraints. Moreover, a linear cost objective and an integer valued objective are introduced. The goal is to deal with the SCQMD problem from a branch-and-bound perspective generating robust solutions. Therefore, an algorithm is outlined which identifies instances where decision makers tighten requirements such that no [epsilon]-robust solution exists. The algorithm is tested on several cases derived from industry.

Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints

The blending problem is studied as a problem of finding cheap robust feasible solutions on the unit simplex fulfilling linear and quadratic inequalities. Properties of a regular grid over the unit simplex are discussed. Several tests based on spherical regions are described and evaluated to check the feasibility of subsets and robustness of products. These tests have been implemented into a Branch-and-Bound algorithm that reduces the set of points evaluated on the regular grid. The whole is illustrated numerically.

Adaptive parallel interval branch and bound algorithms based on their performance for multicore architectures

This work studies how to adapt the number of threads of a parallel Interval Branch and Bound algorithm to the available computational resources based on its current performance. Basically, a thread can create a new thread that will process part of the ancestor workload. In this way, load balancing is inherent to the creation of threads. The applications in which we are interested use branch-and-bound algorithms which are highly irregular and therefore difficult to predict. The proposed methods can be used for more predictable algorithms as well. This research complements and does not substitute other devices that improve the exploitation of the system, such as dynamic scheduling policies or work-stealing. Several approaches are presented. They differ in the metrics used and in the need or not having to modify the Operating System (O.S.). The scenario for this research is just one multithreaded application running in a multicore architecture. Experimental results show that the appropriate number of running threads can be determined at run-time, avoiding having to statically establish the number of threads of an application. Thread creation decisions have to be made frequently to obtain better results, but are time-consuming. One of the presented models uses the existence of an idle processor to carry out these decisions, obtaining the desired results.

Multidimensional Optimization in Image Reconstruction from Projections

A parallel implementation of a global optimization algorithm is described. The algorithm is based on a probabilistic random search method. Computational results are illustrated through application of the algorithm to a time consuming problem, in a multidimensional space, which arises from the field of image reconstruction from projections.