### A Pluto.jl notebook ###
# v1.0.1

using Markdown
using InteractiveUtils

# ╔═╡ 6ae6c41e-70f4-4d0e-b5b2-230143d6845b
using PlutoUI, HypertextLiteral, PlutoTeachingTools

# ╔═╡ a06e1c98-b999-4ae9-8333-c1d45536cbc0
md"""
## Semidefinite Program:

```math
\begin{align}
\min_{X \in \mathbb{R}^{n \times n}} & \;\; \langle C, X \rangle
&
\max_{y \in \mathbb{R}^m, Z \in \mathbb{R}^{n \times n}} & \;\; b^T y
\\
\;\;\text{s.t.} \;\; & \langle A_i, X \rangle = b_i
& \;\;\text{s.t.} \;\; & C - \sum_{i=1}^m A_i y_i = Z\\
& X \text{ is PSD} && Z \text{ is PSD}
\end{align}
```
"""

# ╔═╡ 2857027e-1971-4705-9561-a728b04698e6
TwoColumn(
	md"""
Pros:
* Convex. Guarantee to get optimal solution.
* Polynomial time algorithms.
* Many solvers: SOS, Mosek, Clarabel, CSDP, SDPA, ...
""",
	md"""
Cons:
* Number of variables is ``n^2``.
* Even if we know that there exists a low-rank solution for ``X`` or ``Z``, it cannot be exploited (except [ProxSDP.jl](https://github.com/mariohsouto/ProxSDP.jl) or [Loraine](https://github.com/kocvara/Loraine.jl))
* Geared towards high-rank solutions.
""",
)

# ╔═╡ 1e97c55e-7072-47c3-96dc-eef846677360
md"""
## Burer-Monteiro

```math
\begin{align}
\min_{U \in \mathbb{R}^{n \times r}} & \;\; \langle C, UU^\top \rangle
&
\max_{y \in \mathbb{R}^m, Z \in \mathbb{R}^{n \times n}} & \;\; b^T y
\\
\;\;\text{s.t.} \;\; & \langle A_i, UU^\top \rangle = b_i
& \;\;\text{s.t.} \;\; & C - \sum_{i=1}^m A_i y_i = Z\\
&&& Z \text{ 🤞 PSD}
\end{align}
```
"""

# ╔═╡ dd7c6532-5b3f-4217-aef8-1109c759d18f
TwoColumn(
	md"""
Pros:
* Number of variables is ``nr``.
* Can force the rank of solution to be small.
* **A postiori** guarantee that ``U`` is optimal if ``Z`` is PSD.
* No need to factorize ``X``.
""",
	md"""
Cons:
* Nonconvex:
  - harder to optimize
  - can get stuck at spurious local minima
""",
)

# ╔═╡ 11f6f875-73c0-4d5c-a75a-8a6dcf3e6c8f
md"""
## SDPA format
``X`` block-diagonal (scalars are ``1 \times 1`` blocks), ``A_i`` are symmetric sparse matrices
```math
\begin{align}
\min_{X_j \in \mathbb{R}^{n_j \times n_j}} & \;\; \sum_{j} \langle C_j, X_j \rangle
\\
\;\;\text{s.t.} \;\; & \sum_j \langle A_{ij}, X_j \rangle = b_i\\
& X_j \text{ is PSD}
\end{align}
```
"""

# ╔═╡ 141a9b90-767f-4489-98ac-3f760d6d7e8b
TwoColumn(
	md"""
Pros:
* ``A_{ij}`` matrices easy to obtain from JuMP's affine expressions.
* Similar to LPs, care-free modeling for user.
""",
	md"""
Cons:
* Inefficient representation of dense low-rank matrices
* Cannot do matrix-free, e.g.  ``A x`` → `fft(x)`
""",
)

