ConvexOptimizationII-Lecture07 -...

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ConvexOptimizationII-Lecture07 Instructor (Stephen Boyd) :Well, this is – we’re all supposed to pretend it’s Tuesday. I’ll be in Washington, unfortunately. So today I’ll finish up and give a wrap up on the Analytic Center Cutting-Plane Method, and then we’ll move on to, actually, one of the coolest topics that really kind of finishes this whole section of the class, and that’s the Ellipsoid Method. So we’ll look at this, and I’ll try to make clear what is useful and what’s not. The Analytic Center Cutting-Plane Method is useful. When you have a problem that you need to solve where you really do only have access to something like a cutting plane or sub-gradient oracle, and for whatever reason – you have to look at those reasons very carefully. But if you have such a problem, this is an awfully good choice, and it’s going to beat any sub-gradient type method very much. This is going to have much more computation, obviously, per step, more storage, all sorts of stuff like that, compared to sub-gradient method, which involves essentially zero computation and zero storage. They’re way, way, way, way better than sub-gradient methods. Okay, so here’s the Analytic Center Cutting-Plane Method. You’re given an initial polyhedron known to contain some targets which might be feasible points, might be epsilon sub-optimal points. Doesn’t really matter what it is. You find the analytic center of the polyhedron, which is to say more precisely, you find the analytic center of the linear inequalities that represent the polyhedron. And you query the cutting-plane oracle at that point. If that point is in your target set, you quit. Otherwise, you add, you append this new cutting-plane to the set. Of course, what happens now is your point X (K+1) is not in P (K+1) by definition. Well, sorry. It might be in it if it’s a neutral cut, but it’s on the boundary. What you need to do now is at the next step you’ll need to calculate the analytic center of that new set, and I think – I won’t go through this. There’s a lot of different methods. Infeasible Sartinutin Method is the simplest one. Maybe not the best, but we’ll go on and just go to a problem, and I’ll show how this works. So this is a problem we’ve looked at already several times. It’s piecewise linear minimization. It’s a problem with 20 variables and 100 terms, and an optimal value around one. Let me add, just to make sure this is absolutely certain and clear. If this problem were – if you just had to solve a problem like that, it goes without saying you would not use something like an analytic center. You’d just solve the LP. Let’s bear in mind that every one of these iterations requires an effort that’s at least on the order of magnitude of simply solving this problem to ten significant figures right off the bat by Barrier Method. So your code from last quarter, your homework code which shouldn’t have been too many lines, will actually solve this problem very, very quickly. In fact, anyone want to take a cut at how fast it would be?
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ConvexOptimizationII-Lecture07 -...

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