Abstract
In nonlinear optimization, the dual problem is in general not easier to solve than the primal problem. Convex separable optimization problems, frequently arising in electrical and mechanical engineering, constitute a notable exception to the above rule. The dual problem is to optimize the dual objective functionℓ over a non-negative orthant, and the evaluation ofℓ reduces to the execution of independentlinear searches only. To generalize the idea, we also consider partially-separable problems with objective and constraint functions such that the Hessian matrix of the Lagrange function is a block-diagonal matrix with 2*2 blocks. The evaluation of the dual objective function is accordingly reduced to a number of independentplanar searches. Obviously, 3*3 blocks would lead tospatial searches, etc. We compare the performance of a primal and a dual method on a graded set of artificial test problems with increasing size, increasing degree of degeneracy, and increasing ill-conditioning. The observed speed-up by the dual approach varies between 2 and 30. Finally, we consider the potential of the dual approach for execution on parallel computers.
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Lootsma, F.A. A comparative study of primal and dual approaches for solving separable and partially-separable nonlinear optimization problems. Structural Optimization 1, 73–79 (1989). https://doi.org/10.1007/BF01637663
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DOI: https://doi.org/10.1007/BF01637663