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‎.gitignore‎

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.vscode/
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src/transform/storage/
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Manifest.toml

‎LICENSE‎ renamed to ‎LICENSE.md‎

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‎README.md‎

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# SourceCodeMcCormick.jl
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This package is an experimental approach to use source-code transformation to apply McCormick relaxations
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to symbolic functions for use in deterministic global optimization. While packages like `McCormick.jl` [1]
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take set-valued McCormick objects and utilize McCormick relaxation rules to overload standard math operations,
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`SourceCodeMcCormick.jl` (SCMC) aims to interpret symbolic expressions, apply generalized McCormick rules,
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create source code that computes the McCormick relaxations and natural interval extension of the input,
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and compile the source code into functions that return pointwise values of the natural interval extension
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and convex/concave relaxations. This functionality is designed to be used for both algebraic and dynamic
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(in development) systems.
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| **PSOR Lab** | **Build Status** |
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|:------------:|:-----------------------------------------------------------------------------------------------:|
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| [![](https://img.shields.io/badge/Developed_by-PSOR_Lab-342674)](https://psor.uconn.edu/) | [![Build Status](https://github.com/PSORLab/SourceCodeMcCormick.jl/workflows/CI/badge.svg?branch=master)](https://github.com/PSORLab/SourceCodeMcCormick.jl/actions?query=workflow%3ACI) [![codecov](https://codecov.io/gh/PSORLab/SourceCodeMcCormick.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/PSORLab/SourceCodeMcCormick.jl)|
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This package uses source-code transformation to construct McCormick-based relaxations. Expressions composed
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of `Symbolics.jl`-type variables can be passed into `SourceCodeMcCormick.jl` (`SCMC`) functions, after which
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the expressions are factored, generalized McCormick relaxation rules and inclusion monotonic interval
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extensions are applied to the factors, and the factors are recombined symbolically to create expressions
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representing convex and concave relaxations and inclusion monotonic interval extensions of the original
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expression. The new expressions are compiled into functions that return pointwise values of these
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four elements, which can be used in, e.g., a branch-and-bound routine. These functions can be used with
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floating-point values, vectors of floating-point values, or CUDA arrays of floating point values (using
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`CUDA.jl`) to return outputs of the same type.
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## Algebraic Systems
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‎test/addition.jl‎

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@testset "Addition" begin
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@variables x, y
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to_compute = y+5
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posreal_add_cv, posreal_add_cc, posreal_add_lo, posreal_add_hi, posreal_order = all_evaluators(to_compute)
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to_compute = y-5
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negreal_add_cv, negreal_add_cc, negreal_add_lo, negreal_add_hi, negreal_order = all_evaluators(to_compute)
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to_compute = x+y
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add_cv, add_cc, add_lo, add_hi, add_order = all_evaluators(to_compute)
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# Addition rules are very simple; each component of the McCormick expansion
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# is added separately. No need to test more than one type of McCormick object
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# from McCormick.jl
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y_1D = MC{1,NS}(1.0, 2.0, Interval(0.0, 3.0), SVector{1, Float64}(0.5), SVector{1,Float64}(2.5), false)
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xMC = MC{2,NS}(1.0, 1.5, Interval(0.5, 2.0), SVector{2, Float64}(1.0, 2.0), SVector{2, Float64}(3.0, 4.0), false)
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yMC = MC{2,NS}(-0.5, 0.5, Interval(-2.0, 2.0), SVector{2, Float64}(-0.5, 0.5), SVector{2, Float64}(1.0, -1.0), false)
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@test abs(eval_check(posreal_add_cv, y_1D) - (y_1D+5).cv) <= 1E-15
