dot-qp_refine: Quadratic Programming Refinement for Inequality Constraints

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Quadratic Programming Refinement for Inequality Constraints

Description

After the Lagrangian projection handles smoothness equality constraints, this function refines the estimate to satisfy additional inequality constraints (monotonicity, derivative sign, range bounds, or user-supplied constraints) via quadprog::solve.QP.

Usage

.qp_refine(
  result,
  X,
  y,
  K,
  p_expansions,
  A,
  Lambda,
  Lambda_block,
  family,
  iterate,
  tol,
  qp_Amat,
  qp_bvec,
  qp_meq,
  qp_score_function,
  order_list,
  glm_weight_function,
  schur_correction_function,
  need_dispersion_for_estimation,
  dispersion_function,
  observation_weights,
  VhalfInv,
  ...
)

Arguments

result	List of current coefficient column vectors by partition.
X	List of partition-specific design matrices.
y	List of response vectors by partition.
K	Integer; number of interior knots.
p_expansions	Integer; number of basis terms per partition.
A	Equality constraint matrix \mathbf{A}.
Lambda	Shared penalty matrix \boldsymbol{\Lambda}.
Lambda_block	Full block-diagonal penalty matrix.
family	GLM family object.
iterate	Logical; if TRUE, iterate the SQP loop until convergence rather than taking a single step.
tol	Convergence tolerance.
qp_Amat	Inequality constraint matrix \mathbf{C} for solve.QP.
qp_bvec	Inequality constraint vector \mathbf{c}.
qp_meq	Number of equality constraints within qp_Amat.
qp_score_function	Score function \nabla_{\boldsymbol{\beta}}\ell(\boldsymbol{\beta}^*) for the QP step.
order_list	List of index vectors mapping partition rows to original data ordering.
glm_weight_function	Function computing GLM working weights.
schur_correction_function	Function computing Schur corrections.
need_dispersion_for_estimation	Logical.
dispersion_function	Dispersion estimation function.
observation_weights	List of observation weights by partition.
VhalfInv	Inverse square root of the working correlation matrix in the original observation ordering (or NULL); permuted internally via order_list.
...	Passed to weight, correction, dispersion, and score functions.

Details

The subproblem at each iteration is a second-order Taylor approximation of the penalized log-likelihood around the current iterate \boldsymbol{\beta}^*. Collecting terms, this yields:

\tilde{\boldsymbol{\beta}} = \arg\min_{\boldsymbol{\beta}} \left\{-\mathbf{d}^{\top}\boldsymbol{\beta} + \frac{1}{2} \boldsymbol{\beta}^{\top}\mathbf{G}^{-1}\boldsymbol{\beta}\right\} \quad \text{s.t.} \quad \mathbf{A}^{\top}\boldsymbol{\beta} = \mathbf{0}, \quad \mathbf{C}^{\top}\boldsymbol{\beta} \succeq \mathbf{c}

where \mathbf{d} is the score adjusted by the current iterate and \mathbf{c} is the constraint value vector. Step acceptance uses damped updates with deviance monitoring; see Nocedal and Wright (2006) for the general SQP framework.

Value

A list with components:

result

List of refined coefficient column vectors by partition.

qp_info

List with QP solve metadata including Lagrangian multipliers and the active constraint matrix.

lgspline documentation built on May 8, 2026, 5:07 p.m.