--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/subproblem/nonneg.rs Sun Dec 11 23:25:53 2022 +0200
@@ -0,0 +1,333 @@
+/*!
+Iterative algorithms for solving the finite-dimensional subproblem with non-negativity constraints.
+*/
+
+use nalgebra::{DVector, DMatrix};
+use numeric_literals::replace_float_literals;
+use itertools::{izip, Itertools};
+use colored::Colorize;
+
+use alg_tools::iter::Mappable;
+use alg_tools::error::NumericalError;
+use alg_tools::iterate::{
+ AlgIteratorFactory,
+ AlgIteratorState,
+ Step,
+};
+use alg_tools::linops::GEMV;
+use alg_tools::nalgebra_support::ToNalgebraRealField;
+
+use crate::types::*;
+use super::{
+ InnerMethod,
+ InnerSettings
+};
+
+/// Compute the proximal operator of $x \mapsto x + \delta\_{[0, \infty)}$, i.e.,
+/// the non-negativity contrained soft-thresholding operator.
+#[inline]
+#[replace_float_literals(F::cast_from(literal))]
+fn nonneg_soft_thresholding<F : Float>(v : F, λ : F) -> F {
+ (v - λ).max(0.0)
+}
+
+/// Returns the ∞-norm minimal subdifferential of $x ↦ x^⊤Ax - g^⊤ x + λ\vec 1^⊤ x δ_{≥ 0}(x)$
+/// at $x$.
+///
+/// `v` will be modified and cannot be trusted to contain useful values afterwards.
+#[replace_float_literals(F::cast_from(literal).to_nalgebra_mixed())]
+fn min_subdifferential<F : Float + ToNalgebraRealField>(
+ v : &mut DVector<F::MixedType>,
+ mA : &DMatrix<F::MixedType>,
+ x : &DVector<F::MixedType>,
+ g : &DVector<F::MixedType>,
+ λ : F::MixedType
+) -> F {
+ v.copy_from(g);
+ mA.gemv(v, 1.0, x, -1.0); // v = Ax - g
+ let mut val = 0.0;
+ for (&v_i, &x_i) in izip!(v.iter(), x.iter()) {
+ // The subdifferential of the objective is $Ax - g + λ + ∂ δ_{≥ 0}(x)$.
+ let d = v_i + λ;
+ if x_i > 0.0 || d < 0.0 {
+ val = val.max(d.abs());
+ }
+ }
+ F::from_nalgebra_mixed(val)
+}
+
+/// Forward-backward splitting implementation of [`quadratic_nonneg`].
+/// For detailed documentation of the inputs and outputs, refer to there.
+///
+/// The `λ` component of the model is handled in the proximal step instead of the gradient step
+/// for potential performance improvements.
+#[replace_float_literals(F::cast_from(literal).to_nalgebra_mixed())]
+pub fn quadratic_nonneg_fb<F, I>(
+ mA : &DMatrix<F::MixedType>,
+ g : &DVector<F::MixedType>,
+ //c_ : F,
+ λ_ : F,
+ x : &mut DVector<F::MixedType>,
+ τ_ : F,
+ iterator : I
+) -> usize
+where F : Float + ToNalgebraRealField,
+ I : AlgIteratorFactory<F>
+{
+ let mut xprev = x.clone();
+ //let c = c_.to_nalgebra_mixed();
+ let λ = λ_.to_nalgebra_mixed();
+ let τ = τ_.to_nalgebra_mixed();
+ let τλ = τ * λ;
+ let mut v = DVector::zeros(x.len());
+ let mut iters = 0;
+
+ iterator.iterate(|state| {
+ // Replace `x` with $x - τ[Ax-g]= [x + τg]- τAx$
+ v.copy_from(g); // v = g
+ v.axpy(1.0, x, τ); // v = x + τ*g
+ v.sygemv(-τ, mA, x, 1.0); // v = [x + τg]- τAx
+ let backup = state.if_verbose(|| {
+ xprev.copy_from(x)
+ });
+ // Calculate the proximal map
+ x.iter_mut().zip(v.iter()).for_each(|(x_i, &v_i)| {
+ *x_i = nonneg_soft_thresholding(v_i, τλ);
+ });
+
+ iters +=1;
+
+ backup.map(|_| {
+ min_subdifferential(&mut v, mA, x, g, λ)
+ })
+ });
+
+ iters
+}
+
+/// Semismooth Newton implementation of [`quadratic_nonneg`].
+///
+/// For detailed documentation of the inputs, refer to there.
