ANALYSIS OF A QUADRATIC PROGRAMMING DECOMPOSITION ALGORITHM

• Published in: SIAM journal on numerical analysis Vol. 47; no. 6; pp. 4517 - 4539
• Main Authors: BENCTEUX, G., CANCÉS, E., HAGER, W. W., LE BRIS, C.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
• Subjects: Algorithms; Continuity; Convergence; Decomposition; Eigenvalues; Eigenvectors; Electronic structure; Exact sciences and technology; Mathematical analysis; Mathematical minima; Mathematical models; Mathematical vectors; Mathematics; Numerical Analysis; Numerical analysis. Scientific computation; Objective functions; Optimization; Orthogonality; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Quadratic programming; Sciences and techniques of general use; Unit vectors; Variables; domain decomposition method; 65K05; Necessary and sufficient condition; Initial value problem; Decomposition method; Optimization method; Global optimum; Orthogonality; Quadratic programming; Algorithm; Partial differential equation; Constrained optimization; Sphere; 90C20; electronic structure calculations; Numerical analysis; Boundary value problem; 90C46; Multigrid; orthogonality constraints; Domain decomposition; quadratic programming
• ISSN: 0036-1429; 1095-7170; 1095-7170
• DOI: 10.1137/070701728

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.