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Some convex programs without a duality gap. (English) Zbl 1176.90464
The author studies the convex programming problem:$$ v_{P} = \min f_o(x) \text{ s.t. } f_{i}(x) \le 0,\ i=1,\ldots,m\tag{P}$$ where each $f_i:\bbfR^{n}\rightarrow (-\infty,\infty)$ is a proper convex lower semicontinuous function and its associated dual: $$v_{D} = \max q(\mu) \text{ where } q(\mu) = \inf_{x}\{f_o(x)+ \sum _{i=1}^m f_{i}(x)\}, \mu \ge 0.\tag{D}$$ The main result concerns the non-existence of the duality gap i.e. $v_{P} = v_{D}$. It is shown that for separable functions $f_o, f_1,\ldots,f_m$ the relation $v_{P} = v_{D}$ holds under the assumption of $\text{dom } f_o \subseteq \bigcap_{1=1}^{m} \text{ dom } f_{i}$ where $\text{dom } f = \{x|f(x) < \infty\}$. The author gives also a sufficient condition involving weakly analytic functions.

90C25Convex programming
90C46Optimality conditions, duality
Full Text: DOI
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