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Positive definite separable quadratic programs for non-convex problems

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Abstract

We propose to enforce positive definiteness of the Hessian matrix in a sequence of separable quadratic programs, without demanding that the individual contributions from the objective and the constraint functions are all positive definite. For problems characterized by non-convex objective or constraint functions, this may result in a notable computational advantage. Even though separable quadratic programs are of interest in their own right, they are of particular interest in structural optimization, due to the so-called ‘approximated-approximations’ approach. This approach allows for the construction of quadratic approximations to the reciprocal-like approximations used, for example, in CONLIN and MMA. To demonstrate some of the ideas proposed, the optimal topology design of a structure subject to local stress constraints is studied as one of the examples.

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Notes

  1. Fleury (1979) and Wood and Groenwold (2009, 2010) have discussed nonconvex SAO algorithms, but these algorithms are all problem specific and they can hardly be considered ‘general’.

  2. It is in fact in general not possible to develop analytical relationships between the primal and the dual variables for weight minimization problems subject to multiple constraints, should a dual method be opted for, e.g. see (Groenwold and Etman 2008).

  3. If anything, the use of a continuation strategy seems to emphasize the computational advantage of exploiting non-convexity.

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Acknowledgements

I am grateful indeed to my former student, Dr. Derren W. Wood, for enlightening discussions about some of the intricacies associated with the singularities that arise due to local stress constraints.

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  1. Department of Mechanical and Mechatronic Engineering, University of Stellenbosch, Stellenbosch, South Africa

    Albert A. Groenwold

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  1. Albert A. Groenwold
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Correspondence to Albert A. Groenwold.

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Groenwold, A.A. Positive definite separable quadratic programs for non-convex problems. Struct Multidisc Optim 46, 795–802 (2012). https://doi.org/10.1007/s00158-012-0810-8

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