Exterior minimum-penalty path-following methods in semidefinite programming

Exterior minimum-penalty path-following methods in semidefinite programming

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Article ID: iaor2000481
Country: United States
Volume: 100
Issue: 2
Start Page Number: 327
End Page Number: 348
Publication Date: Feb 1999
Journal: Journal of Optimization Theory and Applications
Authors: ,
Keywords: semidefinite programming
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 behaviour around the minimal-penalty path. Our second method eliminates the above drawback by following efficiently the minimum-penalty path through the centring 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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