+/// This function returns the number of iterations taken if there was no inversion failure,
+///
+/// ## Method derivation
+///
+/// **The below may look like garbage. Sorry, but rustdoc is obsolete rubbish
+/// that doesn't directly support by-now standard-in-markdown LaTeX math. Instead it
+/// forces one into unreliable KaTeX autorender postprocessing andescape hell and that
+/// it doesn't even process correctly.**
+///
+/// <p>
+/// For the objective
+/// $$
+/// J(x) = \frac{1}{2} x^⊤Ax - g^⊤ x + λ{\vec 1}^⊤ x + c + δ_{≥ 0}(x),
+/// $$
+/// we have the optimality condition
+/// $$
+/// x - \mathop{\mathrm{prox}}_{τλ{\vec 1}^⊤ + δ_{≥ 0}}(x - τ[Ax-g^⊤]) = 0,
+/// $$
+/// which we write as
+/// $$
+/// x - [G ∘ F](x)=0
+/// $$
+/// for
+/// $$
+/// G(x) = \mathop{\mathrm{prox}}_{λ{\vec 1}^⊤ + δ_{≥ 0}}
+/// \quad\text{and}\quad
+/// F(x) = x - τ Ax + τ g^⊤
+/// $$
+/// We can use Newton derivative chain rule to compute
+/// $D_N[G ∘ F](x) = D_N G(F(x)) D_N F(x)$, where
+/// $D_N F(x) = \mathop{\mathrm{Id}} - τ A$,
+/// and $[D_N G(F(x))]_i = 1$ for inactive coordinates and $=0$ for active coordinates.
+/// </p>
+///
+/// <p>
+/// The method itself involves solving $D_N[Id - G ∘ F](x^k) s^k = - [Id - G ∘ F](x^k)$ and
+/// updating $x^{k+1} = x^k + s^k$. Consequently
+/// $$
+/// s^k - D_N G(F(x^k)) [s^k - τ As^k] = - x^k + [G ∘ F](x^k)
+/// $$
+/// For $𝒜$ the set of active coordinates and $ℐ$ the set of inactive coordinates, this
+/// expands as
+/// $$
+/// [τ A_{ℐ × ℐ}]s^k_ℐ = - x^k_ℐ + [G ∘ F](x^k)_ℐ - [τ A_{ℐ × 𝒜}]s^k_𝒜
+/// $$
+/// and
+/// $$
+/// s^k_𝒜 = - x^k_𝒜 + [G ∘ F](x^k)_𝒜.
+/// $$
+/// Thus on $𝒜$ the update $[x^k + s^k]_𝒜 = [G ∘ F](x^k)_𝒜$ is just the forward-backward update.
+/// </p>
+///
+/// <p>
+/// We need to detect stopping by a subdifferential and return $x$ satisfying $x ≥ 0$,
+/// which is in general not true for the SSN. We therefore use that $[G ∘ F](x^k)$ is a valid
+/// forward-backward step.
+/// </p>
+#[replace_float_literals(F::cast_from(literal).to_nalgebra_mixed())]
+pub fn quadratic_nonneg_ssn<F, I>(
+ mA : &DMatrix<F::MixedType>,
+ g : &DVector<F::MixedType>,
+ //c_ : F,
+ λ_ : F,
+ x : &mut DVector<F::MixedType>,
+ τ_ : F,
+ iterator : I
+) -> Result<usize, NumericalError>
+where F : Float + ToNalgebraRealField,
+ I : AlgIteratorFactory<F>
+{
+ let n = x.len();
+ let mut xprev = x.clone();
+ let mut v = DVector::zeros(n);
+ //let c = c_.to_nalgebra_mixed();
+ let λ = λ_.to_nalgebra_mixed();
+ let τ = τ_.to_nalgebra_mixed();
+ let τλ = τ * λ;
+ let mut inact : Vec<bool> = Vec::from_iter(std::iter::repeat(false).take(n));
+ let mut s = DVector::zeros(0);
+ let mut decomp = nalgebra::linalg::LU::new(DMatrix::zeros(0, 0));
+ let mut iters = 0;
+
+ let res = iterator.iterate_fallible(|state| {
+ // 1. Perform delayed SSN-update based on previously computed step on active
+ // coordinates. The step is delayed to the beginning of the loop because
+ // the SSN step may violate constraints, so we arrange `x` to contain at the
+ // end of the loop the valid FB step that forms part of the SSN step
+ let mut si = s.iter();
+ for (&ast, x_i, xprev_i) in izip!(inact.iter(), x.iter_mut(), xprev.iter_mut()) {
+ if ast {
+ *x_i = *xprev_i + *si.next().unwrap()
+ }
+ *xprev_i = *x_i;
+ }
+
+ //xprev.copy_from(x);
+
+ // 2. Calculate FB step.
