forward decl for psd More...

#include <psd_cone.hpp>

Inheritance diagram for conic::PositiveSemidefiniteCone< Slack, N >:

Public Types
using	ConeBase = Cone< Slack, PositiveSemidefiniteCone< Slack, N >, ConeType::SYMMETRIC >
	bring into scope alias from cone base

using	ambient_space = linalg::SymmetricMatrixSpace< T, N >
	Ambient vector space of real symmetric matrices.

using	element_type = linalg::space_element_t< ambient_space >
	Canonical matrix representation of an ambient-space element.

Public Types inherited from conic::Cone< Slack, PositiveSemidefiniteCone< Slack, N >, ConeType::SYMMETRIC >
using	T = linalg::vector_element_t< Slack >
	alias for mat type

using	HessianMat = linalg::Matrix< T, M, M >
	hessian type

Public Member Functions
constexpr Slack	x0_impl () const
	default x in K

constexpr Slack	y0_impl () const
	default y in K^*

constexpr bool	in_cone_impl (const Slack &x) const
	x in K check

constexpr bool	in_dual_cone_impl (const Slack &y) const
	y in K^* check

constexpr T	alpha_max_impl (const Slack &x, const linalg::Vector< T, M > &p) const
	alpha_max (fraction to boundary)

constexpr T	alpha_max_dual_impl (const Slack &y, const linalg::Vector< T, M > &p) const
	alpha_max dual (fraction to dK^*)

constexpr auto	H_impl (const Slack &x, const Slack &y) const
	hessian

constexpr auto	grad_impl (const Slack &x) const
	grad (jacobian)

constexpr T	nu_impl () const
	barrier param

constexpr auto	e_impl () const
	e (jordan identity)

constexpr auto	sqrt_impl (const Slack &x) const
	Jordan square root in symmetric-vector coordinates.

constexpr auto	w_impl (const Slack &x, const Slack &y) const
	w (self-scaled point)

constexpr auto	circ_impl (const Slack &x, const Slack &y) const
	circ

constexpr T	trace_impl (const Slack &x, const Slack &y) const
	trace

constexpr auto	gamma_F_impl (const Slack &x, const Slack &y) const
	gamma_F

constexpr auto	inv_impl (const Slack &x) const
	inv

Public Member Functions inherited from conic::Cone< Slack, PositiveSemidefiniteCone< Slack, N >, ConeType::SYMMETRIC >
constexpr Slack	x0 () const
	default interior point (initial slack in cone space)

constexpr Slack	y0 () const
	default interior point (initial slack in dual cone)

constexpr bool	in_cone (const Slack &x) const
	in the interior of k

constexpr bool	in_dual_cone (const Slack &y) const
	in the interior of K^*

constexpr auto	H (const Slack &x, const Slack &y) const
	hessian

constexpr T	alpha_max (const Slack &x, const linalg::Vector< T, M > &p) const
	fraction to boundary dk

constexpr T	alpha_max_dual (const Slack &y, const linalg::Vector< T, M > &p) const
	fraction to boundary dk^*

constexpr T	nu () const
	barrier parameter

constexpr auto	grad (const Slack &x) const
	gradient

constexpr auto	e () const
	e (jordan identity)

constexpr auto	w (const Slack &x, const Slack &y) const
	w (self-scaling point)

constexpr auto	circ (const Slack &x, const Slack &y) const
	circ (jordan map/product operator)

constexpr auto	trace (const Slack &x, const Slack &y) const
	trace

constexpr auto	inv (const Slack &x) const
	inv (jordan inverse element)

constexpr T	trace_weight () const
	Convert Euclidean complementarity to this cone's Jordan trace units.

constexpr auto	scaled_affine_correction (const Slack &x, const Scaling &P, const Slack &px, const Slack &py) const
	Affine second-order correction in dual coordinates under NT scaling.

constexpr auto	scaled_affine_correction_impl (const Slack &x, const Scaling &P, const Slack &px, const Slack &py) const
	Common symmetric-cone implementation using Jordan operations.

constexpr auto	gamma_F (const Slack &x, const Slack &y) const
	gamma_F (functional proximity neighborhood metric)

constexpr auto	self () const
	self function which returns a const ref to the derived object

Static Public Member Functions
static constexpr bool	contains (const element_type &element)
	Closed-cone membership, including semidefinite boundary elements.