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@test abs(eval_check(posreal_add_cc, y_1D) - (y_1D+5).cc) <= 1E-15
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@test abs(eval_check(posreal_add_lo, y_1D) - (y_1D+5).Intv.lo) <= 1E-15
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@test abs(eval_check(posreal_add_hi, y_1D) - (y_1D+5).Intv.hi) <= 1E-15
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@test abs(eval_check(negreal_add_cv, y_1D) - (y_1D-5).cv) <= 1E-15
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@test abs(eval_check(negreal_add_cc, y_1D) - (y_1D-5).cc) <= 1E-15
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@test abs(eval_check(negreal_add_lo, y_1D) - (y_1D-5).Intv.lo) <= 1E-15
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@test abs(eval_check(negreal_add_hi, y_1D) - (y_1D-5).Intv.hi) <= 1E-15
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@test abs(eval_check(add_cv, xMC, yMC) - (xMC+yMC).cv) <= 1E-15
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@test abs(eval_check(add_cc, xMC, yMC) - (xMC+yMC).cc) <= 1E-15
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@test abs(eval_check(add_lo, xMC, yMC) - (xMC+yMC).Intv.lo) <= 1E-15
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@test abs(eval_check(add_hi, xMC, yMC) - (xMC+yMC).Intv.hi) <= 1E-15
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end

‎test/division.jl‎

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mid_expr(x, y, z) = IfElse.ifelse(x >= y, IfElse.ifelse(y >= z, y, IfElse.ifelse(y == x, y, IfElse.ifelse(z >= x, x, z))),
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IfElse.ifelse(z >= y, y, IfElse.ifelse(x >= z, x, z)))
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function div_cv_case(xcv, xcc, xL, xU, ycv, ycc, yL, yU)
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yL_inv = inv(yU)
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yU_inv = inv(yL)
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ycv_inv = ifelse(yL > 0.0, ifelse(yU <= ycv, 1.0 ./ ycv,
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ifelse(yU >= ycc, 1.0 ./ ycc, 1.0 ./ yU)),
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ifelse(yU < 0.0, ifelse(yL == yU, mid_expr(ycc, ycv, yL).^(-1),
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((yL.^(-1))*(yU - mid_expr(ycc, ycv, yL)) + (yU.^(-1))*(mid_expr(ycc, ycv, yL) - yL))./(yU - yL)),
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NaN))
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ycc_inv = ifelse(yL > 0.0, ifelse(yL <= ycv, (yU + yL - ycv)./(yL*yU),
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ifelse(yL >= ycc, (yU + yL - ycc)./(yL*yU), 1.0 ./ yL)),
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ifelse(yU < 0.0, mid_expr(ycc, ycv, yU).^(-1),
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NaN))
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if xL >= 0.0
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if yL >= 0.0
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if yU_inv*xcv + xU*ycv_inv - xU*yU_inv > yL_inv*xcv + xL*ycv_inv - xL*yL_inv
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return 1
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else
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return 2
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end
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elseif yU <= 0.0
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if (-yU_inv)*xcc + xU*(-ycv_inv) - xU*(-yU_inv) > (-yL_inv)*xcc + xL*(-ycv_inv) - xL*(-yL_inv)
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return 3
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else
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return 4
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end
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else
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return 5
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end
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elseif xU <= 0.0
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if yL >= 0.0
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if yL_inv*(-xcv) + (-xL)*ycc_inv - (-xL)*yL_inv > yU_inv*(-xcv) + (-xU)*ycc_inv - (-xU)*yU_inv
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return 6
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else
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return 7
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end
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elseif yU <= 0.0
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if yL_inv*xcc + xL*ycc_inv - xL*yL_inv > yU_inv*xcc + xU*ycc_inv - xU*yU_inv
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return 8
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else
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return 9
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end
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else
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return 10
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end
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else
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if yL >= 0.0
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if xU*ycv_inv + yU_inv*xcv - yU_inv*xU > xL*ycc_inv + yL_inv*xcv - yL_inv*xL
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return 11
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else
