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- W2483291736 abstract "In this chapter, we examine the use of linear and quadratic programming techniques to solve applications in approximation/regression and for classification problems of machine learning. In approximation problems, our aim is to find a vector x that solves a system of equalities and/or inequalities “as nearly as possible” in some sense. Various “loss functions” for measuring the discrepancies in each equality and inequality give rise to different regression techniques. We also examine classification problems in machine learning, in which the aim is to find a function that distinguishes between two sets of labeled points in Rn. Throughout this chapter, we use the concepts discussed in Appendix A, and so a review of that material may be appropriate.9.1 Minimax ProblemsIn this section, we consider the solution of a modification of linear programming in which the linear objective function is replaced by a convex piecewise-linear function. Such a function can be represented as the pointwise maximum of a set of linear functions that we can reduce to a linear programming problem and solve using the techniques of this book. Recall that we have already seen an example of piecewise-linear convex functions during our study of parametric linear programming—the optimal objective value is piecewise linear when considered as a function of linear variations in the right-hand side.Consider first the function ƒ defined as follows:ƒ(x)≔maxi=1,2,…,m (ci)′x+di,(9.1)where ci ∈ Rn and di ∈ R, i = 1, 2, …, m. We can prove that this function is convex by showing that its epigraph (defined in (A.3)) is a convex set." @default.
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- W2483291736 date "2007-01-01" @default.
- W2483291736 modified "2023-10-16" @default.
- W2483291736 title "9. Approximation and Classification" @default.
- W2483291736 doi "https://doi.org/10.1137/1.9780898718775.ch9" @default.
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