S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6(3): 382--392, 1954.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problem [12] given a closed subset S R and a point y = 2 S can one nd a point x 2 R (8 x 2 S) kx xk ky xk : Inspired by this work, T.S. Motzkin and I.J. Schoenberg adopted in their 1954 paper [19] the term Fej er monotone to describe sequences satisfying (1) In this paper (see also [1]) an algorithm was developed to solve systems of linear inequalities in R by successive projections onto the half spaces de ning the polyhedral solution set S. The concept of Fej er monotonicity was shown to be an adequate tool to study convergence of this algorithm. Basic facts such as (5) ....

....systems of linear inequalities in R by successive projections onto the half spaces de ning the polyhedral solution set S. The concept of Fej er monotonicity was shown to be an adequate tool to study convergence of this algorithm. Basic facts such as (5) and (9) can already be found in [19] and [1], respectively. In the 1960s, I.I. Eremin extended the use of Fej er monotonicity to more general convex problems in Hilbert spaces. A summary of his publications covering the period 1961 1967 is given in [9] By the end of the 1960 s, most results on Fej er monotonicity in Hilbert spaces were ....

Agmon, S.: `The relaxation method for linear inequalities ', Canad. J. Math. 6 (1954), 382-392.

....an m n matrix game to within a multiple # of the range of payo#s requires at most m n 2 iterations, each requiring about 4mn flops. If only the 2 factor could be replaced by ln( this would yield a polynomial algorithm 6.3. Relaxation methods The classical relaxation method of Agmon [4] and Motzkin and Schoenberg [76] closely related to iterative methods for linear equations, was of great interest in the 1950s, partly because each iteration requires minimal computation, merely some matrix vector multiplications. To find a point in Y : c , at every iteration the current ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382--392, 1954.

....experiments (see Remark 6. 2 for speci c references) This type of extrapolated scheme was rst employed in the parallel projection method of Merzlyakov [52] to solve systems of ane inequalities in R ; the resulting algorithm was shown to be faster than the sequential projection algorithms of [1] and [54] An alternative interpretation of Pierra s algorithm 22 is the following: it can be obtained by taking T in Algorithm 5.1 as the subgradient projector de ned in (9) where f is the proximity function de ned in (74) A generalization of Pierra s algorithm will be proposed in Section ....

S. Agmon, The relaxation method for linear inequalities, Canadian Journal of Mathematics 6 (1954) 382-392.

....CCR 9732705 and CCR 0105488. Email: avrim cs.cmu.edu Department of Mathematics, MIT, Cambridge MA, 02139. Supported in part by NSF Career Award CCR 9875024. Email: jdunagan math.mit.edu In this paper, we show that a simple greedy linear programming algorithm known as the perceptron algorithm[2, 3], commonly used in machine learning, also has polynomial smoothed complexity (in a high probability sense) The problem being solved is identical to that considered by Spielman and Teng, except that we replace the objective function max c x by a constraint c x c 0 . In addition to ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6(3):382-392, 1954.

....system whose iteration complexity is bounded appropriately by a function of the condition number. The rst aspect has to do with the development and study of elementary algorithms for nding a point in a suitably described convex set, such as re ection algorithms for linear inequality systems (see [1], 15] 8] 13] the perceptron algorithm [21] 22] 23] 24] and the recently revived algorithm of von Neumann (see [6] 7] When applied to linear inequality systems, these algorithms share the following desirable properties, namely: the work per iteration is extremely low (typically ....

....2 (0; 1] and the dimension of the set P , the sequence generated by the relaxation scheme either converges to a point in P or to a spherical surface having the ane hull of P as its axis. Finite termination and or a geometric rate of convergence have been established for some of these methods (see [1], 15] 8] 13] for details) The von Neumann algorithm can be viewed (in the dual) as a relaxation algorithm with parameter 2 (0; 1) being chosen dynamically at each iteration, with a particularly intelligent choice of the parameter at each iteration. Elementary Algorithm for a Conic ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382-392, 1954.

....to the solution of unfeasible inequalities, while the second one is appropriate for the feasible case. A deterministic counterpart of the second method can be traced back to Kaczmarz method for solving linear equations [11] and to Agmon MotzkinShoenberg method for solving linear inequalities, [1], 12] For non linear convex inequalities, it has been proposed in [15] Aversion with finite convergence for linear inequalities was used by V. Yakubovich in the middle of the sixties for solving adaptive control problems (see [5] and references therein) and extended to nonlinear inequalities by ....

S. Agmon, "The Relaxation Method for Linear Inequalities, " Canad. J. Math., vol. 6, pp. 382-393, 1954.

....game to within a multiple # of the range of payo#s requires at most m n # 2 iterations, each requiring about 4mn flops. If only the 1 # 2 factor could be replaced by ln( 1 # ) this would yield a polynomial algorithm 6.3. Relaxation methods The classical relaxation method of Agmon [4] and Motzkin and Schoenberg [76] closely related to iterative methods for linear equations, was of great interest in the 1950s, partly because each iteration requires minimal computation, merely some matrix vector multiplications. To find a point in Y : y : A T y # c , at every iteration ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382--392, 1954.

