Types

struct MC{N, T<:RelaxTag} <: Real

MC{N, T <: RelaxTag} <: Real is the McCormick (w/ (sub)gradient) structure which is used to overload standard calculations. The fields are:

• cv::Float64: Convex relaxation

• cc::Float64: Concave relaxation

• Intv::Interval{Float64}: Interval bounds

• cv_grad::SVector{N, Float64} where N: (Sub)gradient of convex relaxation

• cc_grad::SVector{N, Float64} where N: (Sub)gradient of concave relaxation

• cnst::Bool: Boolean indicating whether the relaxations are constant over the domain. True if bounding an interval/constant. False, otherwise. This may change over the course of a calculation cnst for zero(x) is true even if x.cnst is false.

abstract type RelaxTag

An abstract type the subtypes of which define the manner of relaxation that will be performed for each operator applied to the MC object. Currently, the struct NS which specifies that standard (Mitsos 2009) are to be used is fully supported. Limited support is provided for differentiable McCormick relaxations specified by struct Diff (Khan 2017) and struct MV struct MV (Tsoukalas 2011.) A rounding-safe implementation of the standard McCormick relaxations is specified by the struct NSSafe which is work in progress.

MC{N,T}(y::Interval{Float64})

Constructs a McCormick relaxation with the convex relaxation equal to y.lo and concave relaxation equal to y.hi.

MC{N,T}(y::Float64)

Constructs a McCormick relaxation with the convex relaxation equal to y and concave relaxation equal to y.

mid3(
    x::Float64,
    y::Float64,
    z::Float64
) -> Tuple{Float64, Int64}

Calculates the middle of three numbers returning the value and the index where x >= y.

mid3v(x::Float64, y::Float64, z::Float64) -> Float64

Calculates the middle of three numbers (x,y,z) returning the value where x <= y.

mid_grad(
    cc_grad::SArray{Tuple{N}, Float64, 1, N},
    cv_grad::SArray{Tuple{N}, Float64, 1, N},
    id::Int64
) -> Any

Takes the concave relaxation gradient cc_grad, the convex relaxation gradient cv_grad, and the index of the midpoint returned id and outputs the appropriate gradient according to McCormick relaxation rules.

dline_seg(
    f::Function,
    df::Function,
    x::Float64,
    xL::Float64,
    xU::Float64
) -> Tuple{Any, Any}

Calculates the value and the slope of the line segment between (xL, f(xL)) and (xU, f(xU)). Defaults to evaluating the derivative of the function if the interval is tight.

seed_gradient(j::Int64, x::Val{N}) -> Any

Creates a x::SVector{N,Float64} object that is one at x[j] and zero everywhere else.

cut(
    xL::Float64,
    xU::Float64,
    cv::Float64,
    cc::Float64,
    cv_grad::SArray{Tuple{N}, Float64, 1, N},
    cc_grad::SArray{Tuple{N}, Float64, 1, N}
) -> Tuple{Float64, Float64, Any, Any}

Refines convex/concave relaxations cv and cc with associated subgradients cv_grad and cc_grad by intersecting them with the interval bounds xL and xU.

secant(
    x0::Float64,
    x1::Float64,
    xL::Float64,
    xU::Float64,
    f::Function,
    envp1::Float64,
    envp2::Float64
) -> Tuple{Float64, Bool}

Defines a local 1D secant method to solve for the root of f between the bounds xL and xU using x0 and x1 as a starting points. The inputs envp1 and envp2 are the envelope calculation parameters.

newton(
    x0::Float64,
    xL::Float64,
    xU::Float64,
    f::Function,
    df::Function,
    envp1::Float64,
    envp2::Float64
) -> Tuple{Any, Bool}

Defines a local 1D newton method to solve for the root of f between the bounds xL and xU using x0 as a starting point. The derivative of f is df. The inputs envp1 and envp2 are the envelope calculation parameters.

golden_section_it(
    init::Int64,
    a::Float64,
    fa::Float64,
    b::Float64,
    fb::Float64,
    c::Float64,
    fc::Float64,
    f::Function,
    envp1::Float64,
    envp2::Float64
) -> Float64

Define iteration used in golden section method. The inputs fa,fb, and fc, are the function f evaluated at a,b, and c respectively. The inputs envp1 and envp2 are the envelope calculation parameters. The value init is the iteration number of the golden section method.

golden_section(
    xL::Float64,
    xU::Float64,
    f::Function,
    envp1::Float64,
    envp2::Float64
) -> Float64

Defines a local 1D golden section method to solve for the root of f between the bounds xL and xU using x0 as a starting point. Define iteration used in golden section method. The inputs envp1 and envp2 are the envelope calculation parameters.

struct MCNoGrad <: Real

MCNoGrad <: Real is a McCormick structure without RelaxType Tag or subgradients. This structure is used for source-code transformation approaches to constructing McCormick relaxations. Methods definitions and calls should specify the relaxation type used (i.e.) +(::NS, x::MCNoGrad, y::MCNoGrad).... Moreover, the kernel associated with this returns all intermediate calculations necessary to compute subgradient information whereas the overloading calculation simply returns the MCNoGrad object. For univariate calculations without tiepoints such as we log2(::NS, x::MCNoGrad)::MCNoGrad whereas log2_kernel(::NS, x::MCNoGrad, ::Bool) = (::MCNoGrad, cv_id::Int, cc_id::Int, dcv, dcc). Univariate NS functions follow convention (MCNoGrad, cvid, ccid, dcv, dcc, tp1cv, tp1cc, .... tpncv, tpncc) where cv_id is the subgradient selected (1 = cv, 2 = cc, 3 = 0), dcv and dcc are derivatives (or elements of subdifferential) of the outside function evaluated per theorem at the point being evaluated and tpicv, tpicc are the ith tiepoints associated with computing the envelope of the outside function. .

• cv::Float64: Convex relaxation

• cc::Float64: Concave relaxation

• Intv::Interval{Float64}: Interval bounds

• cnst::Bool: Boolean indicating whether the relaxations are constant over the domain. True if bounding an interval/constant. False, otherwise. This may change over the course of a calculation cnst for zero(x) is true even if x.cnst is false.

MCNoGrad(y::Float64)

Constructs McCormick relaxation with convex relaxation equal to y and concave relaxation equal to y.

MCNoGrad(y::Interval{Float64})

Constructs McCormick relaxation with convex relaxation equal to y.lo and concave relaxation equal to y.hi.

MCNoGrad(cv::Float64, cc::Float64)

Constructs McCormick relaxation with convex relaxation equal to cv and concave relaxation equal to cc.