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return 12
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end
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elseif yU <= 0.0
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if xL*(-ycc_inv) + (-yL_inv)*xcc - (-yL_inv)*xL > xU*(-ycv_inv) + (-yU_inv)*xcc - (-yU_inv)*xU
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return 13
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else
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return 14
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end
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else
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return 15
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end
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end
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end
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function div_cc_case(xcv, xcc, xL, xU, ycv, ycc, yL, yU)
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yL_inv = inv(yU)
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yU_inv = inv(yL)
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ycv_inv = ifelse(yL > 0.0, ifelse(yU <= ycv, 1.0 ./ ycv,
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ifelse(yU >= ycc, 1.0 ./ ycc, 1.0 ./ yU)),
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ifelse(yU < 0.0, ifelse(yL == yU, mid_expr(ycc, ycv, yL).^(-1),
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((yL.^(-1))*(yU - mid_expr(ycc, ycv, yL)) + (yU.^(-1))*(mid_expr(ycc, ycv, yL) - yL))./(yU - yL)),
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NaN))
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ycc_inv = ifelse(yL > 0.0, ifelse(yL <= ycv, (yU + yL - ycv)./(yL*yU),
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ifelse(yL >= ycc, (yU + yL - ycc)./(yL*yU), 1.0 ./ yL)),
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ifelse(yU < 0.0, mid_expr(ycc, ycv, yU).^(-1),
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NaN))
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if xL >= 0.0
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if yL >= 0.0
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if yL_inv*xcc + xU*ycc_inv - xU*yL_inv > yU_inv*xcc + xL*ycc_inv - xL*yU_inv
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return 1
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else
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return 2
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end
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elseif yU <= 0.0
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if (-yL_inv)*xcv + xU*(-ycc_inv) - xU*(-yL_inv) > (-yU_inv)*xcv + xL*(-ycc_inv) - xL*(-yU_inv)
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return 3
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else
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return 4
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end
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else
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return 5
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end
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elseif xU <= 0.0
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if yL >= 0.0
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if yU_inv*(-xcc) + (-xL)*ycv_inv - (-xL)*yU_inv > yL_inv*(-xcc) + (-xU)*ycv_inv - (-xU)*yL_inv
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return 6
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else
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return 7
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end
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elseif yU <= 0.0
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if yU_inv*xcv + xL*ycv_inv - xL*yU_inv > yL_inv*xcv + xU*ycv_inv - xU*yL_inv
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return 8
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else
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return 9
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end
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else
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return 10
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end
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else
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if yL >= 0.0
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if xL*ycv_inv + yU_inv*xcc - yU_inv*xL > xU*ycc_inv + yL_inv*xcc - yL_inv*xU
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return 11
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else
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return 12
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end
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elseif yU <= 0.0
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if xU*(-ycc_inv) + (-yL_inv)*xcv - (-yL_inv)*xU > xL*(-ycv_inv) + (-yU_inv)*xcv - (-yU_inv)*xL
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return 13
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else
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return 14
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end
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else
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return 15
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end
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end
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end
