The Riemannian Interior Point Newton Method

.zenodo.json

@@ -53,6 +53,11 @@
             "name": "Kjemsås, Even Stephansen",
             "type": "ProjectMember"
         },
+        {
+            "affiliation": "NTNU Trondheim",
+            "name": "Stokkenes, Markus A.",
+            "type": "ProjectMember"
+        },
         {
             "name": "Daniel VandenHeuvel",
             "type": "other",

Changelog.md

@@ -5,6 +5,24 @@ All notable Changes to the Julia package `Manopt.jl` will be documented in this
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [0.4.68] – August 2, 2024
+
+### Added
+
+* an Interior Point Newton Method, the `interior_point_newton`
+* a `conjugate_residual` Algorithm to solve a linear system on a tangent space.
+* `ArmijoLinesearch` now allows for additional `additional_decrease_condition` and `additional_increase_condition` keywords to add further conditions to accept additional conditions when to accept an decreasing or increase of the stepsize.
+* add a `DebugFeasibility` to have a debug print about feasibility of points in constrained optimisation employing the new `is_feasible` function
+* add a `InteriorPointCentralityCondition` check that can be added for step candidates within the line search of `interior_point_newton`
+* Add Several new functors
+  * the `LagrangianCost`, `LagrangianGradient`, `LagrangianHessian`, that based on a constrained objective allow to construct the hessian objective of its Lagrangian
+  * the `CondensedKKTVectorField` and its `CondensedKKTVectorFieldJacobian`, that are being used to solve a linear system within `interior_point_newton`
+  * the `KKTVectorField` as well as its `KKTVectorFieldJacobian` and ``KKTVectorFieldAdjointJacobian`
+  * the `KKTVectorFieldNormSq` and its `KKTVectorFieldNormSqGradient` used within the Armijo line search of `interior_point_newton`
+* New stopping criteria
+  * A `StopWhenRelativeResidualLess` for the `conjugate_residual`
+  * A `StopWhenKKTResidualLess` for the `interior_point_newton`
+
 ## [0.4.67] – July 25, 2024
 
 ### Added

Project.toml

@@ -1,7 +1,7 @@
 name = "Manopt"
 uuid = "0fc0a36d-df90-57f3-8f93-d78a9fc72bb5"
 authors = ["Ronny Bergmann <[email protected]>"]
-version = "0.4.67"
+version = "0.4.68"
 
 [deps]
 ColorSchemes = "35d6a980-a343-548e-a6ea-1d62b119f2f4"

docs/make.jl

@@ -168,13 +168,15 @@ makedocs(;
             "Chambolle-Pock" => "solvers/ChambollePock.md",
             "CMA-ES" => "solvers/cma_es.md",
             "Conjugate gradient descent" => "solvers/conjugate_gradient_descent.md",
+            "Conjugate Residual" => "solvers/conjugate_residual.md",
             "Convex bundle method" => "solvers/convex_bundle_method.md",
             "Cyclic Proximal Point" => "solvers/cyclic_proximal_point.md",
             "Difference of Convex" => "solvers/difference_of_convex.md",
             "Douglas—Rachford" => "solvers/DouglasRachford.md",
             "Exact Penalty Method" => "solvers/exact_penalty_method.md",
             "Frank-Wolfe" => "solvers/FrankWolfe.md",
             "Gradient Descent" => "solvers/gradient_descent.md",
+            "Interior Point Newton" => "solvers/interior_point_Newton.md",
             "Levenberg–Marquardt" => "solvers/LevenbergMarquardt.md",
             "Nelder–Mead" => "solvers/NelderMead.md",
             "Particle Swarm Optimization" => "solvers/particle_swarm.md",

