Heuristic Rejection in Interval Global Optimization

Based on the investigation carried out in Ref. 1, this paper incorporates new studies about the properties of inclusion functions on subintervals while a branch-and-bound algorithm is solving global optimization problems. It is found that the relative place of the global minimum value within the inclusion function value of the objective function at the current interval indicates mostly whether the given interval is close to a minimizer point. This information is used in a heuristic interval rejection rule that can save a considerable amount of computation. Illustrative examples are discussed and an extended numerical study shows the advantages of the new approach.     

Studying the Complexity of Global Verification for NP-Hard Discrete Optimization Problems

This paper examines the complexity of global verification for MAX-SAT, MAX-k-SAT (for k3), Vertex Cover, and Traveling Salesman Problem. These results are obtained by adaptations of the transformations that prove such problems to be NP-complete. The class of problems PGS is defined to be those discrete optimization problems for which there exists a polynomial time algorithm such that given any solution , either a solution can be found with a better objective function value or it can be concluded that no such solution exists and is a global optimum. This paper demonstrates that if any one of MAX-SAT, MAX-k-SAT (for k3), Vertex Cover, or Traveling Salesman Problem are in PGS, then P=NP.     

On the Pointwise Limit of Vector Charges with the Saks Property

The well-known theorem about the density of the space of probability charges with the Saks property in the space of all probability charges in the pointwise topology is proved in the vector case. New features are the uniform Saks property for the family of charges and sufficient conditions for the pointwise limit of a sequence of charges to have the Saks property.     

Ab initio Tertiary Structure Prediction of Proteins

A daunting challenge in the area of computational biology has been to develop a method to theoretically predict the correct three-dimensional structure of a protein given its linear amino acid sequence. The ability to surmount this challenge, which is known as the protein folding problem, has tremendous implications. We introduce a novel ab initio approach for the protein folding problem. The accurate prediction of the three-dimensional structure of a protein relies on both the mathematical model used to mimic the protein system and the technique used to identify the correct structure. The models employed are based solely on first principles, as opposed to the myriad of techniques relying on information from statistical databases. The framework integrates our recently proposed methods for the prediction of secondary structural features including helices and strands, as well as -sheet and disulfide bridge formation. The final stage of the approach, which culminates in the tertiary structure prediction of a protein, utilizes search techniques grounded on the foundations of deterministic global optimization, powerful methods which can potentially guarantee the correct identification of a protein's structure. The performance of the approach is illustrated with bovine pancreatic trypsin inhibitor protein and the immunoglobulin binding domain of protein G.     

Prefiltration technique via aggregation for constructing low‐density high‐quality factorized sparse approximate inverse preconditionings

This paper considers a new approach to a priori sparsification of the sparsity pattern of the factorized approximate inverses (FSAI) preconditioner using the so‐called vector aggregation technique. The suggested approach consists in construction of the FSAI preconditioner to the aggregated matrix with a prescribed sparsity pattern. Then small entries of the computed ‘aggregated’ FSAI preconditioning matrix are dropped, and the resulting pointwise sparsity pattern is used to construct the low‐density block sparsity pattern of the FSAI preconditioning matrix to the original matrix. This approach allows to minimize (sometimes significantly) the construction costs of low‐density high‐quality FSAI preconditioners. Numerical results with sample matrices from structural mechanics and thin shell problems are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces

We introduce the notion of the -prolongation of Lie algebras of differential operators on homogeneous spaces. The -prolongations are topological invariants that coincide with one-dimensional cohomologies of the corresponding Lie algebras in the case where V is a homogeneous space. We apply the obtained results to the spaces S ¹ (the Virasoro algebra) and .     

On Global Optimality Conditions and Cutting Plane Algorithms

We discuss global optimality conditions and cutting plane algorithms for DC optimization. The discussion is motivated by certain incorrect results that have appeared recently in the literature on these topics. Incidentally, we investigate the relation of the Tikhonov reciprocity theorem to the optimality conditions for general nonconvex global optimization problems and show how the outer-approximation scheme developed earlier for DC programming can be used to solve a wider class of problems.     

Shortest Path Auction Algorithm Without Contractions Using Virtual Source Concept

In this paper, the problem of finding a shortest path tree rooted at a given source node on a directed graph (SPT) is considered. A new efficient algorithm based on a primal-dual approach is presented, which improves both the convergence and the complexity of the best known auction-like algorithm. It uses the virtual source (VS) concept based on the following consideration: when a node i is visited for the first time by any algorithm which preserves verified the dual admissibility conditions, then the shortest path (SP) from the source node to i is found. Therefore, the SP from the source to the remaining nodes may be computed by considering i as a virtual source.We propose a very efficient implementation of an auction-like algorithm that uses this concept and enables us to obtain a computational cost of O(n ²), where n is the number of nodes.Numerical experimentsare reported showing that the new method outdoes previously proposed auction-like algorithms and is highly competitive with other state-of-art SP approaches.     

Rational Functions with Prescribed Global and Local Minimizers

A family of multivariate rational functions is constructed. It has strong local minimizers with prescribed function values at prescribed positions. While there might be additional local minima, such minima cannot be global. A second family of multivariate rational functions is given, having prescribed global minimizers and prescribed interpolating data.