Date: Thu, 28 Mar 2024 18:35:52 +0000 (UTC) Message-ID: <1286820015.215.1711650952498@1a5b09e72cd8> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_214_858018247.1711650952497" ------=_Part_214_858018247.1711650952497 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html
Using tools provided by the theory of abstract convexity, we extend= conditions for zero duality gap to the context of nonconvex and nonsmooth = optimization. Substituting the classical setting, an abstract convex functi= on is the upper envelope of a subset of a family of abstract affine functio= ns (being conventional vertical translations of the abstract linear functio= ns). We establish new characterizations of the zero duality gap under no as= sumptions on the topology on the space of abstract linear functions. Endowi= ng the latter space with the topology of pointwise convergence, we extend s= everal fundamental facts of the conventional convex analysis. In particular= , we prove that the zero duality gap property can be stated in terms of an = inclusion involving =F0=9D=9C=80-subdifferentials, which are shown to posse= ss a sum rule. These conditions are new even in conventional convex cases. = The Banach-Alaoglu-Bourbaki theorem is extended to the space of abstract li= near functions. The latter result extends a fact recently established by Bo= rwein, Burachik and Yao in the conventional convex case.
This talk is= based on joint work with Regina Burachik, Alex Kruger and David Yost.
<= p>