F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21: 771--785, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of linear inequalities constraining the location and orientation of the optimal separating hyperplane. With a properly defined objective function, a separating hyperplane can be obtained by solving a linear programming problem. Several alternative formulations have been proposed in the past ( 3] [5], 10] 15] 17] employing different objective functions. An early survey of these methods is given in [7] Here we mention a few representative formulations. In a very simple formulation described in [15] the objective function is trivial, so that it is simply a test of linear separability ....

Glover, F., Improved Linear Programming Models for Discriminant Analysis, Decision Sciences, 21, 4, 1990, 771785.

.... also be written as a linear program [3] We will refer to it as the Perturbed Robust Linear Program (RLP P) Our feature minimization method could also be applied 4 to algorithms that minimize the number of points misclassified such as [2, 18] or to other successful linear programming approaches [12, 25], but we leave these extensions for future work. 2.1 Feature Minimization Applied to RLP The following robust linear programming problem, RLP [4] minimizes a weighted average of the sum of the distances from the misclassified points to the separating plane. min w,#,u,v 1 m eu 1 k ev ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

.... elimination method for feature minimization forms the basis of a breast cancer diagnosis system [17, 16] Our feature minimization method could also be applied to algorithms that minimize the number of points misclassified such as [2, 11] or to other successful linear programming approaches [10, 15], but we leave these extensions for future work. 3 The following robust linear programming problem, RLP [3] minimizes a weighted average of the sum of the distances from the misclassified points to the separating plane. min w;fl;u;v 1 m eu 1 k ev subject to u Aw Gamma efl Gamma ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

.... y and z of the inequalities (2) This intuitively plausible linear program has significant theoretical and computational consequences [2] such as naturally eliminating the null point w = 0 from being a solution, a difficulty that other linear programming formulations exclude in an ad hoc manner [9, 10, 14]. Once the plane x T w = fl has been obtained, the same procedure can be applied recursively to one or both of the newly created halfspaces x T w fl and x T w fl, if warranted by the presence of an unacceptable mixture of benign and malignant points in the halfspace. Figure 2 shows an ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771-- 785, 1990.

....(1.2) that is a null solution that does not provide any error minimizing separation. For this purpose, we utilize the linear program introduced recently in [8] and which has the following desirable features not all of which are possessed by any other previous linear programming formulation [11, 27, 28, 45, 20, 19]: i) A strict separating plane (that is neither set lies on the separating plane) for linearly separable sets A and B (ii) An error minimizing plane is obtained when the sets A and B are linearly inseparable. iii) No extraneous constraints are used to exclude the null solution for linearly ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

....plane of points lying on the wrong side of the plane. Unfortunately, if precise distances are used, linearity of the objective function is lost, as will be shown in this work. What many of the previous approaches have tended to use, are distance surrogates that maintained linearity of the problem [9, 8, 19, 2] but did not measure distances of violating points to the separating plane. In contrast, our approach here depends on a closed form formula for the projection of a point in R n onto a given plane using an arbitrary norm, which is given in Theorem 2.2 of Section 2. In Section 3 we formulate the ....

.... k is convex on R n . Thus the objective function of the mathematical program (17) is convex but its feasible region, which is the unit sphere in the dual norm k Delta k 0 , is not convex. It is precisely this essential nonconvex condition that has been either ignored in most previous work [12, 9, 8, 2] or used heuristically [13, 19] to enforce nonzeroness of w but not as a distance normalization constraint. Thus in these papers, the sum of the distances of misclassified points to the separating plane has not been the real objective function that has been minimized. In fact the nonconvexity of ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

....plane of points lying on the wrong side of the plane. Unfortunately, if precise distances are used, linearity of the objective function is lost, as will be shown in this work. What many of the previous approaches have tended to use, are distance surrogates that maintained linearity of the problem [9, 8, 20, 2] but did not measure distances of violating points to the separating plane. Invariably such approaches have to contend with the null solution that does not generate any separating plane. In contrast, our approach here excludes the null solution and depends on a closed form formula given in Theorem ....

....norm k Delta k on R m and any convex function h : R n Gamma R m on R n . The feasible region of (17) which is the unit sphere in the dual norm k Delta k 0 , is however not convex. It is precisely this essential nonconvex condition that has been either ignored in most previous work [12, 9, 8, 2] or used heuristically [13, 20] to enforce nonzeroness of w but not as a distance normalization constraint. Thus in these papers, the sum of the distances of misclassified points to the separating plane has not been the real objective function that has been minimized. In [5] a 2 norm error term is ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

.... T z k fi fi fi fi fi fi fi GammaAw efl e y; Bw Gamma efl e z; y 0; z 0 9 = 7) Robustness here refers to the fact that the useless null vector (w = 0) is naturally excluded as a solution of (7) which is not the case in other linear programming formulations of this problem [41, 66, 26, 25]. Note that because of the constraints of the problem, the variables y and z will satisfy the following conditions: y minf0; GammaAw efl eg and z minf0; Bw Gamma efl eg: Hence minimizing e T m e T z k will force the satisfaction in some best sense of (6) and equivalently (5) ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

....linear functions of the input attributes, exhaustive search is no longer feasible. Linear programming and perceptron algorithms have been used to construct decisions that minimize the distances of the misclassified points from the separating plane. Linear programming approaches [BM92, Ben92, Glo90] find optimal decisions by this criterion in polynomial time. Decision tree methods based on heuristic variants of perceptron algorithms [Utg89, BU92] have worked well in practice, but the algorithms may fail to converge and may not find optimal solutions. The problem of creating a linear ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

....[15] Within the mathematical programming framework, various discrimination functions have been proposed. If the the convex hulls of the two sets of points X and Y intersect, one may wish to determine a hyperplane that minimizes the sum of the deviations of the misclassified points (Glover [6], Lee and Ord [10] to minimize the maximum distance of a misclassified point to the hyperplane (Cavalier and al. 3] Marcotte and Savard [12] to minimize the number of misclassified points (Banks and Abad [1] Koehler and Erenguc [8] Soltysik and Yarnold [14] Marcotte and Savard [12] or any ....

Glover, F., "Improved linear programming models for discriminant analysis", Decision Sciences 21 (1990) 771--785.

.... to inconsistencies that cannot be satisfactorily resolved, in spite of numerous attempts (see Koehler [16] Rubin [21] and Markowski and Markowski [20] A recent paper along this line, which removes some of the difficulties associated with the linear programming formulation, is that of Glover [10] where the normalizing constraint Gammas X i2I p t x i r X j2J p t y j = 1 is substituted to the reverse convex constraint kp 2 k 2 2 1: If a separating hyperplane exists, w is nonnegative at the optimum, and (1) is equivalent to the the relaxed convex programming problem ....

Glover, F. "Improved linear programming models for discriminant analysis", Decision Sciences , 21 (1990) 771--785.

....MotzkinSchoenberg algorithm for finding the solution of linear inequalities [23] A perceptron is also known as a linear discriminant. So any linear discriminant algorithms such as in the book [7] may be used. A single linear program can be used to construct a separating plane in polynomial time [14, 11]. Edelsbrunner proposed an algorithm with O(log m log k) complexity [8] Statistical methods such as Fisher s Linear Discriminant may also be applied [9] 1 When applied to inseparable problems, the solution that misclassified the least number of points is selected. The problems formulated ....

F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21:771--785, 1990.

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F. Glover. Improved linear programming models for discriminant analysis. Decision Sciences, 21: 771--785, 1990.

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F. Glover, Improved Linear Programming Models for Discriminant Analysis, Decision Sciences, 21,4, 1990, pp. 771-785.

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