docs/src/about.md

@@ -11,6 +11,7 @@ The following people contributed
 * Even Stephansen Kjemsås contributed to the implementation of the [Frank Wolfe Method](solvers/FrankWolfe.md) solver
 * Mathias Ravn Munkvold contributed most of the implementation of the [Adaptive Regularization with Cubics](solvers/adaptive-regularization-with-cubics.md) solver
 * [Tom-Christian Riemer](https://www.tu-chemnitz.de/mathematik/wire/mitarbeiter.php) implemented the [trust regions](solvers/trust_regions.md) and [quasi Newton](solvers/quasi_Newton.md) solvers.
+* [Markus A. Stokkenes](https://www.linkedin.com/in/markus-a-stokkenes-b41bba17b/) contributed most of the implementation of the [Interior Point Newton Method](solvers/interior_point_Newton.md)
 * [Manuel Weiss](https://scoop.iwr.uni-heidelberg.de/author/manuel-weiß/) implemented most of the [conjugate gradient update rules](@ref cg-coeffs)
 
 as well as various [contributors](https://github.com/JuliaManifolds/Manopt.jl/graphs/contributors) providing small extensions, finding small bugs and mistakes and fixing them by opening [PR](https://github.com/JuliaManifolds/Manopt.jl/pulls)s.

docs/src/extensions.md

@@ -62,6 +62,14 @@ Manopt.max_stepsize(::TangentBundle, ::Any)
 mid_point
 ```
 
+Internally, `Manopt.jl` provides the two additional functions to choose some
+Euclidean space when needed as
+
+```@docs
+Manopt.Rn
+Manopt.Rn_default
+```
+
 ## JuMP.jl
 
 Manopt can be used using the [JuMP.jl](https://github.com/jump-dev/JuMP.jl) interface.

docs/src/plans/objective.md

@@ -211,6 +211,16 @@ It might be beneficial to use the adapted problem to specify different ranges fo
 ConstrainedManoptProblem
 ```
 
+as well as the helper functions
+
+```@docs
+AbstractConstrainedFunctor
+AbstractConstrainedSlackFunctor
+LagrangianCost
+LagrangianGradient
+LagrangianHessian
+```
+
 #### Access functions
 
 ```@docs
@@ -223,9 +233,16 @@ get_grad_equality_constraint
 get_grad_inequality_constraint
 get_hess_equality_constraint
 get_hess_inequality_constraint
+is_feasible
+```
+
+#### Internal functions
+
+```@docs
+Manopt.get_feasibility_status
 ```
 
-### A vectorial cost function
+### Vectorial objectives
 
 ```@docs
 Manopt.AbstractVectorFunction

docs/src/plans/stepsize.md

@@ -33,6 +33,13 @@ Tangent bundle with the Sasaki metric has 0 injectivity radius, so the maximum s
 `Hyperrectangle` also has 0 injectivity radius and an estimate based on maximum of dimensions along each index is used instead.
 For manifolds with corners, however, a line search capable of handling break points along the projected search direction should be used, and such algorithms do not call `max_stepsize`.
 
+Some solvers have a different iterate from the one used for linesearch. Then the following state can be used to wrap
+these locally
+
+```@docs
+StepsizeState
+```
+
 ## Literature
 
 ```@bibliography

docs/src/references.bib

@@ -341,6 +341,16 @@ @article{DuranMoelleSbertCremers:2016
 % --- E
 %
 %
+@article{El-BakryTapiaTsuchiyaZhang:1996,
+    AUTHOR  = {El-Bakry, A. S. and Tapia, R. A. and Tsuchiya, T. and Zhang, Y.},
+    DOI     = {10.1007/bf02275347},
+    JOURNAL = {Journal of Optimization Theory and Applications},
+    NUMBER  = {3},
+    PAGES   = {507–541},
+    TITLE   = {On the formulation and theory of the Newton interior-point method for nonlinear programming},
+    VOLUME  = {89},
+    YEAR    = {1996}
+}
 