....as a preliminary step for interior point methods (see for example, Chapter 5 of [9] The particular method of solution of this problem called the relaxationprojection method is the main object of the present article. The original research on this method was carried out, around 1954, by S. Agmon [1], T.S. Motzkin and I.J. Schoenberg [17] Associating the inequalities to half spaces, in which lie the points corresponding to the feasible solutions, they proved that such a point can be reached, from an arbitrary outside point, by constructing a trajectory of straight line segments, each of ....

....7 where II(x) is the set of the indices of the constraints which are violated by x, and h i are some positive constants. There are also algorithms which have been developed especially for the solution of the feasibility problem. This is the case for the relaxation projection method of S. Agmon [1], T.S. Motzkin and I.J. Schoenberg [17] mentioned above, and for the simultaneous relaxation projection method, proposed more recently by Y. Censor and T. Elfving [4] This latter method is a variant of the former, in which the steps of the iteration sequence are made in the direction of an ....

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S. Agmon, "The relaxation method for linear inequalities", Can. J. Math., vol. 6, pp. 382-392, 1954.

.... a generalization of the algorithm privately communicated by von Neumann to Dantzig and studied by Dantzig in [6] and [7] and is part of a large class of elementary algorithms for nding a point in a suitably described convex set, such as re ection algorithms for linear inequality systems (see [1,21,8,15]) the perceptron algorithm [31 34] and other so called row action methods. When applied to linear inequality systems, these elementary algorithms share the following desirable properties, namely: the work per iteration is extremely low (typically involving only a few matrix vector or ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382-392, 1954.

....Email: kannan cs.cmu.edu. School of Computer Science, Carnegie Mellon University, Pittsburgh PA 15213. Supported in part by NSF National Young Investigator grant CCR 9357793. Email: svempala cs.cmu.edu. 1 instance, one commonly used greedy algorithm for this task is the Perceptron Algorithm [Ros62, Agm54], described below in Section 3. These algorithms have running times that depend on the amount of wiggle room available to a solution. In particular, the Perceptron Algorithm has the following guarantee [MP69] Given a collection of data points in R n , each labeled as positive or negative, the ....

....2 =144n. If S now satis es the condition of the theorem we stop. Otherwise, we repeat. The dicult issue is proving that this algorithm will in fact halt before removing too many points from S. The proof of this fact is deferred to Section 5. 3 The Perceptron Algorithm The Perceptron Algorithm[Ros62, Agm54] operates on a set S of labeled data points in n dimensional space. Its goal is to nd a vector w such that w x 0 for all positive points x and w x 0 for all negative points x. We will say that such a vector w correctly classi es all points in S. If a non zero threshold value is desired, ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6(3):382-392, 1954.

....Example 3.6 ( Random Agmon Motzkin Schoenberg ) If each set C i is a halfspace, then the random product of relaxed projections converges in norm to some point in QCx 0 . Remark 3. 7 The cyclic control version is due to Gubin et al. 20] whereas the remotest set control version is due to Agmon [1] and to Motzkin and Schoenberg [23] In the field of image reconstruction, these methods are known as AMS relaxation methods or ART for inequalities [9, 8] Example 3.8 ( Random von Neumann Halperin ) Suppose each set C i is a closed subspace and P j2J C j is closed, for every nonempty ....

S. AGMON. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382--392, 1954.

....oldest problems studied in learning theory. The problem can be solved in polynomial time by using a polytime algorithm for linear programming on a sample of O(n) examples. Typically, however, it is solved by using simple greedy methods. A commonly used greedy algorithm is the Perceptron Algorithm [24, 1], which has the following guarantee [19] Given a collection of data points in R n , each labeled as positive or negative, the algorithm will find a vector w such that w Delta x 0 for all positive points x and w Delta x 0 for all negative points x, if such a vector exists (note: a non zero ....

S. Agmon, "The relaxation method for linear inequalities," Canadian Journal of Mathematics, 6(3):382--392, 1954.

....mathematician Eremin ( 4] 5] The monography [6] contains a summarizing description for finitedimensional spaces with various applications to the iterative solution of convex problems. Parallel to this development the basic ideas of Fejer theory also occured in concrete applications (e.g. 9] [1], 7] 12] This process continues up to present time. Often the authors do not know the original literature and the developed theory (e.g. 18] 10] 8] 3] The great interest in Fejer methods is caused by the wide range of applications. At present they play an important part in computerized ....