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div_cv_case(A::MC, B::MC) = div_cv_case(A.cv, A.cc, A.Intv.lo, A.Intv.hi, B.cv, B.cc, B.Intv.lo, B.Intv.hi)
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div_cc_case(A::MC, B::MC) = div_cv_case(A.cv, A.cc, A.Intv.lo, A.Intv.hi, B.cv, B.cc, B.Intv.lo, B.Intv.hi)
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# For division, need to test all cases with MC*MC, then several cases with Real types
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@testset "Division" begin
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@variables x, y
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# Real divided by a McCormick object
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pos = MC{1,NS}(1.0, 1.5, Interval(0.5, 2.0), SVector{1, Float64}(1.0), SVector{1, Float64}(3.0), false)
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mix = MC{1,NS}(-0.5, 0.5, Interval(-2.0, 2.0), SVector{1, Float64}(-0.5), SVector{1, Float64}(1.0), false)
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neg = -pos
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to_compute = 5.0/y
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posreal_div_cv, posreal_div_cc, posreal_div_lo, posreal_div_hi, posreal_order = all_evaluators(to_compute)
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to_compute = -5.0/y
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negreal_div_cv, negreal_div_cc, negreal_div_lo, negreal_div_hi, negreal_order = all_evaluators(to_compute)
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# pos/pos
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@test abs(eval_check(posreal_div_cv, pos) - (5.0/pos).cv) <= 1E-15
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@test abs(eval_check(posreal_div_cc, pos) - (5.0/pos).cc) <= 1E-15
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@test abs(eval_check(posreal_div_lo, pos) - (5.0/pos).Intv.lo) <= 1E-15
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@test abs(eval_check(posreal_div_hi, pos) - (5.0/pos).Intv.hi) <= 1E-15
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# pos/mix
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@test isnan(eval_check(posreal_div_cv, mix))
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@test isnan((5.0/mix).cv)
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@test isnan(eval_check(posreal_div_cc, mix))
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@test isnan((5.0/mix).cc)
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@test isnan(eval_check(posreal_div_lo, mix))
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@test isnan((5.0/mix).Intv.lo)
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@test isnan(eval_check(posreal_div_hi, mix))
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@test isnan((5.0/mix).Intv.hi)
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# pos/neg
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@test abs(eval_check(posreal_div_cv, neg) - (5.0/neg).cv) <= 1E-15
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@test abs(eval_check(posreal_div_cc, neg) - (5.0/neg).cc) <= 1E-15
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@test abs(eval_check(posreal_div_lo, neg) - (5.0/neg).Intv.lo) <= 1E-15
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@test abs(eval_check(posreal_div_hi, neg) - (5.0/neg).Intv.hi) <= 1E-15
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# neg/pos
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@test abs(eval_check(negreal_div_cv, pos) - (-5.0/pos).cv) <= 1E-15
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@test abs(eval_check(negreal_div_cc, pos) - (-5.0/pos).cc) <= 1E-15
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@test abs(eval_check(negreal_div_lo, pos) - (-5.0/pos).Intv.lo) <= 1E-15
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@test abs(eval_check(negreal_div_hi, pos) - (-5.0/pos).Intv.hi) <= 1E-15
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# neg/mix
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@test isnan(eval_check(negreal_div_cv, mix))
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@test isnan((-5.0/mix).cv)
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@test isnan(eval_check(negreal_div_cc, mix))
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@test isnan((-5.0/mix).cc)
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@test isnan(eval_check(negreal_div_lo, mix))
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@test isnan((-5.0/mix).Intv.lo)
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@test isnan(eval_check(negreal_div_hi, mix))
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@test isnan((-5.0/mix).Intv.hi)
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# neg/neg
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@test abs(eval_check(negreal_div_cv, neg) - (-5.0/neg).cv) <= 1E-15
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@test abs(eval_check(negreal_div_cc, neg) - (-5.0/neg).cc) <= 1E-15
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@test abs(eval_check(negreal_div_lo, neg) - (-5.0/neg).Intv.lo) <= 1E-15
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@test abs(eval_check(negreal_div_hi, neg) - (-5.0/neg).Intv.hi) <= 1E-15
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# Note: McCormick object divided by a real automatically converts from (MC/real) to (MC*(real^-1))
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# through Symbolics.jl, so this is simply multiplication.
