A hybrid constraint programming and semidefinite programming approach for the stable set problem

This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming. The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable domain values, based on the solution of a semidefinite relaxation. Using this ranking, we generate the most promising subproblems first, by exploring a search tree using a limited discrepancy strategy. Then the subproblems are being solved using a constraint programming solver. To strengthen the semidefinite relaxation, we propose to infer additional constraints from the discrepancy structure. Computational results show that the semidefinite relaxation is very informative, since solutions of good quality are found in the first subproblems, or optimality is proven immediately.Comment: 14 page

Error analysis for stellar population synthesis as an inverse problem

Stellar population synthesis can be approached as an inverse problem. The physical information is extracted from the observations through an inverse model. The process requires the transformation of the observational errors into model errors. A description is given for the error analysis to obtain objectively the errors in the model. Finding a solution for overdetermined and under-determined case was the purpose of two preceding papers. This new one completes the problem of stellar populations synthesis by means of a data base, by providing practical formul\ae defining the set of acceptable solutions. All solutions within this set are compatible, at a given confidence level, with the observations.Comment: 11 pages, LaTeX, 4 figures, 1 table. M.N.R.A.S.(2000) in pres

Parallel Decomposition Procedures for Large-scale Linear Programming Problems

In practice, many large-scale linear programming problems are too large to be solved effectively due to the computer\u27s speed and/or memory limitation, even though today\u27s computers have many more capabilities than before. Algorithms are exploited to solve such large linear programming problems, either in the sequential or parallel computation environment. This study focuses on two parallel algorithms for solving large-scale linear programming problems efficiently. The first parallel decomposition algorithm discussed in this study is from the theory problems in a special block-angular structure. The theory or the decomposition principle is first examined. Since the subproblems of a linear programming problem can be in any of the three possible cases—optimal solution case, unbounded solution case and no solution case, examples are provided for solving the problem when its subproblems are in any of these cases. The concept of extreme directions is discussed due to its direct connection with the unbounded solution case. A parallel computation code, which can handle all these cases, is implemented in this study with the decomposition principle theory and its performance is tested for large-scale linear programming problems. Only the problems in the special block-angular structure can be solved with the decomposition principle. For general linear programming problems, this study proposed a new decomposition algorithm named “division by the interior point”. The idea of this new algorithm is as follows: with a found interior point inside the feasible region, divide the feasible region into multiple subregions and use multiple processors to solve the problem in each subregion. This new algorithm is first demonstrated with a few small numerical examples. A parallel computation code in this new idea is implemented and tested with large-scale linear programming problems