R Interface to PIQP Solver

Description

PIQP is an Proximal Interior Point Quadratic Programming solver, which can solve dense and sparse quadratic programs described in described in Schwan, Jiang, Kuhn, and Jones (2023) (https://arxiv.org/abs/2304.00290). Combining an infeasible interior point method with the proximal method of multipliers, the algorithm can handle ill-conditioned convex QP problems without the need for linear independence of the constraints. The solver is written in header only 'C++ 14' leveraging the Eigen library for vectorized linear algebra. For small dense problems, vectorized instructions and cache locality can be exploited more efficiently. Allocation free problem updates and re-solves are also provided.

Author(s)

Balasubramanian Narasimhan, Roland Schwan (C), Yuning Jiang, Daniel Kuhn, Colin N. Jones

PIQP Solver object

Description

PIQP Solver object

Usage

piqp(
  P = NULL,
  c = NULL,
  A = NULL,
  b = NULL,
  G = NULL,
  h = NULL,
  x_lb = NULL,
  x_ub = NULL,
  settings = list(),
  backend = c("auto", "sparse", "dense")
)

Arguments

Details

Allows one to solve a parametric problem with for example warm starts between updates of the parameter, c.f. the examples. The object returned by piqp contains several methods which can be used to either update/get details of the problem, modify the optimization settings or attempt to solve the problem.

Value

An R6-object of class "piqp_model" with methods defined which can be further used to solve the problem with updated settings / parameters.

Usage

model = piqp(P = NULL, c = NULL, A = NULL, b = NULL, G = NULL, h = NULL, x_lb = NULL, x_ub = NULL, settings = piqp_settings(), backend = c("auto", "sparse", "dense"))

model$solve()
model$update(P = NULL, c = NULL, A = NULL, b = NULL, G = NULL, h = NULL, x_lb = NULL, x_ub = NULL)
model$get_settings()
model$get_dims()
model$update_settings(new_settings = piqp_settings())

print(model)

See Also

solve_piqp(), piqp_settings()

Examples

## example, adapted from PIQP documentation
library(piqp)
library(Matrix)

P <- Matrix(c(6., 0.,
              0., 4.), 2, 2, sparse = TRUE)
c <- c(-1., -4.)
A <- Matrix(c(1., -2.), 1, 2, sparse = TRUE)
b <- c(1.)
G <- Matrix(c(1., 2., -1., 0.), 2, 2, sparse = TRUE)
h <- c(0.2, -1.)
x_lb <- c(-1., -Inf)
x_ub <- c(1., Inf)

settings <- list(verbose = TRUE)

model <- piqp(P, c, A, b, G, h, x_lb, x_ub, settings)

# Solve
res <- model$solve()
res$x

# Define new data
A_new <- Matrix(c(1., -3.), 1, 2, sparse = TRUE)
h_new <- c(2., 1.)

# Update model and solve again
model$update(A = A_new, h = h_new)
res <- model$solve()
res$x

The PIQP Solver Model Class

Description

This class wraps around the PIQP C++ Solver and exposes methods and fields of the C++ object. Users will never need to directly create instances this class and should use the more user-friendly functions piqp() and solve_piqp().

Methods

Public methods

• piqp_model$new()

• piqp_model$solve()

• piqp_model$update()

• piqp_model$get_settings()

• piqp_model$get_dims()

• piqp_model$update_settings()

• piqp_model$clone()

Method new()

Create a new piqp_model object

Usage

piqp_model$new(
  P,
  c,
  A,
  b,
  G,
  h,
  x_lb,
  x_ub,
  settings = list(),
  dense_backend,
  dims
)

Arguments

Returns

a piqp_model object that can be used to solve the QP

Method solve()

Solve the QP model

Usage

piqp_model$solve()

Returns

a list containing the solution

Method update()

Update the current piqp_model with new data

Usage

piqp_model$update(
  P = NULL,
  c = NULL,
  A = NULL,
  b = NULL,
  G = NULL,
  h = NULL,
  x_lb = NULL,
  x_ub = NULL
)

Arguments

Method get_settings()

Obtain the current settings for this model

Usage

piqp_model$get_settings()

Method get_dims()

Obtain the dimensions of this model

Usage

piqp_model$get_dims()

Method update_settings()

Update the current settings with new values for this model

Usage

piqp_model$update_settings(new_settings = list())

Arguments

Method clone()

The objects of this class are cloneable with this method.

Usage

piqp_model$clone(deep = FALSE)

Arguments

Settings parameters with default values and types in parenthesis

Description

Settings parameters with default values and types in parenthesis

Usage

piqp_settings(
  rho_init = 1e-06,
  delta_init = 1e-04,
  eps_abs = 1e-08,
  eps_rel = 1e-09,
  check_duality_gap = TRUE,
  eps_duality_gap_abs = 1e-08,
  eps_duality_gap_rel = 1e-09,
  reg_lower_limit = 1e-10,
  reg_finetune_lower_limit = 1e-13,
  reg_finetune_primal_update_threshold = 7L,
  reg_finetune_dual_update_threshold = 5L,
  max_iter = 250L,
  max_factor_retires = 10L,
  preconditioner_scale_cost = FALSE,
  preconditioner_iter = 10L,
  tau = 0.99,
  iterative_refinement_always_enabled = FALSE,
  iterative_refinement_eps_abs = 1e-12,
  iterative_refinement_eps_rel = 1e-12,
  iterative_refinement_max_iter = 10L,
  iterative_refinement_min_improvement_rate = 5,
  iterative_refinement_static_regularization_eps = 1e-07,
  iterative_refinement_static_regularization_rel = .Machine$double.eps^2,
  verbose = FALSE,
  compute_timings = FALSE
)

Arguments

Value

a list containing the settings parameters.

PIQP Solver

Description

Solves

arg\min_x 0.5 x'P x + c'x

s.t.

A x = b

G x \leq h

x_{lb} \leq x \leq x_{ub}

for real matrices P (nxn, positive semidefinite), A (pxn) with m number of equality constraints, and G (mxn) with m number of inequality constraints

Usage

solve_piqp(
  P = NULL,
  c = NULL,
  A = NULL,
  b = NULL,
  G = NULL,
  h = NULL,
  x_lb = NULL,
  x_ub = NULL,
  settings = list(),
  backend = c("auto", "sparse", "dense")
)

Arguments

Value

A list with elements solution elements

References

Schwan, R., Jiang, Y., Kuhn, D., Jones, C.N. (2023). “PIQP: A Proximal Interior-Point Quadratic Programming Solver.” doi:10.48550/arXiv.2304.00290

See Also

piqp(), piqp_settings() and the underlying PIQP documentation: https://predict-epfl.github.io/piqp/

Examples

## example, adapted from PIQP documentation
library(piqp)
library(Matrix)

P <- Matrix(c(6., 0.,
              0., 4.), 2, 2, sparse = TRUE)
c <- c(-1., -4.)
A <- Matrix(c(1., -2.), 1, 2, sparse = TRUE)
b <- c(1.)
G <- Matrix(c(1., 2., -1., 0.), 2, 2, sparse = TRUE)
h <- c(0.2, -1.)
x_lb <- c(-1., -Inf)
x_ub <- c(1., Inf)

settings <- list(verbose = TRUE)

# Solve with PIQP
res <- solve_piqp(P, c, A, b, G, h, x_lb, x_ub, settings)
res$x

Return the solver status description string

Description

Return the solver status description string

Usage

status_description(code)

Arguments

Value

a status description string

Examples

status_description(1) ## for solved problem
status_description(-1) ## for max iterations limit reached