If a convex function is bounded below, can we say its minimum is always attained, i.e. there exists a $x$ that always satisfies the optimal condition?
Convex function with lower bounds, is its minimum attained.
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analysis
convex-analysis
convex-optimization
1 Answers
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This not true. Example :
$$f:\mathbb{R}\to\mathbb{R},x\mapsto e^x$$ is convex and has a lower bound but no minimum
However, it can be proved that if $u:\mathbb{R}\to\mathbb{R}$ is convex and if $\lim_{-\infty}u=\lim_{+\infty}u=+\infty$, then $u$ has a minimum.