Concurrent processing of mixed‐integer non‐linear programming problems

Abstract

Purpose

To discuss a new parallel algorithmic platform (minlp_machine) for complex mixed‐integer non‐linear programming (MINLP) problems.

Design/methodology/approach

The platform combines features from classical non‐linear optimization methodology with novel innovations in computational techniques. The system constructs discrete search zones around noninteger discrete‐valued variables at local solutions, which simplifies the local optimization problems and reduces the search process significantly. In complicated problems fast feasibility restoration may be achieved through concentrated Hessians. The system is programmed in strict ANSI C and can be run either stand alone or as a support library for other programs. File I/O is designed to recognize possible usage in both single and parallel processor environments. The system has been tested on Alpha, Sun and Linux mainframes and parallel IBM and Cray XT4 supercomputer environments. The constrained problem can, for example, be solved through a sequence of first order Taylor approximations of the non‐linear constraints and feasibility restoration utilizing Hessian information of the Lagrangian of the MINLP problem, or by invoking a nonlinear solver like SQP directly in the branch and bound tree. minlp_machine( ) has been tested as a support library to genetic hybrid algorithm (GHA). The GHA(minlp_machine) platform can be used to accelerate the performance of any linear or non‐linear node solver. The paper introduces a novel multicomputer partitioning of the discrete search space of genuine MINLP‐problems.

Findings

The system is successfully tested on a small sample of representative MINLP problems. The paper demonstrates that – through concurrent nonlinear branch and bound search – minlp_machine( ) outperforms some recent competing approaches with respect to the number of nodes in the branch and bound tree. Through parallel processing, the computational complexity of the local optimization problems is reduced considerably, an important aspect for practical applications.

Originality/value

This paper shows that binary‐valued MINLP‐problems will reduce to a vector of ordinary non‐linear programming on a suitably sized mesh. Correspondingly, INLP‐ and ILP‐problems will require no quasi‐Newton steps or simplex iterations on a compatible mesh.

Citation

Östermark, R. (2009), "Concurrent processing of mixed‐integer non‐linear programming problems", Kybernetes, Vol. 38 No. 6, pp. 966-989. https://doi.org/10.1108/03684920910973180

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