By Petros Xanthopoulos; Panos M Pardalos; Theodore B Trafalis; et al

Info uncertainty is an idea heavily similar with such a lot genuine lifestyles purposes that contain info assortment and interpretation. Examples are available in info bought with biomedical tools or different experimental recommendations. Integration of strong optimization within the current info mining strategies objective to create new algorithms resilient to mistakes and noise.

This paintings encapsulates all of the most up-to-date purposes of strong optimization in facts mining. This short comprises an summary of the speedily growing to be box of strong info mining learn box and offers the main popular laptop studying algorithms, their powerful counterpart formulations and algorithms for attacking those difficulties.

This short will attract theoreticians and information miners operating during this field.

1. Introduction

2. Least Squares Problems

3. important part Analysis

4. Linear Discriminant Analysis

5. aid Vector Machines

6. end

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**Example text**

12b) i=1 αi ≥ 0 i = 1, . . 12c) Once the optimal dual variables have been obtained, the optimal separation hyperplane can be recovered similar as in hard margin classifier. 13) where sgn(·) is the sign function. To address the problem of nonlinearity that frequently occurs in real world problems, one can use kernel methods. Kernel methods [50] provide an alternative approach by mapping data points x in the input space into a higher dimensional feature space F through a map ϕ such that ϕ : x → ϕ (x).

Di wT xi + b ≥ 1 − ξi, i = 1, . . , n ξi ≥ 0, i = 1, . . 19c) for the robust case we replace each point xi with x˜i = x¯i + σi where x¯i are the nominal (known) values and σi is an additive unknown perturbation that belongs to a well-defined uncertainty set. The objective is to solve the problem for the worst case perturbation. t. min di wT (x¯i + σi ) + b σi ξ ≥ 0, i = 1, . . , n ≥ 1 − ξi, i = 1, . . 20b) corresponds to the distance of the ith point to the separation hyperplane the worst case σi would be the one that minimizes this distance.

2. if f is a continuously differentiable function on an open set containing S, and S is a convex set then x∗ ∈ S is a global minimum if and only if x∗ is a stationary point. Proof. [25] pp. 14–15. The last theorem is a very strong result that connects stationary points with global optimality. Since stationary points are so important for solving convex optimization problems, it is also important to establish a methodology that would allow us to discover such points. This is exactly the goal of Karush–Kuhn–Tucker conditions and method of Lagrangian multipliers.