# ╔═╡ 77650a10-43fa-47fc-855d-04a10f567b12
md"""
## Deviations from the SDPA format

* [SDPLR (2002)](https://github.com/jump-dev/SDPLR.jl), [DSDP (2005)](https://github.com/jump-dev/DSDP.jl), [Loraine (2023)](https://github.com/kocvara/Loraine.jl), [SDPLRPlus (2024)](https://github.com/luotuoqingshan/SDPLRPlus.jl) : Low-rank ``A_i = F_i F_i^\top`` with dense factor ``F_i``
* [Hypatia](https://github.com/jump-dev/Hypatia.jl) : ``X`` only accessed via orthogonal rank-1 ``A_i``.
* [SketchyCGAL (2021)](https://github.com/alpyurtsever/SketchyCGAL) : ``A_i = I``.
* [BMSOS (2022)](https://github.com/JuliaAlgebra/BMSOS.jl) : ``A_i`` matrix-free with FFT.
* [SDPLRPlus (2024)](https://github.com/luotuoqingshan/SDPLRPlus.jl): ``A_i`` in either CSC or COO format. COO better if nnz ``\ll O(n)``.
* [cuHALLaR (2025)](https://arxiv.org/pdf/2505.13719) : ``A_i = F_iD_iF_i^\top``, ``D_i \in \mathbb{R}^{r_i \times r_i}`` **not** necessarily diagonal.
"""

# ╔═╡ 98cdc556-d8c5-4f87-8377-c5b3d216bb52
md"""## The format liberator

[LowRankOpt](github.com/blegat/LowRankOpt.jl/) introduces:
"""

# ╔═╡ 569d52de-1eb0-4a16-86c2-8758557fa23a
TwoColumn(
	md"""
`SetDotProducts{WITHOUT_SET}`
```math
\{(\langle A_1, X \rangle, \ldots, \langle A_m, X \rangle) \mid X \text{ is PSD}\}
```
`SetDotProducts{WITH_SET}`
```math
\{(X, \langle A_1, X \rangle, \ldots, \langle A_m, X \rangle) \mid X \text{ is PSD}\}
```
""",
	md"""
`LinearCombinationInSet{WITHOUT_SET}`
```math
\{ y \in \mathbb{R}^{m} : \sum_{i=1}^m y_i A_i \text{ is PSD} \}.
```
`LinearCombinationInSet{WITH_SET}`
```math
\{ (y, C) \in \mathbb{R}^{m} : \sum_{i=1}^m y_i A_i - C \text{ is PSD} \}.
```
""",
)

# ╔═╡ 2e54be0f-8ba3-4076-940c-810aa740f304
md"""
## Decoupling SDP solvers

[LowRankOpt](github.com/blegat/LowRankOpt.jl/) transforms SDP in MOI + `LinearCombinationInSet{WITH_SET}` into `NLPModels`.
"""

# ╔═╡ 1c8215e4-f0e1-42ec-9231-9037433b66ac
TwoColumn(
	md"""
Solver on ``X``

* Extends `NLPModels` interface for PSD specifics
* Solver never accesses ``A_i`` or ``C`` directly
* Proof of concept with [Loraine](https://github.com/kocvara/Loraine.jl/pull/29)
""",
	md"""
Burer-Monteiro on ``U``

* No need to extend `NLPModels` → existing `NLPModels` can be used
* Implicit dualization: `NLPModels` solver searches ``U``, user formulates with ``y``.
* Solvers like [SDPLRPlus](https://github.com/luotuoqingshan/SDPLRPlus.jl) can also exploit specifics of BM (e.g. easy line-search, rank update).
""",
)

# ╔═╡ 0d7799dc-71cb-41a3-92ca-044e756aeb6e
md"""
> This decoupling makes it **easier** to write an SDP solver, no need for an MOI interface. The solver **exploit new custom** ``A_i`` structure for free.
"""

# ╔═╡ 31f2c438-d1a2-4ebe-8414-0b8c29adf65d
md"## Extending NLPModels"

# ╔═╡ 9bee7b40-b906-430f-9260-ebca684272ca
TwoColumn(
	md"""
Solver on ``X``: [Loraine.jl](https://github.com/kocvara/Loraine.jl)

* KKT → System on ``\Delta X``, ``\Delta y``, ``\Delta Z``
* Nesterov–Todd scaling →
* System on ``\Delta y`` **only** : ``H \Delta y = r``
* Need ``r`` and *Schur complement* ``H``
""",
	md"""
Burer-Monteiro on ``U``: [SDPLRPlus.jl](https://github.com/luotuoqingshan/SDPLRPlus.jl)

* Line-search : Minimizes univariate quartic polynomial with [Polynomials](https://github.com/JuliaMath/Polynomials.jl)
* [Riemannian staircase](https://www.nicolasboumal.net/papers/The_staircase_method.pdf): Change the number of columns of ``U`` **during solve**
""",
)

# ╔═╡ 8fa41b95-f05c-408c-8f66-8f989f50769f
md"""
## Supporting arbitrary matrix types

* Number of matrix types in a given model is small
* Transform `Vector{AbstractMatrix}` -> `Vector{Union{...}}` before calling the solver
* No runtime dispatch thanks to [Julia's union-splitting](https://julialang.org/blog/2018/08/union-splitting/)
```julia
convert(Array{Union{unique(typeof.(A))...}}, A)
```
"""