Additional Inherited Members
Static Public Attributes inherited from conic::Cone< Slack, PositiveSemidefiniteCone< Slack, N >, ConeType::SYMMETRIC >
static constexpr std::size_t	M
	helpers to get M slack size

static constexpr T	eps
	eps constant

Detailed Description

template<linalg::VectorContainer Slack, std::size_t N>
struct conic::PositiveSemidefiniteCone< Slack, N >

forward decl for psd

psd cone class

svec(X) in S^n_+

Definition at line 47 of file psd_cone.hpp.

Member Typedef Documentation

◆ ambient_space

template<linalg::VectorContainer Slack, std::size_t N>

using conic::PositiveSemidefiniteCone< Slack, N >::ambient_space = linalg::SymmetricMatrixSpace<T, N>

Ambient vector space of real symmetric matrices.

Definition at line 64 of file psd_cone.hpp.

◆ ConeBase

template<linalg::VectorContainer Slack, std::size_t N>

using conic::PositiveSemidefiniteCone< Slack, N >::ConeBase = Cone<Slack, PositiveSemidefiniteCone<Slack, N>, ConeType::SYMMETRIC>

bring into scope alias from cone base

Definition at line 51 of file psd_cone.hpp.

◆ element_type

template<linalg::VectorContainer Slack, std::size_t N>

using conic::PositiveSemidefiniteCone< Slack, N >::element_type = linalg::space_element_t<ambient_space>

Canonical matrix representation of an ambient-space element.

Definition at line 66 of file psd_cone.hpp.

Member Function Documentation

◆ alpha_max_dual_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr T conic::PositiveSemidefiniteCone< Slack, N >::alpha_max_dual_impl	(	const Slack &	y,
		const linalg::Vector< T, M > &	p
	)		const

inlineconstexpr

alpha_max dual (fraction to dK^*)

Parameters

y	in K^*
p	in E^*

Returns: alpha_max (dual)

Definition at line 162 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::alpha_max_impl().

◆ alpha_max_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr T conic::PositiveSemidefiniteCone< Slack, N >::alpha_max_impl	(	const Slack &	x,
		const linalg::Vector< T, M > &	p
	)		const

inlineconstexpr

alpha_max (fraction to boundary)

sigma(alpha) = frac{1}{sup{alpha | (x + alpha p in K)}}

X + alpha P in S^n_++ X + alpha P = [ X^{1/2} (I + alpha X^{-1/2} P X^{-1/2}) X^{1/2} ] let U = X^{-1/2} P X^{-1/2} we want I + alpha U >= 0 I + alpha U >= 0 iff 1 + alpha lambda_i(U) >= 0 for all i

alpha_max = cases{ +inf, if lambda_i(U) >= 0 for all i -frac{1}{lambda_min(U)} else }

Parameters

x	in K
p	in E

Returns: alpha_max

Definition at line 138 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::in_cone_impl().

Referenced by conic::PositiveSemidefiniteCone< Slack, N >::alpha_max_dual_impl().

◆ circ_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::circ_impl	(	const Slack &	x,
		const Slack &	y
	)		const

inlineconstexpr

circ

x circ y

Parameters

x	in K
y	in K^*

Returns: x circ y

Definition at line 288 of file psd_cone.hpp.

◆ contains()

template<linalg::VectorContainer Slack, std::size_t N>

static constexpr bool conic::PositiveSemidefiniteCone< Slack, N >::contains ( const element_type & element )

inlinestaticconstexpr

Closed-cone membership, including semidefinite boundary elements.

Uses the element's numerical PSD classification and tolerance. In contrast, in_cone() checks the positive-definite interior required by the solver.

Definition at line 74 of file psd_cone.hpp.

◆ e_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::e_impl ( ) const

inlineconstexpr

e (jordan identity)

e = I

Returns: e

Definition at line 227 of file psd_cone.hpp.

Referenced by conic::PositiveSemidefiniteCone< Slack, N >::H_impl(), conic::PositiveSemidefiniteCone< Slack, N >::x0_impl(), and conic::PositiveSemidefiniteCone< Slack, N >::y0_impl().

◆ gamma_F_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::gamma_F_impl	(	const Slack &	x,
		const Slack &	y
	)		const

inlineconstexpr

gamma_F

gamma_F = -ln(det(X)) - ln(det(Y)) + nu ln(Tr(X^{1/2} Y X^{1/2}) / nu)

Parameters

x	in K
y	in K^*

Returns: gamma_F

Definition at line 318 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::nu_impl().

◆ grad_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::grad_impl ( const Slack & x ) const

inlineconstexpr

grad (jacobian)

F'(x) = -x^{-1}

Parameters

x	in K or y in K^*

Returns: (F'(x), event)

Definition at line 208 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::inv_impl().

