dot-bf_sqp_loop: Damped SQP Refinement on the Full (Optionally Whitened)...
In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

.bf_sqp_loop

R Documentation

Damped SQP Refinement on the Full (Optionally Whitened) System

Description

Runs a damped sequential quadratic programming loop using quadprog::solve.QP at each iteration, enforcing all smoothness equalities, flat-equality constraints, and any user-supplied inequality constraints.

Usage

.bf_sqp_loop(
  X_design,
  y_design,
  X_block_raw,
  beta_init,
  Lambda_block,
  qp_Amat_combined,
  qp_bvec_combined,
  qp_meq_combined,
  K,
  p_expansions,
  family,
  order_list,
  glm_weight_function,
  schur_correction_function,
  qp_score_function,
  need_dispersion_for_estimation,
  dispersion_function,
  observation_weights,
  iterate,
  tol,
  VhalfInv,
  VhalfInv_perm,
  is_gee,
  deviance_fun,
  X_partitions,
  y_partitions,
  verbose,
  ...
)

Arguments

`X_design`	`N \times P` design matrix (whitened for GEE, unwhitened otherwise).
`y_design`	`N \times 1` response vector (whitened for GEE, unwhitened otherwise).
`X_block_raw`	Unwhitened block-diagonal design matrix (used for computing the linear predictor on the original scale). For the non-GEE path this equals `X_design`.
`beta_init`	`P \times 1` initial coefficient vector.
`Lambda_block`	`P \times P` block-diagonal penalty.
`qp_Amat_combined`	Combined constraint matrix (equalities then inequalities).
`qp_bvec_combined`	Combined constraint RHS.
`qp_meq_combined`	Number of leading equality constraints.
`K`, `p_expansions`	Integers.
`family`	GLM family object.
`order_list`	Observation-index lists.
`glm_weight_function`, `schur_correction_function`	qp_score_function Functions.
`need_dispersion_for_estimation`	Logical.
`dispersion_function`	Function.
`observation_weights`	List or vector of weights.
`iterate`	Logical; if FALSE, takes at most two steps.
`tol`	Convergence tolerance.
`VhalfInv`, `VhalfInv_perm`	`\mathbf{V}^{-1/2}` matrices (original and permuted ordering). NULL for non-GEE path.
`is_gee`	Logical; TRUE when called from the GEE refinement path (affects deviance computation).
`deviance_fun`	Function computing deviance; either `.bf_deviance` or `.bf_gee_deviance`.
`X_partitions`	List of per-partition design matrices (unwhitened); needed for Schur correction in non-GEE path.
`y_partitions`	List of per-partition response vectors (unwhitened); needed for Schur correction in non-GEE path.
`verbose`	Logical.
`...`	Forwarded to weight, dispersion, and score functions.

Details

This function serves two roles within blockfit_solve:

GEE Stage 2 (Case c): operates on the whitened system \tilde{\mathbf{X}} = \mathbf{V}^{-1/2} \mathbf{X}_{\mathrm{block}}, initialized from the damped Newton-Raphson backfitting warm start.
Non-GEE SQP refinement: operates on the unwhitened block-diagonal design, applying inequality constraints after backfitting convergence.

Value

A named list with elements beta_block (final coefficient vector), result (per-partition coefficient list), last_qp_sol (last successful solve.QP output or NULL), converged (logical), and final_deviance (scalar).