Mixed constrained quadratic programming problem (MCQPP) formulation. More...

#include <cqpp.hpp>

Public Types
using	backend_t = Backend
using	linear_op_storage_a_t = ::mundy::storage<LinearOpA>
using	linear_op_storage_l_t = ::mundy::storage<LinearOpL>
using	q_vector_storage_t = ::mundy::storage<QVector>
using	f_vector_storage_t = ::mundy::storage<FVector>
using	q_vector_t = typename q_vector_storage_t::value_type
using	f_vector_t = typename f_vector_storage_t::value_type
using	space_t = ConvexSpace
using	a_workspace_t = WorkspaceA
using	l_workspace_t = WorkspaceL
using	value_type = impl::vector_value_type<q_vector_t>

Public Member Functions
	MCQPP (Backend, LinearOpA &&A, QVector &&q, LinearOpL &&L, FVector &&f_b, const space_t &space)
	MCQPP (Backend, LinearOpA &&A, QVector &&q, LinearOpL &&L, FVector &&f_b, const space_t &space, a_workspace_t a_workspace, l_workspace_t l_workspace)
Backend	backend () const
const auto &	A () const
const auto &	q () const
const auto &	L () const
const auto &	f_b () const
const space_t &	space () const
a_workspace_t &	a_workspace () const
	This problem's cached scratch state for evaluating A and L (see Backend::apply and Op::Workspace types).
l_workspace_t &	l_workspace () const

Detailed Description

template<typename Backend, typename LinearOpA, typename QVector, typename LinearOpL, typename FVector, ValidConvexSpace ConvexSpace, typename WorkspaceA = impl::workspace_for_t<std::remove_cvref_t<LinearOpA>>, typename WorkspaceL = impl::workspace_for_t<std::remove_cvref_t<LinearOpL>>>
class mundy::MCQPP< Backend, LinearOpA, QVector, LinearOpL, FVector, ConvexSpace, WorkspaceA, WorkspaceL >

This is for a mixed constrained convex quadratic programming problem such as: x^*, y^* = argmin_{x in Omega_x, y in R^m} q^T x + b^T y + 0.5 (Dx + By)^T M (Dx + By) + 0.5 y^T K^{-1} y where M and K^{-1} are symmetric positive definite matrices, D and B are linear operators, q and b are vectors, and Omega_x is a convex space.

This can be mapped onto a reduced CQPP in x alone via the Schur complement. Define: S^{-1} := B^T M B + K^{-1} (symmetric positive definite) H := D^T M D - D^T M B S B^T M D g := q - D^T M B S b Then the reduced CQPP is: x^* = argmin_{x in Omega_x} 0.5 x^T H x + g^T x y^* = -S^{-1}(b + B^T M D x^*)

In more general terms, the mixed problem exists because of an affine operator mapping x -> f: f = (I + L M) D x + f_b, where f_b = f(x = 0) and L is an spsd linear operator

In the less-general problem above: L = B S B^T and f_b = B S b. In this form, the reduced CQPP is: x^* = argmin_{x in Omega_x} 0.5 x^T H x + g^T x H := D^T M (D - L M D) g := q - D^T M f_b y^*, f^*, and u^* are all intermediates that may or may not be explicitly formed during the solution process depending on the structure of L. We refer to these variables as your "workspace" variables. They are updated by the linear operators during the solution process, and their final values are "committed" at the end of the solve to indicate that they are the correct values corresponding to the optimal x^*.

We may also have the simpler problem where D = I, in which case the reduced CQPP is: x^* = argmin_{x in Omega_x} 0.5 x^T H x + g^T x H := A (I - L A) g := q - A f_b

Just like the CQPP, we will either accept the simple case of (A, q, L, f_b) or the more general case of (DT, M, D, q, BT, S, B, b).