Continuity conditions weaker than lower semi-continuity

TL;DR

Let $f:\mathcal{X}\to \overline{\mathbb{R}}$ be defined on a topological space $\mathcal{X}$. This paper studies relaxations of lower semi-continuity (LSC) for functions on topological spaces in unstructured optimization, surveying weaker notions such as lower quasi-continuity, lower pseudo-continuity, and transfer-type continuities, and it introduces additional continuity conditions. It then proves two comprehensive implication diagrams that relate these conditions, including both pointwise and domain-wide versions, and provides new counterexamples to establish sharpness. By clarifying when minimizers exist and how minimizer sets behave under weaker assumptions, the work broadens the applicability of classical optimization results like the Weierstrass extreme value theorem and related variational principles.

Abstract

Lower semi-continuity (\texttt{LSC}) is a critical assumption in many foundational optimisation theory results; however, in many cases, \texttt{LSC} is stronger than necessary. This has led to the introduction of numerous weaker continuity conditions that enable more general theorem statements. In the context of unstructured optimization over topological domains, we collect these continuity conditions from disparate sources and review their applications. As primary outcomes, we prove two comprehensive implication diagrams that establish novel connections between the reviewed conditions. In doing so, we also introduce previously missing continuity conditions and provide new counterexamples.

Figures (4)

• Figure 1: Implications between continuity conditions which hold at a point $x\in\mathcal{X}$. Solid lines indicate that the implication holds unconditionally, while '$N_1$' indicates that the implication holds when $\mathcal{X}$ is first countable. 'Conv. min. seq/net.' indicates that the implication holds when a minimizing sequence/net converges to $x$, while 'no jump' indicates that the implication holds when $f$ doesn't have a jump at $x$. 'Empty argmin' means the implication holds if $\mathop{\mathrm{arg\,min}}(f) = \emptyset$. '$N_1^*$' indicates the implication holds either when $\mathcal{X}$ is first countable or there is a convergent minimizing sequence.
• Figure 1: Arrows with counter examples preventing inference of the corresponding implication.
• Figure 2: Implications between continuity conditions which hold for $\forall x\in\@fontswitch{}{\mathcal{}} X$. Solid lines indicate that the implication holds unconditionally, while '$N_1$' indicates that the implication holds when $\mathcal{X}$ is first countable. 'Conv. min. seq.' indicate that the implication holds when there is a converging minimizing sequence, while 'no jump' indicates that the implication holds when $f$ has no jumps. 'Empty argmin' means the implication holds if $\mathop{\mathrm{arg\,min}}(f) = \emptyset$.
• Figure 3: Implications and results previously reported regarding continuity conditions that hold $\forall x\in\@fontswitch{}{\mathcal{}} X$. Black arrows indicate that the implication holds in general, while red arrows indicate that counterexamples exist that prevent the corresponding implication. '$N_1$' indicates that the implication holds when $\mathcal{X}$ is first countable.

Theorems & Definitions (13)

• Proposition 2.1
• Proof 1
• Proposition 2.2
• Proposition 2.3
• Proof 2
• Proposition 2.4
• Proof 3
• Proposition 2.5
• Proof 4
• Lemma 4.1