Closed convex function

A closed convex function is a mathematical function f that is both convex and lower semicontinuous. Convexity means that the function's graph lies entirely above the line segment connecting any two points on the graph. Lower semicontinuity means that the function's graph does not have any "jumps" or discontinuities, and that the limit of the function as the input approaches a given point from below is equal to the value of the function at that point. Together, these properties make closed convex functions useful in optimization and other areas of mathematics.From: Nonsmooth Mechanics and Convex Optimization [2019]

Nonlinear Programming

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Published in Albert G. Holzman, Mathematical Programming, 2020

Mordecai Avriel

Theorem 25. Let f be a convex function on Rn. The conjugate function f* is also a convex function and f** = cl f. Moreover, f* is a closed convex function. It is proper if and only if f is proper.

Primal-dual algorithms for multi-agent structured optimization over message-passing architectures with bounded communication delays

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Published in Optimization Methods and Software, 2022

Puya Latafat, Panagiotis Patrinos

For a set C, we denote its relative interior by . Let be a proper closed convex function. Its domain is denoted by . Its subdifferential is the set-valued operator For a positive scalar ρ the proximal map associated with q is the single-valued mapping defined by The Fenchel conjugate of q, denoted by , is defined as The function q is said to be μ-convex with if is convex.

Inexact basic tensor methods for some classes of convex optimization problems

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Published in Optimization Methods and Software, 2022

Yurii Nesterov

In this paper, we consider a convex optimization problem in the following composite form: where is a smooth convex function and is a closed convex function such that Let us assume that the problem (3) is solvable and denote by one of its optimal solutions, .

An efficient augmented Lagrangian method for support vector machine

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Published in Optimization Methods and Software, 2020

Yinqiao Yan, Qingna Li

Let be a closed convex function. The Moreau–Yosida [24,47] regularization of q at is defined by The unique solution of (6), denoted as , is called the proximal point of x associated with q. The following property holds for Moreau–Yosida regularization [21, Proposition 2.1].

Closed convex function

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Nonlinear Programming

Primal-dual algorithms for multi-agent structured optimization over message-passing architectures with bounded communication delays

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An efficient augmented Lagrangian method for support vector machine