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05-localization_methods_notes

05-localization_methods_notes - Localization and...

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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
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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 constraints. Cutting-plane methods can be used to decompose problems into smaller problems that can be solved sequentially or in parallel. To apply these methods to nondifferentiable problems, you need to know about subgra- dients, which are described in a separate set of notes. More details of the analytic center cutting-plane method are given in another separate set of notes. 1 Cutting-planes The goal of cutting-plane and localization methods is to find a point in a convex set X R n , which we call the target set , or, in some cases, to determine that X is empty. In an optimization problem, the target set X can be taken as the set of optimal (or ǫ -suboptimal) points for the problem, and our goal is find an optimal (or ǫ -suboptimal) point for the optimization problem.
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