Affine manifold

Affine manifold

Hahn-Banach and Duality Type Theorems for Vector Lattice-Valued Operators and Applications to Subdifferential Calculus and Optimization

Published in S. A. Mohiuddine, Bipan Hazarika, Sequence Space Theory with Applications, 2023

Let X be a real vector space. An affine combination of elements x1, x2,…,xn of X is any linear combination of the type ∑i=1nλixi, with λ1, λ2,…,λn∈ℝ and ∑i=1nλi=1. An affine manifold of X is a nonempty subset of X, closed under affine combinations.

An extended conjugate duality for generalized semi-infinite programming problems via a convex decomposition

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Published in Optimization, 2020

A. Aboussoror, S. Adly, S. Salim

For the definitions and details concerning such conventions in we refer to [17]. Let A be a nonempty subset of . We shall denote by and the indicator and the support functions of the set A respectively, defined on by where denotes the inner product of two vectors in , i.e. for and , . Let denote the smallest affine manifold containing A. Then the relative interior of A denoted by is the interior of A relative to . Note that if A is convex, then is nonempty [18].

A conjugate directions-type procedure for quadratic multiobjective optimization

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Published in Optimization, 2022

Ellen H. Fukuda, L. M. Graña Drummond, Ariane M. Masuda

Our goal is to find a minimizer of in the affine manifold , which coincides with by Lemma 2.1. We plan to do this by performing Algorithm 1 n + 1 times with cleverly chosen different initial points, and then comparing the values of the respective last iterates. We now present the weak version of this scheme.

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Hahn-Banach and Duality Type Theorems for Vector Lattice-Valued Operators and Applications to Subdifferential Calculus and Optimization

An extended conjugate duality for generalized semi-infinite programming problems via a convex decomposition

A conjugate directions-type procedure for quadratic multiobjective optimization