Version: 0.3-1
Date: 2023-12-09
Title: Cone Constrained Convex Problems
Maintainer: Bernhard Pfaff <bernhard@pfaffikus.de>
Depends: R (≥ 3.0.1), methods
Suggests: RUnit, numDeriv
LazyLoad: yes
Description: Routines for solving convex optimization problems with cone constraints by means of interior-point methods. The implemented algorithms are partially ported from CVXOPT, a Python module for convex optimization (see https://cvxopt.org for more information).
Imports: Rcpp (≥ 0.11.2)
LinkingTo: Rcpp, RcppArmadillo
License: GPL (≥ 3)
RcppModules: CPG
NeedsCompilation: yes
Packaged: 2023-12-09 17:18:24 UTC; bp
Author: Bernhard Pfaff [aut, cre], Lieven Vandenberghe [cph] (copyright holder of cvxopt), Martin Andersen [cph] (copyright holder of cvxopt), Joachim Dahl [cph] (copyright holder of cvxopt)
Repository: CRAN
Date/Publication: 2023-12-09 17:50:02 UTC

Solving linear and quadratic programs with cone constraints

Description

This function is the main function for defining and solving convex problems in the form of either linear or quadratic programs with cone constraints.

Usage

cccp(P = NULL, q = NULL, A = NULL, b = NULL, cList = list(),
     x0 = NULL, f0 = NULL, g0 = NULL, h0 = NULL,
     nlfList = list(), nlgList = list(), nlhList = list(),
     optctrl = ctrl())

Arguments

P

An object of class matrix with dimension N \times N or NULL.

q

An object of class vector with dimension N \times 1 or NULL.

A

An object of class matrix with dimension p \times N.

b

An object of class vector with dimension p \times 1.

cList

A list object containing the cone constraints. Elements must be of either S4-class NNOC, or SOCC, or PSDC.

x0

An object of class vector with dimension n \times 1 for the initial values. The point x0 must be in the domain of the nonlinear constraints.

f0

function: the scalar-valued convex and twice-differentiable objective function (its first argument must be ‘x’).

g0

function: the gradient function of the objective (its first argument must be ‘x’).

h0

function: the Hessian function of the objective (its first argument must be ‘x’).

nlfList

A list object containing the nonlinear constraints as its elements. The functions have to be specified with x as their first argument and must be casted in implicit form, i.e. f(x) \le 0.

nlgList

A list object containing the gradient functions as its elements. The functions have to be specified with x as their first argument.

nlhList

A list object containing the Hessian functions as its elements. The functions have to be specified with x as their first argument.

optctrl

An object of S4-class Rcpp_CTRL.

Value

An object of class Rcpp_CPS.

Class "CPD"

Description

Class union of Rcpp_DLP, Rcpp_DQP, Rcpp_DCP and Rcpp_DNL.

Objects from the Class

A virtual Class: No objects may be created from it.

Methods

No methods defined with class "CPD" in the signature.

Rcpp module: CPG

Description

Module for defining and solving convex programs.

Details

The module contains the following items: classes:

CONEC

Class for inequality (cone) constraints.

CTRL

Class for control parameters used in optimizations.

PDV

Class for primal/dual variables.

DCP

Class for definition of convex programs.

DLP

Class for definition of linear programs.

DNL

Class for definition of linear programs with non-linear constraints.

DQP

Class for definition of quadratic programs.

CPS

Class for solution of convex programs.

functions:

rpp

Function for solving risk parity portfolios.

gpp

Function for solving a geometric program.

Solving a convex program

Description

This function returns an optimal point for a cone constraint convex program.

Usage

## S4 method for signature 'Rcpp_DCP,Rcpp_CTRL'
cps(cpd, ctrl)  
## S4 method for signature 'Rcpp_DLP,Rcpp_CTRL'
cps(cpd, ctrl)  
## S4 method for signature 'Rcpp_DNL,Rcpp_CTRL'
cps(cpd, ctrl)  
## S4 method for signature 'Rcpp_DQP,Rcpp_CTRL'
cps(cpd, ctrl)

Arguments

cpd

An object belonging to the class union CPD.

ctrl

An object of reference-class Rcpp_CTRL.

Value

An object of reference-class Rcpp_CPS.

Creating objects of reference-class CTRL

Description

This function creates an object of reference-class CTRL which contains optimization parameters, e.g. the maximum number of iterations.

Usage

ctrl(maxiters = 100L, abstol = 1e-06, reltol = 1e-06,
     feastol = 1e-06, stepadj = 0.95, beta = 0.5, trace = TRUE)

Arguments

maxiters

integer, the maximum count of iterations.

abstol

numeric, the absolute level for convergence to be achieved.

reltol

numeric, the relative level for convergence to be achieved.

feastol

numeric, the feasable level for convergence to be achieved.

stepadj

numeric, step size adjustment in combined step.

beta

numeric, parameter in backtracking line search.

trace

logical, if TRUE (the default), the solver's progress during the iterations is shown.