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# McCormick object divided by a McCormick object. Verify that cases are satisfied before
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# checking the solution using the cv/cc case checker functions
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pos = MC{2,NS}(1.0, 1.5, Interval(0.5, 2.0), SVector{2, Float64}(1.0, 2.0), SVector{2, Float64}(3.0, 4.0), false)
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pos_lo = MC{2,NS}(0.5, 0.5, Interval(0.5, 2.0), SVector{2, Float64}(1.0, 2.0), SVector{2, Float64}(3.0, 4.0), false)
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pos_hi = MC{2,NS}(1.8, 1.9, Interval(0.5, 2.0), SVector{2, Float64}(1.0, 2.0), SVector{2, Float64}(3.0, 4.0), false)
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mix = MC{2,NS}(-0.5, 0.5, Interval(-2.0, 2.0), SVector{2, Float64}(-0.5, 0.5), SVector{2, Float64}(1.0, -1.0), false)
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neg = -pos
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neg_lo = -pos_lo
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neg_hi = -pos_hi
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@variables x, y
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to_compute = x/y
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div_cv, div_cc, div_lo, div_hi, order = all_evaluators(to_compute)
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@test div_cv_case(pos, pos_lo) == 1
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@test div_cc_case(pos, pos_lo) == 1
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@test abs(eval_check(div_cv, pos, pos_lo) - (pos/pos_lo).cv) <= 1E-15
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@test abs(eval_check(div_cc, pos, pos_lo) - (pos/pos_lo).cc) <= 1E-15
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@test abs(eval_check(div_lo, pos, pos_lo) - (pos/pos_lo).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, pos, pos_lo) - (pos/pos_lo).Intv.hi) <= 1E-15
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@test div_cv_case(pos, pos_hi) == 2
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@test div_cc_case(pos, pos_hi) == 2
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@test abs(eval_check(div_cv, pos, pos_hi) - (pos/pos_hi).cv) <= 1E-15
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@test abs(eval_check(div_cc, pos, pos_hi) - (pos/pos_hi).cc) <= 1E-15
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@test abs(eval_check(div_lo, pos, pos_hi) - (pos/pos_hi).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, pos, pos_hi) - (pos/pos_hi).Intv.hi) <= 1E-15
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@test div_cv_case(pos, neg_lo) == 3
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@test div_cc_case(pos, neg_lo) == 3
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@test abs(eval_check(div_cv, pos, neg_lo) - (pos/neg_lo).cv) <= 1E-15
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@test abs(eval_check(div_cc, pos, neg_lo) - (pos/neg_lo).cc) <= 1E-15
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@test abs(eval_check(div_lo, pos, neg_lo) - (pos/neg_lo).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, pos, neg_lo) - (pos/neg_lo).Intv.hi) <= 1E-15
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@test div_cv_case(pos, neg) == 4
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@test div_cc_case(pos, neg) == 4
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@test abs(eval_check(div_cv, pos, neg) - (pos/neg).cv) <= 1E-15
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@test abs(eval_check(div_cc, pos, neg) - (pos/neg).cc) <= 1E-15
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@test abs(eval_check(div_lo, pos, neg) - (pos/neg).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, pos, neg) - (pos/neg).Intv.hi) <= 1E-15
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@test div_cv_case(pos_hi, mix) == 5
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@test div_cc_case(pos_hi, mix) == 5
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@test isnan(eval_check(div_cv, pos_hi, mix))
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@test isnan((pos_hi/mix).cv)
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@test isnan(eval_check(div_cc, pos_hi, mix))
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@test isnan((pos_hi/mix).cc)
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@test isnan(eval_check(div_lo, pos_hi, mix))
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@test isnan((pos_hi/mix).Intv.lo)
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@test isnan(eval_check(div_hi, pos_hi, mix))
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@test isnan((pos_hi/mix).Intv.hi)
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@test div_cv_case(neg, pos_lo) == 6
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@test div_cc_case(neg, pos_lo) == 6
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@test abs(eval_check(div_cv, neg, pos_lo) - (neg/pos_lo).cv) <= 1E-15
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@test abs(eval_check(div_cc, neg, pos_lo) - (neg/pos_lo).cc) <= 1E-15
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@test abs(eval_check(div_lo, neg, pos_lo) - (neg/pos_lo).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, neg, pos_lo) - (neg/pos_lo).Intv.hi) <= 1E-15
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@test div_cv_case(neg, pos_hi) == 7
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@test div_cc_case(neg, pos_hi) == 7
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@test abs(eval_check(div_cv, neg, pos_hi) - (neg/pos_hi).cv) <= 1E-15
