A nonlinear integer programming problem in general is NP-hard. If the objective function has some special form (e.g., convexity), the Lagrangian relaxation [1] or dynamic programming [2] can be used. If the objective function has a concave or log-concave property, the steepest ascent algorithm [3] can be used.
References
[1] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Univ. Press, 2004.
[2] T. Ibaraki and N. Katoh, Resource Allocation Problems: Algorithmic Approaches. The MIT Press, 1988.
[3] G. P. Akilov, L. V. Kantorovich, Functional Analysis, 2nd edition. Pergamon Press, 1982.
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