forward decl for soc More...

#include <soc.hpp>

Inheritance diagram for conic::SecondOrderCone< Slack >:

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

Public Types inherited from conic::Cone< Slack, SecondOrderCone< Slack >, 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 interior point

constexpr Slack	y0_impl () const
	default dual point

constexpr bool	in_cone_impl (const Slack &x) const
	in interior of K

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

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

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

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

constexpr auto	grad_impl (const Slack &x) const
	grad

constexpr T	nu_impl () const
	barrier param

constexpr auto	e_impl () const
	e (jordan identity)

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

constexpr auto	sqrt_impl (const Slack &x) const
	Event-returning Jordan square root for the common cone implementation.

constexpr auto	sqrt (const Slack &x) const
	sqrt

constexpr auto	eigs (const Slack &x) const
	eigs

Public Member Functions inherited from conic::Cone< Slack, SecondOrderCone< Slack >, 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

Additional Inherited Members
Static Public Attributes inherited from conic::Cone< Slack, SecondOrderCone< Slack >, 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>
struct conic::SecondOrderCone< Slack >

forward decl for soc

soc cone class

K_soc = { (x, t) | ||x|| <= t }

Definition at line 127 of file soc.hpp.

Member Typedef Documentation

◆ ConeBase

template<linalg::VectorContainer Slack>

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

bring into scope alias from cone base

Definition at line 130 of file soc.hpp.

Member Function Documentation

◆ alpha_max_dual_impl()

template<linalg::VectorContainer Slack>

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

inlineconstexpr

fracto to dK^*

Parameters

y	in K
p	in E^*

Returns: alpha_max (dual)

Definition at line 274 of file soc.hpp.

References conic::SecondOrderCone< Slack >::alpha_max_impl().

◆ alpha_max_impl()

template<linalg::VectorContainer Slack>

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

inlineconstexpr

fraction to boundary dK

let x = (v,t) let p = [ dv; dt ] sigma(alpha) = frac{1}{sup{alpha | (v + alpha p in K, t + alpha dt)}} t(alpha) >= ||v(alpha)||

g(alpha) = t(alpha)^2 - ||v(alpha)||^2 >= 0 t(alpha) >= 0

g(alpha) = (t + alpha dt)^2 - ||v + alpha dv||^2 let a = dt^2 - ||dv||^2 let b = 2(t dt - v^T dv) let c = t^2 - ||v||^2... x in K & c > 0 so g(alpha) = a alpha^2 + b alpha + c

x = -b +- sqrt(b^2 - 4 a c) / 2a Delta = b^2 - 4 a c -> Delta <= 0 ... g(alpha) >= 0 True... alpha = 1 -> Delta > 0 ... calculate roots r1 = (-b - sqrt(Delta)) / 2a r2 = -b + sqrt(Delta) / 2a sort r_min <= r_max

alpha = piecewise{ 1, if x + p in K -c / b, if a == 0 and b < 0 min(r1, r2) if Delta > 0 1, else }

Parameters

x	in K
p	in E (delta dir)

Returns: alpha_max

(dakre) take max step if update is in cone

(dakre) take smallest positive root (first boundary hit from alpha=0)

(dakre) ensure t + alpha dt > 0

(dakre) clamp alpha [0, 1]

Definition at line 218 of file soc.hpp.

References conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::eps, conic::SecondOrderCone< Slack >::in_cone_impl(), and conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M.

Referenced by conic::SecondOrderCone< Slack >::alpha_max_dual_impl().

◆ circ_impl()

template<linalg::VectorContainer Slack>

constexpr auto conic::SecondOrderCone< Slack >::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 415 of file soc.hpp.

Referenced by conic::SecondOrderCone< Slack >::trace_impl().

◆ e_impl()

template<linalg::VectorContainer Slack>

constexpr auto conic::SecondOrderCone< Slack >::e_impl ( ) const

inlineconstexpr

e (jordan identity)

e = (0,1) (x,t) circ e = (x,t)

Returns: e

Definition at line 366 of file soc.hpp.

References conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::e(), and conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M.

Referenced by conic::SecondOrderCone< Slack >::H_impl(), and conic::SecondOrderCone< Slack >::sqrt().