Value

An object of reference-class CTRL.

Note

Either abstol or reltol can be set to a negative real number. feastol must be greater than zero.

See Also

Rcpp_CTRL

Creating a member object of the reference-class DCP

Description

This function returns an object containing the definition of a convex program with non-linear constraints and (if provided) cone constraints. The returned object is a member of the reference-class DCP.

Usage

dcp(x0, f0, g0, h0, cList = list(), nlfList = list(), nlgList = list(),
    nlhList = list(), A = NULL, b = NULL)

Arguments

x0

An object of class vector with dimension n \times 1 for the initial values. The point x0 must be in the domain of the nonlinear constraints.

f0

function: the scalar-valued convex and twice-differentiable objective function (its first argument must be ‘x’).

g0

function: the gradient function of the objective (its first argument must be ‘x’); returning a vector.

h0

function: the Hessian function of the objective (its first argument must be ‘x’); returning a matrix.

cList

A list object containing the cone constraints. Elements must be of either S4-class NNOC, or SOCC, or PSDC or an empty list in case of no inequality constraints.

nlfList

A list object containing the nonlinear constraints as its elements. The functions have to be specified with x as their first argument and must be casted in implicit form, i.e. f(x) \le 0.

nlgList

A list object containing the gradient functions as its elements. The functions have to be specified with x as their first argument.

nlhList

A list object containing the Hessian functions as its elements. The functions have to be specified with x as their first argument.

A

An object of class matrix with dimension p \times n or NULL for problems without equality constraints.

b

An object of class vector with dimension p \times 1 or NULL for problems without equality constraints.

Value

An object belonging to the reference-class DCP.

Creating a member object of the reference-class DLP

Description

This function returns an object containing the definition of a cone constrained linear program. The returned object is a member of the reference-class DLP.

Usage

dlp(q, A = NULL, b = NULL, cList = list())

Arguments

q

An object of class vector with dimension n \times 1.

A

An object of class matrix with dimension p \times n or NULL for problems without equality constraints.

b

An object of class vector with dimension p \times 1 or NULL for problems without equality constraints.

cList

A list object containing the cone constraints. Elements must be of either reference-class NNOC, or SOCC, or PSDC or an empty list in case of no inequality constraints.

Value

An object belonging to the reference-class DLP.

Creating a member object of the reference-class DNL

Description

This function returns an object containing the definition of a linear program with non-linear constraints and (if provided) cone constraints. The returned object is a member of the reference-class DNL.

Usage

dnl(q, A = NULL, b = NULL, cList = list(),
    x0, nlfList = list(), nlgList = list(), nlhList = list())

Arguments

q

vector of length n for the coefficients in the objective.

A

An object of class matrix with dimension p \times n or NULL for problems without equality constraints.

b

An object of class vector with dimension p \times 1 or NULL for problems without equality constraints.

cList

A list object containing the cone constraints. Elements must be of either S4-class NNOC, or SOCC, or PSDC or an empty list in case of no inequality constraints.

x0

An object of class vector with dimension n \times 1 for the initial values. The point x0 must be in the domain of the nonlinear constraints.

nlfList

A list object containing the nonlinear constraints as its elements. The functions have to be specified with x as their first argument and must be casted in implicit form, i.e. f(x) \le 0.

nlgList

A list object containing the gradient functions as its elements. The functions have to be specified with x as their first argument.

nlhList

A list object containing the Hessian functions as its elements. The functions have to be specified with x as their first argument.

Value

An object belonging to the reference-class DNL.

Creating a member object of the reference-class DQP

Description

This function returns an object containing the definition of a cone constrained quadratic program. The returned object is a member of the reference-class DQP.

Usage

dqp(P, q, A = NULL, b = NULL, cList = list())

Arguments

P

An object of class matrix with dimension n \times n.

q

An object of class vector with dimension n \times 1.

A

An object of class matrix with dimension p \times n or NULL for problems without equality constraints.

b

An object of class vector with dimension p \times 1 or NULL for problems without equality constraints.

cList

A list object containing the cone constraints. Elements must be of either reference-class NNOC, or SOCC, or PSDC or an empty list in case of no inequality constraints.

Value

An object belonging to the reference-class DQP.

Extractor methods for reference class objects

Description

Returns a member of reference class objects.

