Regularization methods for optimization problems with probabilistic constraints
Considered is the following nonlinear optimization problem with probabilistic constraints: \[ \min f(x) \leqno (1) \text{ subject to }\mathbb{P}[g(x)\geq Y] \geq p,\quad x\in \mathcal{D}, \] where \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) is a convex function, \(g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\) is componentwise concave, i.e., \(g_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}\), \(i=1,\dots ,m\), are concave functions, \(\mathcal{D}\subseteq\mathbb{R}^{n}\) is a closed convex set, \(Y\) is an \(m\)-dimensional random vector, the symbol \(\mathbb{P}\) denotes the probability, and it is required that \(g\left( x\right) \geq Y\) shall hold with some prescribed probability \(p\in (0,1)\). The authors develop two efficient numerical methods for solving (1) using regularization techniques. Finally, numerical results based on a bond portfolio and a supply-chain optimization are presented.
scientific article; zbMATH DE number 6160899
scientific article