[docs] add tutorial on piecewise linear

[odow]

preview: https://jump.dev/JuMP.jl/previews/PR3563/tutorials/linear/piecewise_linear/

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (ec3c779) 98.20% compared to head (c43d014) 98.38%.
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+ Coverage   98.20%   98.38%   +0.17%     
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  Lines        5622     5624       +2     
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+ Hits         5521     5533      +12     
+ Misses        101       91      -10     

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[odow]

I think this is a good addition. I can imagine linking quite a few discourse posts to it.

[jd-foster]

This is nice; I've wanted to write something like this for a while but hadn't got around to it.

[jd-foster]

1. Are you interested in solving the problems and displaying the optimal values?
2. I find it a bit confusing to put outer approximation first up when it's about PWL functions - the OA stuff is 'implicit' PWL while the linear approximation is more 'explicit'.
3. What do you think about an example showing combination PWL constraint + exact function (x^2 >= 0) constraint to give a tight envelope?
4. Could reinforce that SOS2 gets you on the line by showing an optimal solution 'before' vs 'after' add the SOS2() constraint.

BTW: typo here for SOS1, read SOS2:

JuMP.jl/src/sets.jl

Line 218 in 913744a

SOS1 (Special Ordered Sets type 2) object than can be used to constrain a

[odow]

See #3565 for docstrings

[odow]

I find it a bit confusing to put outer approximation first up when it's about PWL functions

Should we just call it # Approximating nonlinear functions then?

[odow]

What do you think about an example showing combination PWL constraint + exact function (x^2 >= 0) constraint to give a tight envelope?

I don't understand. How does one represent x^2 >= 0?

[odow]

Are you interested in solving the problems and displaying the optimal values?

I wasn't quite sure of a good way to do this. Ideally, you'd have some sort of interactive slider, where you could move x around, but we can't do that in a static environment.

[jd-foster]

I don't understand. How does one represent x^2 >= 0?

No idea why I wrote that: I meant the epi-graph{t : t >= x^2 }.

[odow]

I still don't really follow what you're suggesting?

[jd-foster]

Broadly speaking, I was thinking that you can combine both sections of the tutorial: a non-convex inner approximation boundary with a convex function (or concave, depending on the direction of the inequality) constraint to get a tight envelope to a function.

@constraints(model, begin
	y >= f(x),
	x == sum(λ[i] * x̂[i] for i in 1:n)
	y <= sum(λ[i] * ŷ[i] for i in 1:n)
    sum(λ) == 1
    λ in SOS2() 
)

But I'll just wait to read the whole thing again once it settles, or just PR later.

Edit: I guess this becomes nonlinear, so might not be useful within the LP tutorials or HiGHS anyway.

[odow]

@jd-foster I merged because this is an improvement. But feel free to make other edits in a separate PR.