Abstract
A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable but convex. It covers several standard problems, such as linear and quadratic programming, and has many applications in engineering. In this paper, we introduce the notion of minimal-penalty path, which is defined as the collection of minimizers for a family of convex optimization problems, and propose two methods for solving the problem by following the minimal-penalty path from the exterior of the feasible set. Our first method, which is also a constraint-aggregation method, achieves the solution by solving a sequence of linear programs, but exhibits a zigzagging behavior around the minimal-penalty path. Our second method eliminates the above drawback by following efficiently the minimum-penalty path through the centering and ascending steps. The global convergence of the methods is proved and their performance is illustrated by means of an example.
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References
Ermoliev, Y. M., Kryazhimskh, A., and Ruszczynski, A., Constraint Aggregation Principle in Convex Optimization, Preprint, 1995.
Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, 1994.
Golub, G. H., and Van Loan, C. F., Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 1989.
Tsing, N. K., Fan, K. H., and Verriest, E. I., On Analyticity of Functions Involving Eigenvalues, Linear Algebra and Its Applications, Vol. 207, pp. 159–180, 1994.
Diamond, P., and Kloeden, P., Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994.
Luenberger, D. G., Linear and Nonlinear Programming, Addison Wesley, Reading, Massachusetts, 1989.
Royden, H. L., Real Analysis, Macmillan, New York, New York, 1988.
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Fan, M.K.H., Gong, Y. Exterior Minimum-Penalty Path-Following Methods in Semidefinite Programming. Journal of Optimization Theory and Applications 100, 327–348 (1999). https://doi.org/10.1023/A:1021782202975
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DOI: https://doi.org/10.1023/A:1021782202975