05-localization_methods_notes

05-localization_methods_notes - Localization and...

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Unformatted text preview: Localization and Cutting-Plane Methods S. Boyd and L. Vandenberghe April 13, 2008 Contents 1 Cutting-planes 2 2 Finding cutting-planes 3 2.1 Unconstrained minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Feasibility problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Inequality constrained problem . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Localization algorithms 7 3.1 Basic cutting-plane and localization algorithm . . . . . . . . . . . . . . . . . 7 3.2 Measuring uncertainty and progress . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Choosing the query point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Some specific cutting-plane methods 12 4.1 Bisection method on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Center of gravity method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 MVE cutting-plane method . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4 Chebyshev center cutting-plane method . . . . . . . . . . . . . . . . . . . . . 16 4.5 Analytic center cutting-plane method . . . . . . . . . . . . . . . . . . . . . . 16 5 Extensions 16 5.1 Multiple cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Dropping or pruning constraints . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 Nonlinear cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6 Epigraph cutting-plane method 18 7 Lower bounds and stopping criteria 19 1 In these notes we describe a class of methods for solving general convex and quasiconvex optimization problems, based on the use of cutting-planes , which are hyperplanes that sepa- rate the current point from the optimal points. These methods, called cutting-plane methods or localization methods , are quite different from interior-point methods, such as the barrier method or primal-dual interior-point method described in [BV04, § 11]. Cutting-plane meth- ods are usually less efficient for problems to which interior-point methods apply, but they have a number of advantages that can make them an attractive choice in certain situations. • Cutting-plane methods do not require differentiability of the objective and constraint functions, and can directly handle quasiconvex as well as convex problems. Each itera- tion requires the computation of a subgradient of the objective or contraint functions. • Cutting-plane methods can exploit certain types of structure in large and complex problems. A cutting-plane method that exploits structure can be faster than a general- purpose interior-point method for the same problem. • Cutting-plane methods do not require evaluation of the objective and all the constraint functions at each iteration. (In contrast, interior-point methods require evaluating all the objective and constraint functions, as well as their first and second derivatives.) This can make cutting-plane methods useful for problems with a very large number of...
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This note was uploaded on 04/09/2010 for the course EE 364B at Stanford.

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05-localization_methods_notes - Localization and...

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