Quadratic Constrained Minimization Problems
This chapter deals with solving quadratic minimization problems defined from a s.p.d. matrix A subject to box inequality constraints that model Signorini’s problems in contact mechanics. In particular, we investigate the use of preconditioners B for A incorporated in the commonly used projection methods. The latter methods are also quadratic minimization problems involving the preconditioner B to define the quadratic functional. To make the projection methods computationally feasible (for more general than diagonal B) an equivalent dual formulation is introduced that involves the inverse actions of B (and not the actions of B). For the special case when the constrained set involves a small subset of the unknowns a reduced problem formulation is introduced and analyzed. Our presentation of these topics is based on the results by J. Schoeberl in [Sch98] and [Sch01]. We conclude the chapter with a multilevel FAS (full approximation scheme) based on monotone smoothers (such as projected Gauss–Seidel) providing a monotonicity proof from [IoV04].
KeywordsProjection Method Constrain Minimization Problem Coarse Space Monotone Scheme Quadratic Minimization Problem
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