Usage

## S4 method for signature 'Rcpp_PDV'
getx(object)  
## S4 method for signature 'Rcpp_CPS'
getx(object)  
## S4 method for signature 'Rcpp_PDV'
gety(object)  
## S4 method for signature 'Rcpp_CPS'
gety(object)  
## S4 method for signature 'Rcpp_PDV'
gets(object)  
## S4 method for signature 'Rcpp_CPS'
gets(object)  
## S4 method for signature 'Rcpp_PDV'
getz(object)  
## S4 method for signature 'Rcpp_CPS'
getz(object)  
## S4 method for signature 'Rcpp_CPS'
getstate(object)  
## S4 method for signature 'Rcpp_CPS'
getstatus(object)  
## S4 method for signature 'Rcpp_CPS'
getniter(object)  
## S4 method for signature 'Rcpp_CTRL'
getparams(object)

Arguments

object

An object of either reference-class Rcpp_PDV or Rcpp_CPS, or Rcpp_CTRL.

Value

The relevant member object of the class.

Geometric program

Description

This function solves a geometric program.

Usage

gp(F0, g0, FList = list(), gList = list(), nno = NULL,
   A = NULL, b = NULL, optctrl = ctrl())

Arguments

F0

Matrix in the objective function.

g0

Matrix in the objective function (affine terms).

FList

List of matrices in posinomial functions.

gList

List of matrices in posinomial functions (affine terms).

nno

Object created by a call to nnoc().

A

Lefthand-side matrix of equality constraints.

b

Lefthand-side matrix of equality constraints.

optctrl

Object of reference class ‘Rcpp_CTRL’, created by a call to ctrl().

Details

Solves a geometric program casted in its epigraph form.

Value

An object of S4-class Rcpp_CPS.

References

Boyd, S., Kim, S.-J., Vandenberghe, L. and A. Hassibi (2007), A tutorial on geometric programming, Optim Eng, Educational Section, 8:67–127, Springer.

Minimizing L1-norm

Description

This function minimizes a L1-norm of the form ||P u - q||_1, whereby P is a (m \times n) matrix and q is a m \times 1 vector. This function is wrapper function for invoking the cps-method of Linear Programs.

Usage

l1(P, q = NULL, optctrl = ctrl())

Arguments

P

matrix of dimension m \times n.

q

vector of length m.

optctrl

An object of S4-class Rcpp_CTRL.

Value

An object of S4-class Rcpp_CPS.

Definition of nonlinear inequality constraints

Description

This function is the interface to the reference class NLFC for creating nonlinear constraints.

Usage

nlfc(G, h)

Arguments

G

Object of class "matrix": A (m \times n) matrix containing the coefficients of the lefthand-side linear inequality constraints.

h

Object of class NLFV: A (m \times 1) vector containing the coefficients of the righthand-side linear inequality constraints as slot u.

Value

List with elements: conType, G and h.

Definition of linear inequality constraints

Description

This function is the interface to the reference class NNOC for creating linear constraints.

Usage

nnoc(G, h)

Arguments

G

Object of class "matrix": A (m \times n) matrix containing the coefficients of the lefthand-side linear inequality constraints.

h

Object of class NNOV: A (m \times 1) vector containing the coefficients of the righthand-side linear inequality constraints as slot u.

Value

List with elements: conType, G and h.

Definition of positive semidefinite cone inequality constraints

Description

This function is the interface to the reference class PSDC for creating positive semidefinite cone constraints.

Usage

psdc(Flist, F0)

Arguments

Flist

Object of class "list": A list with the matrices appearing on the left-hand side of the matrix inequality.

F0

Object of class "matrix": The matrix appearing on the righthand-side.

Details

A psd-cone constraint is given as \sum_{i = 1}^n x_i F_i \le F_0. The matrix G is created as G = [\textrm{vech}(F_1) | \ldots | \textrm{vech}(F_n)] and the vector h is constructed as h = [\textrm{vech}(F_0)].

Value

List with elements: conType, G and h.

Class "Rcpp_CONEC"

Description

Class for inequality (cone) constraints.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

cone:

Object of class activeBindingFunction: Type of cone constraints.

G:

Object of class activeBindingFunction: Left-hand side of inequality constraints.

h:

Object of class activeBindingFunction: Right-hand side of inequality constraints.

sidx:

Object of class activeBindingFunction: Row index for subsets of cone constraints.

dims:

Object of class activeBindingFunction: Dimension of cone constraints.

K:

Object of class activeBindingFunction: Count of inequality constraints.

n:

Object of class activeBindingFunction: Count of variables in objective.

Examples

showClass("Rcpp_CONEC")

Class "Rcpp_CPS"

Description

Class for solution of convex programs.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

pdv:

Object of class activeBindingFunction: Primal-dual variables.

state:

Object of class activeBindingFunction: Vector of state variables in convex programs.

status:

Object of class activeBindingFunction: Character indicating the status of the returned solution.

niter:

Object of class activeBindingFunction: Integer, count of iterations.

sidx:

Object of class activeBindingFunction: Integer matrix, start and end indices of slack variables.

