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2018

On the Properties of Convex Functions over Open Sets

Drori, Yoel

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We consider the class of smooth convex functions defined over an open convex set.

We show that this class is essentially different than the class of smooth convex functions defined over the entire linear space by exhibiting a function that belongs to the former class but cannot be extended to the entire linear space while keeping its properties.
We proceed by deriving new properties of the class under consideration, including an inequality that is strictly stronger than the classical Descent Lemma.

On the Properties of Convex Functions over Open Sets · Around

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Problem complexity and method efficiency in optimization
A. S. Nemirovsky and D. B. Yudin · 1983
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Introduction to Optimization
B. T. Polyak · 1987
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Interior-point polynomial algorithms in convex programming
Y. Nesterov and A. Nemirovskii · 1994
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