 % --- F
 %
@@ -532,6 +542,18 @@ @article{Karcher:1977
 % --- L
 %
 %
+@article{LaiYoshise:2024,
+    AUTHOR  = {Lai, Zhijian and Yoshise, Akiko},
+    DOI     = {10.1007/s10957-024-02403-8},
+    EPRINT     = {2203.09762},
+    EPRINTTYPE = {arXiv},
+    JOURNAL = {Journal of Optimization Theory and Applications},
+    NUMBER  = {1},
+    PAGES   = {433–469},
+    TITLE   = {Riemannian Interior Point Methods for Constrained Optimization on Manifolds},
+    VOLUME  = {201},
+    YEAR    = {2024}
+}
 @article{LausNikolovaPerschSteidl:2017,
     AUTHOR       = {Laus, F. and Nikolova, M. and Persch, J. and Steidl, G.},
     YEAR         = {2017},

docs/src/solvers/conjugate_residual.md

@@ -0,0 +1,34 @@
+# Conjugate Residual Solver in a Tangent space
+
+```@meta
+CurrentModule = Manopt
+```
+
+```@docs
+conjugate_residual
+```
+
+## State
+
+```@docs
+ConjugateResidualState
+```
+
+## Objetive
+
+```@docs
+SymmetricLinearSystemObjective
+```
+
+## Additional stopping criterion
+
+```@docs
+StopWhenRelativeResidualLess
+```
+
+## Literature
+
+```@bibliography
+Pages = ["conjugate_residual.md"]
+Canonical=false
+```

docs/src/solvers/index.md

@@ -88,16 +88,24 @@ For these you can use
 
 * The [Augmented Lagrangian Method](augmented_Lagrangian_method.md) (ALM), where both `g` and `grad_g` as well as `h` and `grad_h` are keyword arguments, and one of these pairs is mandatory.
 * The [Exact Penalty Method](exact_penalty_method.md) (EPM) uses a penalty term instead of augmentation, but has the same interface as ALM.
+* The [Interior Point Newton Method](interior_point_Newton.md) (IPM) rephrases the KKT system of a constrained problem into an Newton iteration being performed in every iteration.
 * [Frank-Wolfe algorithm](FrankWolfe.md), where besides the gradient of ``f`` either a closed form solution or a (maybe even automatically generated) sub problem solver for ``\operatorname*{arg\,min}_{q ∈ C} ⟨\operatorname{grad} f(p_k), \log_{p_k}q⟩`` is required, where ``p_k`` is a fixed point on the manifold (changed in every iteration).
 
-# Alphabetical list List of algorithms
+## On the tangent space
+
+* [Conjugate Residual](conjugate_residual.md) a solver for a linear system ``\mathcal A[X] + b = 0`` on a tangent space.
+* [Steihaug-Toint Truncated Conjugate-Gradient Method](truncated_conjugate_gradient_descent.md) a solver for a constrained problem defined on a tangent space.
+
+
+## Alphabetical list List of algorithms
 
