Projection on a Cone, Penalty Functionals and Duality Theory for Problems with Inequaltity Constraints in Hilbert Space

Each element p of a real Hilbert space H can be uniquely decomposed into two orthogonal components, $p = p^D + p^{ - D^ * } $ where $p^D \in D$ is the projection of p on a closed convex cone D and $p^{ - D^ * } $ is the projection of p on the minus dual cone $ - D^ * $. Hence, if the cone D generates a partial order in H, then the positive part $p^D $ and the negative part $p^{ - D^ * } $ of each $p \in H$ can be distinguished. For a general optimization problem: minimize $Q(y)$ over $Y_p = \{ y \in E:p - P(y) \in D \subset H\} $, where $Q:E \to R$, $P:E \to H$, E is Banach, H is Hilbert: the violation of the constraint can be determined by ($(p - P(y))^{ - D^ * } $. Hence a generalized penalty functional and an augmented Lagrange functional can be defined for this problem. The paper presents a short review of known penalty techniques, some properties of the projection on a cone, basic properties of penalty functionals for a general optimization problem and duality theory for nonconvex problems in infinit...