Duality theory is pervasive in finite dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented...

We show that the stellar subdivisions of a simplex are extendably shellable. These polytopes appear as the facets of the dual of a hypersimplex. Using this fact, we calculate the simplicial and toric h-vector of the dual of a hypersimplex. Finally, we calculate the contribution of each shelling component to the toric h-vector.

In this paper, we extend a type of Strassen’s theorem for the existence of probability measures with given marginals to positive vector measures with values in the dual of a barreled locally convex space which has certain order conditions. In this process of the extension we also give some useful properties for vector measures with values in dual spaces.

We review the notion of a vector space, basis and dimension, linear transformations between vector spaces, dual vector spaces and transformations, spectral decomposition for normal operators (which includes symmetric, Hermitian, orthogonal, and unitary operators), and determinants. Along the way we review direct-sum decompositions, bilinear forms and inner product spaces, adjoints, characterist...

Universum-support vector machine (U-SVM) is an elegant method for 2-class classification problem. It is systematically studied in this paper, including the existence and uniqueness of the primal problem as well as the relation between the solutions of primal problem and dual problem. We find that U-SVM uses 3-class classification approach to solve the 2-class classification problem. So we have ...

The standard SVR formulation for real-valued function approximation on multidimensional spaces is based on the -insensitive loss function, where errors are considered not correlated. Due to this, local information in the feature space which can be useful to improve the prediction model is disregarded. In this paper we address this problem by defining a generalized quadratic loss where the co-oc...