dot-check_kkt_partitionwise: Check KKT Conditions for Partition-Wise Active-Set Method

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Check KKT Conditions for Partition-Wise Active-Set Method

Description

Given current constrained coefficient estimates and a set of active inequality constraints (treated as equalities in the last Lagrangian projection), checks primal feasibility of inactive constraints and dual feasibility (non-negative multipliers) of active constraints.

Usage

.check_kkt_partitionwise(
  result,
  Ghalf,
  GhalfInv,
  Xy_or_uncon,
  is_path3,
  A_aug,
  n_eq_orig,
  qp_Amat,
  qp_bvec,
  active_ineq,
  K,
  p_expansions,
  family,
  parallel_matmult,
  parallel_aga,
  cl,
  chunk_size,
  num_chunks,
  rem_chunks,
  tol,
  parallel_qr = FALSE
)

Arguments

result	List of K+1 coefficient column vectors.
Ghalf	List of \mathbf{G}^{1/2}_k matrices.
GhalfInv	List of \mathbf{G}^{-1/2}_k matrices.
Xy_or_uncon	Either the list of cross-products \mathbf{X}_k^{\top}\mathbf{y}_k (Path 2) or unconstrained estimates (Path 3).
is_path3	Logical; if TRUE, Xy_or_uncon contains unconstrained estimates and GhalfInv is used for transformation.
A_aug	Augmented constraint matrix (original A plus active inequality columns).
n_eq_orig	Integer; number of original equality constraints (columns of A before augmentation).
qp_Amat	Full inequality constraint matrix.
qp_bvec	Full inequality constraint RHS.
active_ineq	Integer vector; indices into columns of qp_Amat that are currently active.
K, p_expansions	Integer dimensions.
family	GLM family object.
parallel_matmult, parallel_aga	Logical flags.
cl, chunk_size, num_chunks, rem_chunks	Parallel parameters.
tol	Numeric tolerance for feasibility and multiplier checks.

Details

Multipliers for active inequality constraints are recovered from the OLS fit used in the Lagrangian projection: the fitted coefficients on \mathbf{X}^* = \mathbf{G}^{1/2}\mathbf{A}_{\mathrm{aug}} give the Lagrangian multipliers (up to sign and scaling).

Value

A list with components:

feasible

Logical; TRUE if all inactive inequality constraints are satisfied within tolerance.

dual_feasible

Logical; TRUE if all active inequality multipliers are non-negative within tolerance.

violated

Integer vector; indices of violated inactive constraints (into qp_Amat columns).

drop

Integer vector; indices of active constraints with negative multipliers that should be dropped.

multipliers

Numeric vector of Lagrangian multipliers for the active inequality constraints.

lgspline documentation built on Aug. 5, 2026, 1:10 a.m.