Stochastic algorithm for solving convex semi-infinite programming problems with equality and inequality constraints
The authors consider the problem of the semi-infinite optimization, i.e. the problem of mathematical programming with an infinite number of constraints. The problem \(P(Y^0,\tilde Y^0)\). \[ \text{Find} \quad x \in \Xi^0_{\text{opt.}}, \quad \Xi^0_{\text{opt.}}= \{x \in X^0 \mid f(x)= \min_{x'\in X^0}f(x')\}, \] \[ X^0 = \{x \in X^0 \mid g(x,y)\leq 0\;\forall y \in Y^0,\;v(x, \widetilde y)= 0\;\forall \widetilde y \in \widetilde Y^0\},\quad X^0 \ni \mathbb{R}^k,\;Y^0 \ni \mathbb{R}^l,\;\widetilde Y^0 \ni \mathbb{R}^m. \] The functions \(f(x)\), \(g(x,y)\) are supposed continuously differentiable and convex with respect to \(x\) for any \(\widetilde y \in \widetilde Y^0\). The function \(v(x, \widetilde y)\) is linear with respect to \(x\) for any \(\widetilde y \in \widetilde Y^0\). For solving the problem \(P(Y^0, \widetilde Y^0)\) a stochastic method of outer approximations is used. The idea of the method consists in the substitution of the problem of semi-infinite optimization \(P(Y,\widetilde Y^0)\) by a sequence of approximation problems \(P(Y,\widetilde Y)\), where instead of infinite sets of constraints \(Y^0, \tilde Y^0\) their finite subsets are used. The proposed algorithm SMETH.ACTIVE.sip.eq. [see \textit{Y. V. Volkov} and \textit{S. K. Zavriev}, SIAM J. Control Optim. 35, 1387-1421 (1997; Zbl 0899.90159)] can essentially simplify the analytical work.
scientific article; zbMATH DE number 1827445