Examples

showClass("Rcpp_CPS")

Class "Rcpp_CTRL"

Description

Class for control options used in optimization routines.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

ctrlparams:

Object of class activeBindingFunction: List of control parameters.

Examples

showClass("Rcpp_CTRL")

Class "Rcpp_DCP"

Description

Class for definition of convex programs with non-linear constraints.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

x0:

Object of class activeBindingFunction: Initial values.

cList:

Object of class activeBindingFunction: Inequality constraints, class CONEC.

nList:

Object of class activeBindingFunction: List with elements of functions for evaluating non-linear constraints, their associated gradients and their associated Hessians.

A:

Object of class activeBindingFunction: Left-hand side of equality cosntraints.

b:

Object of class activeBindingFunction: Right-hand side of equality cosntraints.

Examples

showClass("Rcpp_DCP")

Class "Rcpp_DLP"

Description

Class for definition of linear programs.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

q:

Object of class activeBindingFunction: Matrix related to linear term in objective.

A:

Object of class activeBindingFunction: Left-hand side of equality cosntraints.

b:

Object of class activeBindingFunction: Right-hand side of equality cosntraints.

cList:

Object of class activeBindingFunction: Inequality constraints, class CONEC.

Examples

showClass("Rcpp_DLP")

Class "Rcpp_DNL"

Description

Class for definition of linear programs with non-linear constraints.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

q:

Object of class activeBindingFunction: Matrix related to linear term in objective.

A:

Object of class activeBindingFunction: Left-hand side of equality cosntraints.

b:

Object of class activeBindingFunction: Right-hand side of equality cosntraints.

cList:

Object of class activeBindingFunction: Inequality constraints, class CONEC.

x0:

Object of class activeBindingFunction: Initial values.

nList:

Object of class activeBindingFunction: List with elements of functions for evaluating non-linear constraints, their associated gradients and their associated Hessians.

Examples

showClass("Rcpp_DNL")

Class "Rcpp_DQP"

Description

Class for definition of quadratic programs.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

P:

Object of class activeBindingFunction: Matrix related to quadratic term in objective.

q:

Object of class activeBindingFunction: Matrix related to linear term in objective.

A:

Object of class activeBindingFunction: Left-hand side of equality cosntraints.

b:

Object of class activeBindingFunction: Right-hand side of equality cosntraints.

cList:

Object of class activeBindingFunction: Inequality constraints, class CONEC.

Examples

showClass("Rcpp_DQP")

Class "Rcpp_PDV"

Description

Class for primal/dual variables in convex programs.

Extends

Class "C++Object", directly. All reference classes extend and inherit methods from "envRefClass".

Fields

x:

Object of class activeBindingFunction: Primal variables.

y:

Object of class activeBindingFunction: Dual variables.

s:

Object of class activeBindingFunction: Primal slack variables.

z:

Object of class activeBindingFunction: Dual slack variables.

kappa:

Object of class activeBindingFunction: Self-dual embedding variable; used in LPs, only.

tau:

Object of class activeBindingFunction: Self-dual embedding variable; used in LPs, only.

Examples

showClass("Rcpp_PDV")

Risk-parity optimization

Description

This function determines a risk-parity solution of a long-only portfolio with a budget-constraint.

Usage

rp(x0, P, mrc, optctrl = ctrl())

Arguments

x0

matrix of dimension n \times 1; starting values.

P

matrix of dimension n \times n; dispersion matrix.

mrc

matrix of dimension n \times 1; the marginal risk contributions.

optctrl

An object of S4-class Rcpp_CTRL.

Value

An object of S4-class Rcpp_CPS.

References

Spinu, F. (2013), An Algorithm for Computing Risk Parity Weights, SSRN, OMERS Capital Markets, July 2013.

Definition of second-oder cone inequality constraints

Description

This function is the interface to the reference class SOCC for creating second-oder cone constraints.

Usage

socc(F, g, d, f)

Arguments

F

Object of class "matrix": The matrix appearing in the norm-expression on the left-hand side of a second-order cone constraint.

g

Object of class "numeric": The vector appearing in the norm-expression on the left-hand side of a second-order cone constraint.

d

Object of class "numeric": The vector appearing on the right-hand side of a second-order cone constraint.

f

Object of class "numeric": The scalar appearing on the right-hand side of a second-order cone constraint.

Details

A second-order cone constraint is given as || Fx + g ||_2 \le d'x + f. The matrix G is created as G = [-d, -F] and the vector h is constructed as h = [f, g].

Value

List with elements: conType, G and h.