# ╔═╡ 64a3ccea-60c8-42a5-890c-d60e6454d6ba
md"## Benchmark"

# ╔═╡ 5466dcad-20c2-443c-9b3f-75d467a74bc1
html"<p align=center style=\"font-size: 20px; margin-bottom: 5cm; margin-top: 5cm;\">The End</p>"

# ╔═╡ dd939c2f-93a2-46e1-be93-490eb99f74ca
begin
struct Path
    path::String
end

function imgpath(path::Path)
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struct URL
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end;

# ╔═╡ b0789776-5c1a-11f1-9355-0b1c7aa428ee
@htl("""
<p align=center style=\"font-size: 30px;\">Second-Order GPU solver for Burer-Monteiro</p>
<p align=right><i>Benoît Legat</i>, 1st June 2026</p>
$(img("https://upload.wikimedia.org/wikipedia/commons/7/72/UCLouvain_logo.svg", :height => 32))
$(img("https://jump.dev/assets/jump-dev-workshops/2026/logo.png", :height => 64))
$(PlutoTeachingTools.ChooseDisplayMode())
$(PlutoUI.TableOfContents(depth=1))
""")

# ╔═╡ d917bc6f-89c8-4ba2-b110-f26a91ee0016
TwoColumnWideLeft(
	img("$(pwd())/bench.png"),
	md"""
Benchmark instance of Section 7 of [arxiv#2205.11466](https://arxiv.org/abs/2205.11466), rank-4. [SumOfSquares](https://github.com/jump-dev/SumOfSquares.jl/) Trigo + Lagrange bases
* NLopt → [JSOSolvers](https://github.com/JuliaSmoothOptimizers/JSOSolvers.jl)' `lbfgs` and `trunk`.
* FFT uses [AbstractFFTs](https://github.com/JuliaMath/AbstractFFTs.jl) : [FFTW.jl](https://github.com/JuliaMath/FFTW.jl) on CPU and [`CUDA.CUFFT`](https://github.com/JuliaGPU/CUDA.jl) on GPU
* NVIDIA RTX PRO 1000 Blackwell Laptop gen
* Done with François Pacaud
""",
)

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uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
version = "1.11.0"

[[deps.Tricks]]
git-tree-sha1 = "311349fd1c93a31f783f977a71e8b062a57d4101"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.13"

[[deps.URIs]]
git-tree-sha1 = "bef26fb046d031353ef97a82e3fdb6afe7f21b1a"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.6.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"
version = "1.11.0"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
version = "1.11.0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.3.1+2"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.15.0+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.64.0+1"
"""

# ╔═╡ Cell order:
# ╟─b0789776-5c1a-11f1-9355-0b1c7aa428ee
# ╟─a06e1c98-b999-4ae9-8333-c1d45536cbc0
# ╟─2857027e-1971-4705-9561-a728b04698e6
# ╟─1e97c55e-7072-47c3-96dc-eef846677360
# ╟─dd7c6532-5b3f-4217-aef8-1109c759d18f
# ╟─11f6f875-73c0-4d5c-a75a-8a6dcf3e6c8f
# ╟─141a9b90-767f-4489-98ac-3f760d6d7e8b
# ╟─77650a10-43fa-47fc-855d-04a10f567b12
# ╟─98cdc556-d8c5-4f87-8377-c5b3d216bb52
# ╟─569d52de-1eb0-4a16-86c2-8758557fa23a
# ╟─2e54be0f-8ba3-4076-940c-810aa740f304
# ╟─1c8215e4-f0e1-42ec-9231-9037433b66ac
# ╟─0d7799dc-71cb-41a3-92ca-044e756aeb6e
# ╟─31f2c438-d1a2-4ebe-8414-0b8c29adf65d
# ╟─9bee7b40-b906-430f-9260-ebca684272ca
# ╟─8fa41b95-f05c-408c-8f66-8f989f50769f
# ╟─64a3ccea-60c8-42a5-890c-d60e6454d6ba
# ╟─d917bc6f-89c8-4ba2-b110-f26a91ee0016
# ╟─5466dcad-20c2-443c-9b3f-75d467a74bc1
# ╟─6ae6c41e-70f4-4d0e-b5b2-230143d6845b
# ╟─dd939c2f-93a2-46e1-be93-490eb99f74ca
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002