B. Gartner. A subexponential algorithm for abstract optimization problems. SIAM J. Comput., 24:1018-1035, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....smallest ball of n points in (n 1) space. Since the necessary vertex evaluations can be performed in polynomial time, we have obtained an O(1:61 n poly(n) algorithm for that problem. Our randomized bound is not relevant here, since a randomized procedures of complexity e O( p n) is known [2]. They are relevant in the context of linear complementarity problems, though: For P matrix problems nothing better than O(2 n ) was known [13] Findings. To the best of our knowledge all previous algorithms searching for the sink work in a simplex like fashion. Start at some vertex, and ....

BERND G ARTNER. A subexponential algorithm for abstract optimization problems, SIAM J. Comput. 24 (1995), 1018--1035.

....margin classi er with or without explicit feature vectors is computed in a Lagrangian formulation with about n d variables and linear constraints. The jump from d to n d variables can have a great impact on the running time and choice of QP algorithm. Recent results in computational geometry [8, 11] give fast QP algorithms for the case of large n and small d , algorithms requiring about O(nd) log n) exp(O( p d ) arithmetic operations. The best bound on the number of arithmetic operations for a QP with n d variables and constraints is about O( n d) 3 L) where L is the precision ....

....to the exterior, meaning line segments with i weights set to either 0 or C . Computational geometry may have a practical algorithm to contribute for the case of n large and d small, say n 100; 000 and d 20. For this case, the generalized linear programming (GLP) paradigm of Matou sek et al. [8, 11]. The training vectors need not actually live in R d for small d , so long as the GLP dimension of the problem is small, where the GLP dimension is the number of support vectors in any subproblem de ned by a subset of the training vectors. On the theoretical side, we are wondering about the ....

B. Gartner. A subexponential algorithm for abstract optimization problems. SIAM J. Computing 24 (1995), 1018-1035.

....needed to determine the smallest enclosing ball. This problem, however, is not basis regular (the smallest enclosing ball may be determined by any number, between 2 and d 1, of points) and a naive implementation of the basis changing operation may be quite costly (in d) Nevertheless, G artner [84] showed that this operation can be performed in this case using expected e O( p d) arithmetic operations. Hence, the expected running time of the algorithm is O(d 2 n) e O( p d log d) A natural extension of the 1 center problem is to nd a disk of the smallest radius that contains k ....

....expected running time is O(n log n) There are several other extensions of the smallest enclosing ball problem. They include (i) computing the smallest enclosing ellipsoid of a point set [41, 66, 153, 176] ii) computing the largest ellipsoid (or ball) inscribed inside a convex polytope in R d [84], iii) computing a smallest ball that intersects (or contains) a given set of convex objects in R d (see [142] and (iv) computing a smallest area annulus containing a given planar point set. All these problems are known to be LP type, and thus can be solved using the algorithm described in ....

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B. Gartner, A subexponential algorithm for abstract optimization problems, SIAM J. Comput. , 24 (1995), 1018-1035.

....can be modeled directly as low dimensional linear programs. Many other problems, such as the circumcircle of a point set, are not linear programs, but the same techniques often apply to them. To explain this phenomenon, various authors have formulated a theory of generalized linear programming [3, 22, 28]. A generalized linear program (GLP, also known as an LP type problem) consists of a finite set S of constraints and an objective function f mapping subsets of S to some totally ordered space and satisfying the following properties: 1. For any A # B, f (A) # f (B) 2. For any A, p, and ....

....objective function defined over X . The problem is to compute f (S) using only evaluations of f on small subsets of S. A basis of a GLP is a set B such that for any A ( B, f (A) f (B) The dimension d of a GLP is the maximum cardinality of a basis. A number of efficient GLP algorithms are known [1, 3, 10, 15, 22, 28], the best running time of which is O(dnT f (d)E log n) where n is the number of constraints, T measures the time to test a proposed solution against a constraint (typically this is O(d) f is exponential or subexponential, and E is the time to perform a single basis evaluation. Indeed, these ....

B. G artner. A subexponential algorithm for abstract optimization problems. SIAM J. Comput. 24, 1995, pp. 1018--1035; http://www.inf.fu-berlin.de/inst/pubs/ tr-b-93-05.abstract.html.

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B. Gartner. A subexponential algorithm for abstract optimization problems. SIAM J. Comput., 24:1018-1035, 1995.

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B. Gartner. A subexponential algorithm for abstract optimization problems. SIAM J. Comput., 24:1018-1035, 1995.

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B. Gartner. A subexponential algorithm for abstract optimization problems. SIAM J. Comput., 24:1018-1035, 1995.

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