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@test abs(eval_check(div_cc, neg, pos_hi) - (neg/pos_hi).cc) <= 1E-15
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@test abs(eval_check(div_lo, neg, pos_hi) - (neg/pos_hi).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, neg, pos_hi) - (neg/pos_hi).Intv.hi) <= 1E-15
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@test div_cv_case(neg, neg_lo) == 8
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@test div_cc_case(neg, neg_lo) == 8
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@test abs(eval_check(div_cv, neg, neg_lo) - (neg/neg_lo).cv) <= 1E-15
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@test abs(eval_check(div_cc, neg, neg_lo) - (neg/neg_lo).cc) <= 1E-15
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@test abs(eval_check(div_lo, neg, neg_lo) - (neg/neg_lo).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, neg, neg_lo) - (neg/neg_lo).Intv.hi) <= 1E-15
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@test div_cv_case(neg, neg_hi) == 9
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@test div_cc_case(neg, neg_hi) == 9
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@test abs(eval_check(div_cv, neg, neg_hi) - (neg/neg_hi).cv) <= 1E-15
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@test abs(eval_check(div_cc, neg, neg_hi) - (neg/neg_hi).cc) <= 1E-15
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@test abs(eval_check(div_lo, neg, neg_hi) - (neg/neg_hi).Intv.lo) <= 1E-15
273+
@test abs(eval_check(div_hi, neg, neg_hi) - (neg/neg_hi).Intv.hi) <= 1E-15
274+
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@test div_cv_case(neg, mix) == 10
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@test div_cc_case(neg, mix) == 10
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@test isnan(eval_check(div_cv, neg, mix))
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@test isnan((neg/mix).cv)
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@test isnan(eval_check(div_cc, neg, mix))
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@test isnan((neg/mix).cc)
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@test isnan(eval_check(div_lo, neg, mix))
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@test isnan((neg/mix).Intv.lo)
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@test isnan(eval_check(div_hi, neg, mix))
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@test isnan((neg/mix).Intv.hi)
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@test div_cv_case(mix, pos_lo) == 11
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@test div_cc_case(mix, pos_lo) == 11
288+
@test abs(eval_check(div_cv, mix, pos_lo) - (mix/pos_lo).cv) <= 1E-15
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@test abs(eval_check(div_cc, mix, pos_lo) - (mix/pos_lo).cc) <= 1E-15
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@test abs(eval_check(div_lo, mix, pos_lo) - (mix/pos_lo).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, mix, pos_lo) - (mix/pos_lo).Intv.hi) <= 1E-15
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@test div_cv_case(mix, pos) == 12
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@test div_cc_case(mix, pos) == 12
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@test abs(eval_check(div_cv, mix, pos) - (mix/pos).cv) <= 1E-15
296+
@test abs(eval_check(div_cc, mix, pos) - (mix/pos).cc) <= 1E-15
297+
@test abs(eval_check(div_lo, mix, pos) - (mix/pos).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, mix, pos) - (mix/pos).Intv.hi) <= 1E-15
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@test div_cv_case(mix, neg) == 13
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@test div_cc_case(mix, neg) == 13
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@test abs(eval_check(div_cv, mix, neg) - (mix/neg).cv) <= 1E-15
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@test abs(eval_check(div_cc, mix, neg) - (mix/neg).cc) <= 1E-15
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@test abs(eval_check(div_lo, mix, neg) - (mix/neg).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, mix, neg) - (mix/neg).Intv.hi) <= 1E-15
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@test div_cv_case(mix, neg_lo) == 14
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@test div_cc_case(mix, neg_lo) == 14
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@test abs(eval_check(div_cv, mix, neg_lo) - (mix/neg_lo).cv) <= 1E-15
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@test abs(eval_check(div_cc, mix, neg_lo) - (mix/neg_lo).cc) <= 1E-15
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@test abs(eval_check(div_lo, mix, neg_lo) - (mix/neg_lo).Intv.lo) <= 1E-15
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@test abs(eval_check(div_hi, mix, neg_lo) - (mix/neg_lo).Intv.hi) <= 1E-15
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@test div_cv_case(mix, mix) == 15
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@test div_cc_case(mix, mix) == 15
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@test isnan(eval_check(div_cv, mix, mix))
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@test isnan((mix/mix).cv)
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@test isnan(eval_check(div_cc, mix, mix))
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@test isnan((mix/mix).cc)
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@test isnan(eval_check(div_lo, mix, mix))
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@test isnan((mix/mix).Intv.lo)
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@test isnan(eval_check(div_hi, mix, mix))
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@test isnan((mix/mix).Intv.hi)
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end

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