New glimpses on convex infinite optimization duality
Let \(X\) be a separated locally convex topological space, \(C\subset X\) be a nonempty closed convex set, \(T\) be a possibly infinite index set, \(\mathbb{R}^{(T)}_+:= \{\lambda: T\to R_+\mid\text{supp\,}\lambda\text{ is finite}\}\) and let \(f\) and \(f_t\) \((t\in T)\) be convex lsc proper extended real-valued functions with \(M:=\bigcap_{t\in T}\text{dom\,}f_t\). For the convex optimization problem (P) \(\inf_x f(x)\) s.t. \(x\in C\), \(f_t(x)\leq 0\), \(t\in T\), the authors discuss the dual problems, (D) \(\sup_\lambda\inf_{x\in C}(f(x)+ \sum_{t\in T}\lambda_t f_t(x))\) s.t. \(\lambda= (\lambda_t)_{t\in T}\in\mathbb{R}^{(T)}_+\), (Q) \(\sup_\lambda\inf_{x\in C\cap M}(f(x)+\sum_{t\in T}\lambda_t f_t(x))\) s.t. \(\lambda=(\lambda_t)_{t\in T}\in\mathbb{R}^{(T)}_+\), \((\Delta)\) \(\sup_\lambda\inf_{x\in C}(f(x)+ \sum_{t\in T}\lambda_t f_t(x))\) s.t. \(\lambda= (\lambda_t)_{t\in T}\in \mathbb{R}^{(T)}_+\setminus\{0_T\}\). Obviously, it is \(\sup(\Delta)\leq \sup(\text{D})\leq \sup(\text{Q})\leq\inf(\text{P})\). Sufficient conditions (essentially closedness conditions to associated epigraphical sets) are provided such that strong duality holds in sense of \(\inf(\text{P})= \max(\text{D})\), \(\inf(\text{P})= \max(\text{Q})\) and \(\inf(\text{P})= \max(\Delta)\), respectively. The second part of the paper is devoted to the minsup property. Sufficient conditions (based on the notion of quasicontinuity and recession assumptions) are proposed which guarantee \(\min(\text{P})= \sup(\Delta)\). The paper is finished with some applications of the obtained results.
scientific article; zbMATH DE number 6483920