proximal distance algorithms

proximal distance algorithms implement penalized optimization by combining the classical penalty method with distance majorization to solve constrained and structured optimization problems.

Key Features:

• Proximal distance principle: Minimizes a penalized objective combining a loss function f(x) with a squared Euclidean distance penalty scaled by parameter rho.
• Distance majorization: Majorizes the squared Euclidean distance term by a spherical quadratic surrogate to produce an easier-to-minimize upper bound.
• Surrogate function and proximal map: Iteratively minimizes the surrogate via proximal mappings, e.g., x_{k+1} = prox_f^rho(P_C(x_k)), where P_C denotes projection onto the constraint set C.
• Iterative nature and convergence: Employs an iterative update scheme that reduces convex instances to proximal-gradient-like algorithms and allows convergence analysis for nonconvex cases under general conditions such as Zangwill-type global convergence arguments.

Scientific Applications:

• Linear programming: Efficient solution of linear programs under constraints.
• Constrained least squares: Least-squares optimization with feasibility constraints.
• Kinship matrix projection: Computation of the nearest valid kinship matrix under structural constraints.
• Second-order cone constraints: Projection and optimization with second-order cone feasibility regions.
• Horn’s copositive matrix index calculation: Computation of indices associated with copositive matrices.
• Linear complementarity programming: Solution of linear complementarity formulations.
• Sparse principal components analysis: Accelerates sparse principal components analysis via proximal distance formulations.

Methodology:

Combines the classical penalty method with distance majorization, majorizes the squared Euclidean distance by a spherical quadratic surrogate, performs iterative minimization via proximal mappings and projections P_C, and analyzes convergence with Zangwill-type global convergence arguments.