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Stochastic Optimization Theory
Published in Soumya D. Mohanty, Swarm Intelligence Methods for Statistical Regression, 2018
A set D⊂ℝD is said to be a convex set if for any two elements a¯,b¯∈D and 0 < λ < 1, the element c¯=(1−λ)a¯+λb¯ also belongs to D. In other words, all the points on the straight line joining any two points in D also lie within D. As an example, the points constituting an elliptical area in a two dimensional plane form a convex set.
Pontryagin’s Minimum Principle
Published in Simant Ranjan Upreti, Optimal Control for Chemical Engineers, 2016
Now for a convex set, there exists a supporting hyperplane at any boundary point of the set (see Appendix 5.B, p. 149). We apply this result to our convex set of final states, in which the final optimal state y^f≡[y^0(tf)??y^1(tf)?…?y^n(tf)]⊤ is a boundary point since it cannot be an interior point. Otherwise, ỹf would have been surrounded in all directions by other points of the set, including those with y0 (t̂f) components less than ŷ0 (t̂f). This situation is contradictory since no point in the set can have the y0(t̂f) component — the objective function value I — less than ŷ0(t̂f), which is the minimum I.
Introduction to Linear Programming
Published in Shashi Kant Mishra, Bhagwat Ram, Introduction to Linear Programming with MATLAB®, 2017
Shashi Kant Mishra, Bhagwat Ram
In other words, if x∈Ω $ x\in \Omega $ and y∈Ω $ y\in \Omega $ and also if z∈Ω $ z\in \Omega $ for all values of λ∈[0,1] $ \lambda \in [0,1] $ and z=λx+(1-λ)y $ z=\lambda x+(1-\lambda )y $ , then Ω $ \Omega $ is called a “convex set". For example, a triangle and its interior form a convex set. See Figure 4.2.
Prox-regular sets and Legendre-Fenchel transform related to separation properties
Published in Optimization, 2022
Samir Adly, Florent Nacry, Lionel Thibault
It has been well-recognized that the geometric Hahn-Banach theorems are among of the most important and powerful principles of functional analysis (see, e.g. [5,6,29,30] and the references therein). Roughly speaking, the geometric Hahn-Banach theorem for closed convex sets asserts that a compact convex set A and a closed convex set B of X (or more generally of a locally convex space) with can be separated by a hyperplane/half-space. The case where A is reduced to a singleton (so A is an exterior point of B) is of a great interest and can be stated as follows (see also [29, Theorem 6.23]).
Formation-containment control of heterogeneous linear multi-agent systems with adaptive event-triggered strategies
Published in International Journal of Systems Science, 2022
Chen Yuan, Huaicheng Yan, Yuan Wang, Yufang Chang, Xisheng Zhan
A set is said to be convex if whenever , and . The convex hull of a finite set of points , is the minimal convex set containing all points , , denoted by .