 | Solver   | Function        | State   |
 |:---------|:----------------|:---------|
 | [Adaptive Regularisation with Cubics](adaptive-regularization-with-cubics.md) | [`adaptive_regularization_with_cubics`](@ref) | [`AdaptiveRegularizationState`](@ref) |
 | [Augmented Lagrangian Method](augmented_Lagrangian_method.md) | [`augmented_Lagrangian_method`](@ref) | [`AugmentedLagrangianMethodState`](@ref) |
 | [Chambolle-Pock](ChambollePock.md) | [`ChambollePock`](@ref) | [`ChambollePockState`](@ref) |
 | [Conjugate Gradient Descent](conjugate_gradient_descent.md) | [`conjugate_gradient_descent`](@ref) | [`ConjugateGradientDescentState`](@ref) |
+| [Conjugate Residual](conjugate_residual.md) | [`conjugate_residual`](@ref) | [`ConjugateResidualState`](@ref) |
 | [Convex Bundle Method](convex_bundle_method.md) | [`convex_bundle_method`](@ref) |  [`ConvexBundleMethodState`](@ref) |
 | [Cyclic Proximal Point](cyclic_proximal_point.md) | [`cyclic_proximal_point`](@ref) |  [`CyclicProximalPointState`](@ref) |
 | [Difference of Convex Algorithm](@ref solver-difference-of-convex) | [`difference_of_convex_algorithm`](@ref) | [`DifferenceOfConvexState`](@ref) |
@@ -106,6 +114,7 @@ For these you can use
 | [Exact Penalty Method](exact_penalty_method.md) | [`exact_penalty_method`](@ref) |  [`ExactPenaltyMethodState`](@ref) |
 | [Frank-Wolfe algorithm](FrankWolfe.md) | [`Frank_Wolfe_method`](@ref) | [`FrankWolfeState`](@ref) |
 | [Gradient Descent](gradient_descent.md) | [`gradient_descent`](@ref) |  [`GradientDescentState`](@ref) |
+| [Interior Point Newton](interior_point_Newton.md) | [`interior_point_Newton`](@ref) | |
 | [Levenberg-Marquardt](LevenbergMarquardt.md) | [`LevenbergMarquardt`](@ref) | [`LevenbergMarquardtState`](@ref) | ``f = \sum_i f_i`` ``\operatorname{grad} f_i`` (Jacobian)|
 | [Nelder-Mead](NelderMead.md) | [`NelderMead`](@ref) | [`NelderMeadState`](@ref) |
 | [Particle Swarm](particle_swarm.md) | [`particle_swarm`](@ref) | [`ParticleSwarmState`](@ref) |
@@ -189,4 +198,12 @@ also use the third (lowest level) and just call
 
 ```
 solve!(problem, state)
-```
+```
+
+### Closed-form subsolvers
+
+If a subsolver solution is available in closed form, `ClosedFormSubSolverState` is used to indicate that.
+
+```@docs
+Manopt.ClosedFormSubSolverState
+```

docs/src/solvers/interior_point_Newton.md

@@ -0,0 +1,48 @@
+# Interior Point Newton method
+
+```@meta
+CurrentModule = Manopt
+```
+
+```@docs
+interior_point_Newton
+interior_point_Newton!
+```
+
+## State
+
+```@docs
+InteriorPointNewtonState
+```
+
+## Subproblem functions
+
+```@docs
+CondensedKKTVectorField
+CondensedKKTVectorFieldJacobian
+KKTVectorField
+KKTVectorFieldJacobian
+KKTVectorFieldAdjointJacobian
+KKTVectorFieldNormSq
+KKTVectorFieldNormSqGradient
+```
+
+## Helpers
+
+```@docs
+InteriorPointCentralityCondition
+Manopt.calculate_σ
+```
+
+## Additional stopping criteria
+
+```@docs
+StopWhenKKTResidualLess
+```
+
+## References
+
+```@bibliography
+Pages = ["interior_point_Newton.md"]
+Canonical=false
+```

docs/styles/config/vocabularies/Manopt/accept.txt

@@ -49,6 +49,7 @@ Jasa
 Jax
 JuMP.jl
 kwargs
+Lai
 Levenberg
 Lagrangian
 Lanczos
@@ -112,5 +113,6 @@ tridiagonal
 Weinmann
 Willem
 Wolfe
+Yoshise
 Yuan
 Zhang

ext/ManoptManifoldsExt/ManoptManifoldsExt.jl

@@ -10,7 +10,9 @@ import Manopt:
     get_gradient!,
     set_manopt_parameter!,
     reflect,
-    reflect!
+    reflect!,
+    Rn,
+    Rn_default
 using LinearAlgebra: cholesky, det, diag, dot, Hermitian, qr, Symmetric, triu, I, Diagonal
 import ManifoldsBase: copy, mid_point, mid_point!
 
@@ -24,6 +26,8 @@ else
     using ..Manifolds
 end
 
+Rn(::Val{:Manifolds}, args...; kwargs...) = Euclidean(args...; kwargs...)
+
 const NONMUTATINGMANIFOLDS = Union{Circle,PositiveNumbers,Euclidean{Tuple{}}}
 include("manifold_functions.jl")
 include("ChambollePockManifolds.jl")