Computational Optimization: A Tribute to Olvi Mangasarian by Jong-Shi Pang (auth.), Jong-Shi Pang (eds.)

By Jong-Shi Pang (auth.), Jong-Shi Pang (eds.)

Computational Optimization: A Tribute to Olvi Mangasarian serves as a great reference, supplying perception into essentially the most not easy learn matters within the box.
This choice of papers covers a large spectrum of computational optimization themes, representing a mix of wide-spread nonlinear programming subject matters and such novel paradigms as semidefinite programming and complementarity-constrained nonlinear courses. Many new effects are offered in those papers that are absolute to encourage additional learn and generate new avenues for purposes. an off-the-cuff categorization of the papers comprises:

  • Algorithmic advances for targeted sessions of limited optimization difficulties
  • Analysis of linear and nonlinear courses
  • Algorithmic advances
  • B- desk bound issues of mathematical courses with equilibrium constraints
  • Applications of optimization
  • Some mathematical themes
  • Systems of nonlinear equations.

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The methods were compared in terms of generalization (testing set accuracy), number of support vectors, and computational time. 56 BREDENSTEINER AND BENNETT 0 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 0 Figure 1. Piecewise-polynomial separation of three classes in two dimensions The following notation will be used throughout this paper. Mathematically we can abstract the problem as follows: Given the elements of the sets, Ai, i = 1, ... , k, in the n-dimensional real space Rn, construct a discriminant function that separates these points into distinct regions.

T. L but 01 and 02 may be any positive weights for the ml m2 misclassification costs. The dual SVM problem and its extension to nonlinear discriminants are given in the next section. 2. Nonlinear Classifiers Using Support Vector Machines The primary advantage of the SVM (6) over RLP (7) is that efficient methods based on the dual of SVM (6) exist for constructing nonlinear discriminants [29, 11]. These methods with minor modifications can produce polynomial separators, radial basis functions, neural networks, etc.

1 is a fixed constant. Note that Problem (6) is equivalent to RLP with the addition of a regularization term ~wTW and with misclassification costs 6] = 62 = l. Statistical Learning Theory shows that this regularization term is essential for good generalization. A linear programming version of (6) can be constructed by replacing the norm used to minimize the weights 11' [3 J. Recall that the SVM objective minimizes the square of the 2 2-norm of /11, Ilwll = Il,T/I'. The I-norm of w, 1111'111 = eTlwl, can be used instead [3, 10,9, 7).

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