Package 'cccp' reference manual

Title:	Cone Constrained Convex Problems
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).
Authors:	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)
Maintainer:	Bernhard Pfaff <[email protected]>
License:	GPL (>= 3)
Version:	0.3-1
Built:	2024-11-04 04:33:35 UTC
Source:	https://github.com/cran/cccp

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())
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.

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)
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.

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)
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())
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())
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())
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)  
## 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())
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())
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)
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)
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)
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")
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")
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")
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")
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")
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")
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")
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")
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())
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)
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.

`cpd`	An object belonging to the class union `CPD`.
`ctrl`	An object of reference-class `Rcpp_CTRL`.

Package 'cccp'

Help Index

Solving linear and quadratic programs with cone constraints

Description

Usage

Arguments

Value

Class "CPD"

Description

Objects from the Class

Methods

Rcpp module: CPG

Description

Details

Solving a convex program

Description

Usage

Arguments

Value

Creating objects of reference-class CTRL

Description

Usage

Arguments

Value

Note

See Also

Creating a member object of the reference-class DCP

Description

Usage

Arguments

Value

Creating a member object of the reference-class DLP

Description

Usage

Arguments

Value

Creating a member object of the reference-class DNL

Description

Usage

Arguments

Value

Creating a member object of the reference-class DQP

Description

Usage

Arguments

Value

Extractor methods for reference class objects

Description

Usage

Arguments

Value

Geometric program

Description

Usage

Arguments

Details

Value

References

Minimizing L1-norm

Description

Usage

Arguments

Value

Definition of nonlinear inequality constraints

Description

Usage

Arguments

Value

Definition of linear inequality constraints

Description

Usage

Arguments

Value

Definition of positive semidefinite cone inequality constraints

Description

Usage

Arguments

Details

Value

Class "Rcpp_CONEC"

Class `"CPD"`

Creating a member object of the reference-class `DCP`

Creating a member object of the reference-class `DLP`

Creating a member object of the reference-class `DNL`

Creating a member object of the reference-class `DQP`

Class `"Rcpp_CONEC"`

Class `"Rcpp_CPS"`

Class `"Rcpp_CTRL"`

Class `"Rcpp_DCP"`

Class `"Rcpp_DLP"`

Class `"Rcpp_DNL"`

Class `"Rcpp_DQP"`

Class `"Rcpp_PDV"`