JulieSliwak/MathProgComplex.jl

MathProgComplex.jl

Dev:

The MathProgComplex module is a tool for polynomial optimization problems with complex variables. These problems consist in optimizing a generic complex multivariate polynomial function, subject to some complex polynomial equality and inequality constraints. The MathProgComplex module enables:

• the manipulation of multivariate polynomials with complex numbers to construct polynomial optimization problems with complex variables (POP-C).
• the evaluation of polynomials, for example the objective and the constraints of a (POP-C) from points
• the resolution of a (POP-C) via a JuMP model
• the export of a (POP-C) to be solved using another language

Setting Julia for custom modules

The modules need to be accessible from julia's path to be loaded with the using command. This can be done by running push!(LOAD_PATH, "location/of/modules"), which will update the path for the current session.

The path can be updated at every start of a Julia session by adding the command to the .juliarc.jl file, which should be located (or created) at the location given by homedir().

Structures

The MathProgComplex environment provides a structure and methods for working with complex polynomial optimization problems subject to polynomial constraints. It is based on a polynomial environment that allows to work on polynomial objects with natural operations (+, -, *, conj, |.|^2).

Polynomials

• The base type is Variable: it is a structure with a name (a string) and a type (Complex, Real or Bool).

using MathProgComplex
a = Variable("a", Complex)
b = Variable("b", Real)
c = Variable("c", Bool)

From Variable type, Exponent and Polynomial can be constructed by calling the respective constructors or with algebraic operations (+, -, *, conj, |.|).

• An Exponent is a product of Variables.

expo1 = a*b
expo2 = conj(a)^3*b^5
expo3 = abs2(a) # =a*conj(a)

• A Polynomial is a sum of Exponents times complex numbers.

p = 3*expo1 + (4+2im)*expo2 +2im*expo3

• The Point type holds the variables at which polynomials can be evaluated.

Implemented methods

- isconst, isone
- evaluate
- abs2, conj
- is_homogeneous: tests if p(exp(iϕ)X) = p(X) ∀X∈R^n, Φ∈R (phaseinvariant  equation)
- cplx2real: convert the provided object to a tuple of real and imaginary part, expressed with real and imaginary part variables.

using MathProgComplex

a = Variable("a", Complex)
b = Variable("b", Real)
p = a*conj(a) + b + 2

print(p)
# 2 + b + conj(a) * a

pt = Point([a, b], [1+2im, 1+im])
print(pt)
# a 1 + 2im
# b 1

pt = pt - Point([a], [1])
print(pt)
# a 0 + 2im
# b 1

val = evaluate(p, pt) # 7.0 + 0.0im

p_real, p_imag = cplx2real(p)
pt_r = cplx2real(pt)
val_real = evaluate(p_real, pt_r) # 7.0
val_imag = evaluate(p_imag, pt_r) # 0

Polynomial optimization problems

• A Constraint structure holds a Polynomial and complex upper and lower bounds.

• A Problem is made up of:

  • the collection of Variables of the problem (updated internally),
  • a Polynomial objective,
  • several named constraints.

Implemented methods

• get_objective, set_objective!
• get_variables, get_variabletype, has_variable, add_variable!
• get_constraint, get_constraints, has_constraint, constraint_type, add_constraint!, rm_constraint!
• get_slacks, get_minslack
• pb_cplx2real: converts the problem variables, objective and constraints to real expressions function of real and imaginary part of the original problem variables.

Here is a full example, more can be found in the examples and test directories.

using MathProgComplex

a = Variable("a", Complex)
b = Variable("b", Real)
p_obj = abs2(a) + abs2(b) + 2
p_cstr1 = 3a + b + 2
p_cstr2 = abs2(b) + 5a*b + 2

pb = Problem()

set_objective!(pb, p_obj)

add_constraint!(pb, "Cstr 1", p_cstr1 << (3+5im)) # 2 + (3.0)*a + b < 3 + 5im

add_constraint!(pb, "Cstr 2", (2-im) << p_cstr2 << (3+7im)) # 2 - 1im < 2 + (5.0)*a * b + b^2 < 3 + 7im

add_constraint!(pb, "Cstr 3", p_cstr2 == 0) # 2 + (5.0)*a * b + b^2 = 0


print(pb)
# ▶ variables: a b 
# ▶ objective: 2 + conj(a) * a + b^2
# ▶ constraints: 
#  →     Cstr 1: 2 + (3.0)*a + b < 3 + 5im
#  →     Cstr 2: 2 - 1im < 2 + (5.0)*a * b + b^2 < 3 + 7im
#  →     Cstr 3: 2 + (5.0)*a * b + b^2 = 0

pt_sol = Point([a, b], [1, 1+2im])
print("slack by constraint at given point:\n", get_slacks(pb, pt_sol))
# slack by constraint at given point:
# Cstr 1 -3.0 + 5.0im
# Cstr 2 -5.0 + 1.0im
# Cstr 3 -8.0 - 0.0im

Resolution

The polynomial optimization problems can be converted into JuMP models or be exported into formatted text files to be used in another language.

Using JuMP

m, JuMPvar = get_JuMP_cartesian_model(pb, solver)
solve(m)

Using file export

The .dat text format used in this module allows storing polynomial optimization problems in inary, Real or Complex variables, along with a scalar value for each variable (a point).

export_to_dat(pb, amplexportpath, point)
run_knitro(amplexportpath, amplscriptpath)