◆ eigs()

template<linalg::VectorContainer Slack>

constexpr auto conic::SecondOrderCone< Slack >::eigs ( const Slack & x ) const

inlineconstexpr

eigs

(v,t) in K or K^*

Parameters

v	(vector)
t	(scalar)

Returns: tuple(lam1,lam2)

Definition at line 531 of file soc.hpp.

References conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M.

Referenced by conic::SecondOrderCone< Slack >::gamma_F_impl(), and conic::SecondOrderCone< Slack >::sqrt().

◆ gamma_F_impl()

template<linalg::VectorContainer Slack>

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

inlineconstexpr

gamma_F

Parameters

x	in K
y	in K^*

Returns: gamma_F

Definition at line 441 of file soc.hpp.

References conic::SecondOrderCone< Slack >::eigs(), conic::SecondOrderCone< Slack >::nu_impl(), and conic::SecondOrderCone< Slack >::trace_impl().

◆ grad_impl()

template<linalg::VectorContainer Slack>

constexpr auto conic::SecondOrderCone< Slack >::grad_impl ( const Slack & x ) const

inlineconstexpr

grad

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

Parameters

x	in K or y in K^*

Returns: (F'(x), event)

Definition at line 339 of file soc.hpp.

References conic::SecondOrderCone< Slack >::inv_impl().

◆ H_impl()

template<linalg::VectorContainer Slack>

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

inlineconstexpr

hessian

w = X^{1/2} (X^{1/2} S X^{1/2})^{-1/2} X^{1/2} PSD congruence map X^{1/2} S X^{1/2} equivalent to quad rep P(x^{1/2}) y for SOC then another congruence map X^{1/2} inner X^{1/2} equivalent to same quad rep

w = P(x^{1/2})(P(x^{-1/2})y)^{1/2}

let w = (x,t) ** cone of squares ** x circ x = hat{x} circ hat{x} = (2xt, x^T x + t^2); t^2 - x^T x >= 0 ** 2-eig / rank-2 jordan algebra ** two eigs: t +- x^T x det(hat{x}) = lam1*lam2 = t^2 - x^T x > 0 ** self-concordant barrier ** F(w) = F(x,t) = -log det(w) F'(w) = -w^{-1} F''(w) = P(w^{-1})

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

P(w)y = x

let z = w^{-1} P(z) = 2L(z)^2 - L(z^2)

Parameters

x	in K
y	in K^*

Returns: (H, event)

Definition at line 312 of file soc.hpp.

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

◆ in_cone_impl()

template<linalg::VectorContainer Slack>

constexpr bool conic::SecondOrderCone< Slack >::in_cone_impl ( const Slack & x ) const

inlineconstexpr

in interior of K

Parameters

slack vector

Returns: bool

Definition at line 163 of file soc.hpp.

References conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M.

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

◆ in_dual_cone_impl()

template<linalg::VectorContainer Slack>

constexpr bool conic::SecondOrderCone< Slack >::in_dual_cone_impl ( const Slack & y ) const

inlineconstexpr

in interior of K^*

Parameters

y in K^*

Returns: bool

Definition at line 177 of file soc.hpp.

References conic::SecondOrderCone< Slack >::in_cone_impl().

◆ inv_impl()

template<linalg::VectorContainer Slack>

constexpr auto conic::SecondOrderCone< Slack >::inv_impl ( const Slack & x ) const

inlineconstexpr

inv

Jordan Inverse of x where x = (v,t) x^{-1} = frac{1}{t^2 - v^Tv}(-v,t)... frac{1}{det(A)}*adj(A)

Parameters

x	(slack vector)

Returns: (inv(x), event)

Definition at line 466 of file soc.hpp.

References conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::eps, and conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M.

Referenced by conic::SecondOrderCone< Slack >::grad_impl(), conic::SecondOrderCone< Slack >::H_impl(), and conic::SecondOrderCone< Slack >::w_impl().

◆ nu_impl()

template<linalg::VectorContainer Slack>

constexpr T conic::SecondOrderCone< Slack >::nu_impl ( ) const

inlineconstexpr

barrier param

K = {(x,t) | t >= ||x||} F(x,t) = -log(t^2 - ||x||^2) t^2 - ||x||^2 -> lam1 = t + ||x||, lam2 = t - ||x|| det(x) = lam1*lam2 -> F = -log lam1 - log lam2

^ 2 log terms -> 2 barrier params

Returns: nu

Definition at line 356 of file soc.hpp.

