ANALYSIS OF A QUADRATIC PROGRAMMING DECOMPOSITION ALGORITHM - BENCTEUX, G.; CANCÉS, E.; HAGER, W. W.; LE BRIS, C. (SIAM journal on numerical analysis, 2010)
We analyze a decomposition algorithm for minimizing a quadratic objective function, separable in x₁ and x₂, subject to the constraint that x₁ and x₂ are orthogonal vectors on the unit sphere. Our algorithm consists of a local step where we minimize the objective function in either variable separately, while enforcing the constraints, followed by a global step where we minimize over a subspace generated by solutions to the local subproblems. We establish a local convergence result when the global minimizers are nondegenerate. Our analysis employs necessary and sufficient conditions and continuity properties for a global optimum of a quadratic objective function subject to a sphere constraint and a linear constraint. The analysis is connected with a new domain decomposition algorithm for electronic structure calculations.