Agmon, S. : The relaxation method for linear inequalities. Canad. J. Math. 6, 382-392 (1954)

....x = hx 1 ; x n i 2 R n . Let A i be the ith row of A. The projection of a point x to a half space A i x Gamma b i 0 is defined as P i (x) x if A i x Gamma b i 0 x Gamma cA T i otherwise where c = A i x Gamma b i ) jA T i j 2 . This reduces to the method described in [Agm54]. Without any modification, this method can be also applied to a set of linear equalities, by simply replacing each linear equality g i (x) 0 with two linear inequalities: g i (x) 0 and Gammag i (x) 0. CHAPTER 14. CONSTRAINT BASED DYNAMIC SYSTEMS 151 There are various ways to modify this ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382--392, 1954.

....depends on fn i ; i 2 Jg such that for each K : 6= K ae J and FK 6= and each x 2 S, d(x; FK ) C X i2K (n i Delta x Gamma b i ) B.1) Proof. For fixed K such that ; 6= K ae J and FK 6= the estimate (B. 1) follows from a theorem of Hoffman [25] with supporting lemmas proved by Agmon [1]. To see how Hoffman s theorem applies one needs to let his matrix A have rows given by f Gamman i ; i 2 J; n i ; i 2 Kg and his b have entries f Gammab i ; i 2 J; b i ; i 2 Kg. His functions Fm , F n can be taken to equal the usual Euclidean distance functions on IR m , IR n , respectively. ....

Agmon, S. The relaxation method for linear inequalities. Canadian J. Math. 6, 382--392 (1954).

....conformal mappings) to name only a few. II. Image Reconstruction: Discrete Models. Properties: Each set C i is a halfspace or a hyperplane. X is a Euclidean space (i.e. a finite dimensional Hilbert space) Very flexible algorithmic schemes. Basic results: Kaczmarz [64] Cimmino [29] Agmon [1], Motzkin and Schoenberg [76] Comments: Behaviour in general Hilbert space and quality of convergence only partially understood. References: Censor [21, 23, 24] Censor and Herman [27] Viergever [99] Sezan [88] Areas of application: Medical imaging and radiation therapy treatment planning ....

....of d( Delta; C j ) for every index j, the weak limit of (x (n k ) lies in C. By Theorem 2.16. ii) x (n) has at most one weak cluster point in C; therefore, i) is verified. ii) is proved similarly. Remark 4. 27 Remotest set control is an old and successful concept: In 1954, Agmon [1] and Motzkin and Schoenberg [76] studied projection algorithms for solving linear inequalities using remotest set control. Bregman [16] has considered the situation when there is an arbitrary collection of intersecting closed convex sets. We will recapture Agmon s main result [1, Theorem 3] and ....

[Article contains additional citation context not shown here]

S. AGMON. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382--392, 1954.

....Email: kannan cs.cmu.edu. School of Computer Science, Carnegie Mellon University, Pittsburgh PA 15213. Supported in part by NSF National Young Investigator grant CCR 9357793. Email: svempala cs.cmu.edu. instance, one commonly used greedy algorithm for this task is the Perceptron Algorithm [Ros62, Agm54], described below in Section 3. These algorithms have running times that depend on the amount of wiggle room available to a solution. In particular, the Perceptron Algorithm has the following guarantee [MP69] Given a collection of data points in R n , each labeled as positive or negative, the ....

....fi=144n. If S now satisfies the condition of the theorem we stop. Otherwise, we repeat. The difficult issue is proving that this algorithm will in fact halt before removing too many points from S. The proof of this fact is deferred to Section 5. 3 The Perceptron Algorithm The Perceptron Algorithm[Ros62, Agm54] operates on a set S of labeled data points in n dimensional space. Its goal is to find a vector w such that w Delta x 0 for all positive points x and w Delta x 0 for all negative points x. We will say that such a vector w correctly classifies all points in S. If a non zero threshold value ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6(3):382--392, 1954.

....within some constant factors [17] those thermal 3 perceptron variants [20, 16, 18] provide good solutions in a short amount of time. In this subsection we show that a thermal version of the Agmon Motzkin Schoenberg relaxation method (AMS) for solving consistent systems of linear inequalities [21, 22, 23] can be used to find large consistent subsystems of (2) The AMS method is a simple iterative procedure that generates a sequence of estimates. Given a consistent system Ax b, start with an arbitrary initial guess x 0 2 R n and consider each inequality cyclically but in a random order. Suppose ....

S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382--392, 1954.

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S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6(3): 382--392, 1954.

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S. Agmon, The relaxation method for linear inequalities, Canadian J. of Math., 6(3), 382--392, 1954.

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S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382{ 392, 1954. 27

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S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6:382-392, 1954.

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S. Agmon. The relaxation method for linear inequalities. Canadian Journal of Mathematics, 6(3):382-392, 1954.

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S. Agmon, The relaxation method for linear inequalities, Can. J. Math. 6

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Agmon, S. (1954). "The Relaxation Method for Linear Inequalities." Canadian Journal of Mathematics 6: 382-392.

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