◆ H_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::H_impl	(	const Slack &	x,
		const Slack &	y
	)		const

inlineconstexpr

hessian

F(w) = -log det(w) F'(w) = - w^{-1} F''(w) = P(w^{-1})

F''(w^{-1}) x = y

Parameters

x	in K
y	in K^*

Returns: (H, event)

Definition at line 181 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::e_impl(), conic::PositiveSemidefiniteCone< Slack, N >::inv_impl(), conic::Cone< Slack, PositiveSemidefiniteCone< Slack, N >, ConeType::SYMMETRIC >::w(), and conic::PositiveSemidefiniteCone< Slack, N >::w_impl().

◆ in_cone_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr bool conic::PositiveSemidefiniteCone< Slack, N >::in_cone_impl ( const Slack & x ) const

inlineconstexpr

x in K check

Parameters

x in K

Returns: T/F

Definition at line 99 of file psd_cone.hpp.

Referenced by conic::PositiveSemidefiniteCone< Slack, N >::alpha_max_impl(), and conic::PositiveSemidefiniteCone< Slack, N >::in_dual_cone_impl().

◆ in_dual_cone_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr bool conic::PositiveSemidefiniteCone< Slack, N >::in_dual_cone_impl ( const Slack & y ) const

inlineconstexpr

y in K^* check

Parameters

y in K^*

Returns: T/F

Definition at line 112 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::in_cone_impl().

◆ inv_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::inv_impl ( const Slack & x ) const

inlineconstexpr

inv

Parameters

x in K

Returns: (inv(x), event)

Definition at line 350 of file psd_cone.hpp.

Referenced by conic::PositiveSemidefiniteCone< Slack, N >::grad_impl(), and conic::PositiveSemidefiniteCone< Slack, N >::H_impl().

◆ nu_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr T conic::PositiveSemidefiniteCone< Slack, N >::nu_impl ( ) const

inlineconstexpr

barrier param

Returns: nu (matrix order N, the degree of det(X))

Definition at line 218 of file psd_cone.hpp.

Referenced by conic::PositiveSemidefiniteCone< Slack, N >::gamma_F_impl().

◆ sqrt_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::sqrt_impl ( const Slack & x ) const

inlineconstexpr

Jordan square root in symmetric-vector coordinates.

Definition at line 237 of file psd_cone.hpp.

◆ trace_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr T conic::PositiveSemidefiniteCone< Slack, N >::trace_impl	(	const Slack &	x,
		const Slack &	y
	)		const

inlineconstexpr

trace

tr(XY)

Parameters

x	in K
y	in K^*

Returns: tr(XY)

Definition at line 302 of file psd_cone.hpp.

◆ w_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr auto conic::PositiveSemidefiniteCone< Slack, N >::w_impl	(	const Slack &	x,
		const Slack &	y
	)		const

inlineconstexpr

w (self-scaled point)

W = X^{1/2} (X^{1/2} Y X^{1/2})^{-1/2} X^{1/2}

Parameters

x	in K
y	in K^*

Returns: (w, event)

Definition at line 254 of file psd_cone.hpp.

References conic::Cone< Slack, PositiveSemidefiniteCone< Slack, N >, ConeType::SYMMETRIC >::w().

Referenced by conic::PositiveSemidefiniteCone< Slack, N >::H_impl().

◆ x0_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr Slack conic::PositiveSemidefiniteCone< Slack, N >::x0_impl ( ) const

inlineconstexpr

default x in K

Returns: x

Definition at line 83 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::e_impl().

◆ y0_impl()

template<linalg::VectorContainer Slack, std::size_t N>

constexpr Slack conic::PositiveSemidefiniteCone< Slack, N >::y0_impl ( ) const

inlineconstexpr

default y in K^*

Returns: y

Definition at line 90 of file psd_cone.hpp.

References conic::PositiveSemidefiniteCone< Slack, N >::e_impl().

The documentation for this struct was generated from the following files:

include/akreon/convex/cone_traits.hpp
include/akreon/convex/cones/psd_cone.hpp

Public Types

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ ambient_space

◆ ConeBase

◆ element_type

Member Function Documentation

◆ alpha_max_dual_impl()

◆ alpha_max_impl()

◆ circ_impl()

◆ contains()

◆ e_impl()

◆ gamma_F_impl()

◆ grad_impl()

◆ H_impl()

◆ in_cone_impl()

◆ in_dual_cone_impl()

◆ inv_impl()

◆ nu_impl()

◆ sqrt_impl()

◆ trace_impl()

◆ w_impl()

◆ x0_impl()

◆ y0_impl()