Referenced by conic::SecondOrderCone< Slack >::gamma_F_impl(), and conic::SecondOrderCone< Slack >::trace_impl().

◆ sqrt()

template<linalg::VectorContainer Slack>

constexpr auto conic::SecondOrderCone< Slack >::sqrt ( const Slack & x ) const

inlineconstexpr

sqrt

Jordan Square Root lam1 = t + ||v||_2, lam2 = t - ||v||_2 x^{1/2} = frac{sqrt(lam1) + sqrt(lam2)}{2}e + frac{sqrt(lam1) - sqrt(lam2)}{2}u

Parameters

x	(slack vector)

Returns: sqrt(x)

Definition at line 499 of file soc.hpp.

References conic::SecondOrderCone< Slack >::e_impl(), conic::SecondOrderCone< Slack >::eigs(), conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::eps, and conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M.

Referenced by conic::SecondOrderCone< Slack >::sqrt_impl(), and conic::SecondOrderCone< Slack >::w_impl().

◆ sqrt_impl()

template<linalg::VectorContainer Slack>

constexpr auto conic::SecondOrderCone< Slack >::sqrt_impl ( const Slack & x ) const

inlineconstexpr

Event-returning Jordan square root for the common cone implementation.

Definition at line 483 of file soc.hpp.

References conic::SecondOrderCone< Slack >::sqrt().

◆ trace_impl()

template<linalg::VectorContainer Slack>

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

inlineconstexpr

trace

trace(x circ y) = last elem

Parameters

x	in K
y	in K^*

Returns: (x circ y)(-1)

Definition at line 429 of file soc.hpp.

References conic::SecondOrderCone< Slack >::circ_impl(), conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M, and conic::SecondOrderCone< Slack >::nu_impl().

Referenced by conic::SecondOrderCone< Slack >::gamma_F_impl().

◆ w_impl()

template<linalg::VectorContainer Slack>

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

inlineconstexpr

w (self-scaled point)

w = X^{1/2}(X^{1/2} S X^{1/2})^{−1/2} X^{1/2}

Euclidean Jordan Algebra P(x) = 2L(x)^2 - L(x^2) L(x)^2 = L(x) circ L(x) L(x)y = x circ y

where u = (v / ||v||, 0)... primitive idempotent (directional term)

x = lam1 c1 + lam2 c2... c1 & c2 are idempotents (rank-1 dir updates) where x^{1/2} = sqrt{lam1}c1 + sqrt{lam2}c2

w = P(x^{1/2})(P(x^{1/2})y)^{-1/2}

Parameters

x	in K
y	in K^*

Returns: (w, event)

Definition at line 394 of file soc.hpp.

References conic::SecondOrderCone< Slack >::inv_impl(), and conic::SecondOrderCone< Slack >::sqrt().

Referenced by conic::SecondOrderCone< Slack >::H_impl().

◆ x0_impl()

template<linalg::VectorContainer Slack>

constexpr Slack conic::SecondOrderCone< Slack >::x0_impl ( ) const

inlineconstexpr

default interior point

Returns: x0

Definition at line 143 of file soc.hpp.

References conic::Cone< Slack, SecondOrderCone< Slack >, ConeType::SYMMETRIC >::M.

Referenced by conic::SecondOrderCone< Slack >::y0_impl().

◆ y0_impl()

template<linalg::VectorContainer Slack>

constexpr Slack conic::SecondOrderCone< Slack >::y0_impl ( ) const

inlineconstexpr

default dual point

Returns: y0

Definition at line 154 of file soc.hpp.

References conic::SecondOrderCone< Slack >::x0_impl().

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

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

Public Types

Public Member Functions

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ ConeBase

Member Function Documentation

◆ alpha_max_dual_impl()

◆ alpha_max_impl()

◆ circ_impl()

◆ e_impl()

◆ eigs()

◆ gamma_F_impl()

◆ grad_impl()

◆ H_impl()

◆ in_cone_impl()

◆ in_dual_cone_impl()

◆ inv_impl()

◆ nu_impl()

◆ sqrt()

◆ sqrt_impl()

◆ trace_impl()

◆ w_impl()

◆ x0_impl()

◆ y0_impl()