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Multicategory Classification by Support Vector Machines
 Computational Optimizations and Applications
, 1999
"... We examine the problem of how to discriminate between objects of three or more classes. Specifically, we investigate how twoclass discrimination methods can be extended to the multiclass case. We show how the linear programming (LP) approaches based on the work of Mangasarian and quadratic programm ..."
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Cited by 56 (0 self)
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We examine the problem of how to discriminate between objects of three or more classes. Specifically, we investigate how twoclass discrimination methods can be extended to the multiclass case. We show how the linear programming (LP) approaches based on the work of Mangasarian and quadratic programming (QP) approaches based on Vapnik's Support Vector Machines (SVM) can be combined to yield two new approaches to the multiclass problem. In LP multiclass discrimination, a single linear program is used to construct a piecewise linear classification function. In our proposed multiclass SVM method, a single quadratic program is used to construct a piecewise nonlinear classification function. Each piece of this function can take the form of a polynomial, radial basis function, or even a neural network. For the k > 2 class problems, the SVM method as originally proposed required the construction of a twoclass SVM to separate each class from the remaining classes. Similarily, k twoclass linear programs can be used for the multiclass problem. We performed an empirical study of the original LP method, the proposed k LP method, the proposed single QP method and the original k QP methods. We discuss the advantages and disadvantages of each approach. 1 1
Mathematical Programming in Neural Networks
 ORSA Journal on Computing
, 1993
"... This paper highlights the role of mathematical programming, particularly linear programming, in training neural networks. A neural network description is given in terms of separating planes in the input space that suggests the use of linear programming for determining these planes. A more standard d ..."
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Cited by 40 (13 self)
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This paper highlights the role of mathematical programming, particularly linear programming, in training neural networks. A neural network description is given in terms of separating planes in the input space that suggests the use of linear programming for determining these planes. A more standard description in terms of a mean square error in the output space is also given, which leads to the use of unconstrained minimization techniques for training a neural network. The linear programming approach is demonstrated by a brief description of a system for breast cancer diagnosis that has been in use for the last four years at a major medical facility. 1 What is a Neural Network? A neural network is a representation of a map between an input space and an output space. A principal aim of such a map is to discriminate between the elements of a finite number of disjoint sets in the input space. Typically one wishes to discriminate between the elements of two disjoint point sets in the ndim...
Constraint Programming and Operations Research
"... f constraints by interpreting the logical connectives as in logic programming .Supports procedural programming for mixed discrete and continuous methods .Supports new search strategies .Supports disjunctive linear programming .Expands to parallel and distributed computing Constraint Programmin ..."
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f constraints by interpreting the logical connectives as in logic programming .Supports procedural programming for mixed discrete and continuous methods .Supports new search strategies .Supports disjunctive linear programming .Expands to parallel and distributed computing Constraint Programming and Operations Research 6 of 100 Creating the Feasible Region 2lp_main() { continuous X,Y,Z; X + Y + Z <= 1; X + Y == 1; X == Y; printf("The coordinates of the wp are\n"); printf("( %.1f, %.1f, %.1f )\n",X,Y,Z); } .Output The coordinates of the witness point are ( 0.5, 0.5, 0.0 ) Constraint Programming and Operations Research 7 of 100 The Witness Point Z Y X witness point (a) (0,1,0) (0,0,1) (b) (1,0,0) (1,0,0) (c) (.5,.5,0) (d) FIGURE 10.1 (a) The positive