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

Abstract

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 infinite-dimensional spaces. Properties of minimizing sequences in constrained optimization are discussed and the convergence of increased and shifted penalty techniques is studied in detail. Conditions of stability of the optimization problem, implying convergence conditions, are discussed in the closing section.