Some notes on continuity in convex optimization

TL;DR

This paper proves that the optimal value function in convex parametric optimization problems is convex, enhancing understanding of the mathematical properties underlying convex optimization.

Contribution

It provides a formal proof that the optimal value function is convex in Euclidean spaces, clarifying a fundamental property of convex optimization problems.

Findings

Optimal value function is convex in Euclidean spaces.

The proof applies to convex parametrized optimization problems.

Supports theoretical foundations of convex optimization.

Abstract

We provide proof that the optimal value function of a convex parametrized optimization problem in Euclidean spaces is itself a convex function onto the extended real line.