+ // 2.1. Replace `x` with $x⁻ - τ[Ax⁻-g]= [x⁻ + τg]- τAx⁻$
+ x.axpy(τ, g, 1.0); // x = x⁻ + τ*g
+ x.sygemv(-τ, mA, &xprev, 1.0); // x = [x⁻ + τg]- τAx⁻
+ // 2.2. Calculate prox and set of active coordinates at the same time
+ let mut act_changed = false;
+ let mut n_inact = 0;
+ for (x_i, ast) in izip!(x.iter_mut(), inact.iter_mut()) {
+ if *x_i > τλ {
+ *x_i -= τλ;
+ if !*ast {
+ act_changed = true;
+ *ast = true;
+ }
+ n_inact += 1;
+ } else {
+ *x_i = 0.0;
+ if *ast {
+ act_changed = true;
+ *ast = false;
+ }
+ }
+ }
+
+ // *** x now contains forward-backward step ***
+
+ // 3. Solve SSN step `s`.
+ // 3.1 Construct [τ A_{ℐ × ℐ}] if the set of inactive coordinates has changed.
+ if act_changed {
+ let decomp_iter = inact.iter().cartesian_product(inact.iter()).zip(mA.iter());
+ let decomp_constr = decomp_iter.filter_map(|((&i_inact, &j_inact), &mAij)| {
+ //(i_inact && j_inact).then_some(mAij * τ)
+ (i_inact && j_inact).then_some(mAij) // 🔺 below matches removal of τ
+ });
+ let mat = DMatrix::from_iterator(n_inact, n_inact, decomp_constr);
+ decomp = nalgebra::linalg::LU::new(mat);
+ }
+
+ // 3.2 Solve `s` = $s_ℐ^k$ from
+ // $[τ A_{ℐ × ℐ}]s^k_ℐ = - x^k_ℐ + [G ∘ F](x^k)_ℐ - [τ A_{ℐ × 𝒜}]s^k_𝒜$.
+ // With current variable setup we have $[G ∘ F](x^k) = $`x` and $x^k = x⁻$ = `xprev`,
+ // so the system to solve is $[τ A_{ℐ × ℐ}]s^k_ℐ = (x-x⁻)_ℐ - [τ A_{ℐ × 𝒜}](x-x⁻)_𝒜$
+ // The matrix $[τ A_{ℐ × ℐ}]$ we have already LU-decomposed above into `decomp`.
+ s = if n_inact > 0 {
+ // 3.2.1 Construct `rhs` = $(x-x⁻)_ℐ - [τ A_{ℐ × 𝒜}](x-x⁻)_𝒜$
+ let inactfilt = inact.iter().copied();
+ let rhs_iter = izip!(x.iter(), xprev.iter(), mA.row_iter()).filter_zip(inactfilt);
+ let rhs_constr = rhs_iter.map(|(&x_i, &xprev_i, mAi)| {
+ // Calculate row i of [τ A_{ℐ × 𝒜}]s^k_𝒜 = [τ A_{ℐ × 𝒜}](x-xprev)_𝒜
+ let actfilt = inact.iter().copied().map(std::ops::Not::not);
+ let actit = izip!(x.iter(), xprev.iter(), mAi.iter()).filter_zip(actfilt);
+ let actpart = actit.map(|(&x_j, &xprev_j, &mAij)| {
+ mAij * (x_j - xprev_j)
+ }).sum();
+ // Subtract it from [x-prev]_i
+ //x_i - xprev_i - τ * actpart
+ (x_i - xprev_i) / τ - actpart // 🔺 change matches removal of τ above
+ });
+ let mut rhs = DVector::from_iterator(n_inact, rhs_constr);
+ assert_eq!(rhs.len(), n_inact);
+ // Solve the system
+ if !decomp.solve_mut(&mut rhs) {
+ return Step::Failure(NumericalError(
+ "Failed to solve linear system for subproblem SSN."
+ ))
+ }
+ rhs
+ } else {
+ DVector::zeros(0)
+ };
+
+ iters += 1;
+
+ // 4. Report solution quality
+ state.if_verbose(|| {
+ // Calculate subdifferential at the FB step `x` that hasn't yet had `s` yet added.
+ min_subdifferential(&mut v, mA, x, g, λ)
+ })
+ });
+
+ res.map(|_| iters)
+}
+
+/// This function applies an iterative method for the solution of the quadratic non-negativity
+/// constrained problem
+/// <div>$$
+/// \min_{x ∈ ℝ^n} \frac{1}{2} x^⊤Ax - g^⊤ x + λ{\vec 1}^⊤ x + c + δ_{≥ 0}(x).
+/// $$</div>
+/// Semismooth Newton or forward-backward are supported based on the setting in `method`.
+/// The parameter `mA` is matrix $A$, and `g` and `λ` are as in the mathematical formulation.
+/// The constant $c$ does not need to be provided. The step length parameter is `τ` while
+/// `x` contains the initial iterate and on return the final one. The `iterator` controls
+/// stopping. The “verbose” value output by all methods is the $ℓ\_∞$ distance of some
+/// subdifferential of the objective to zero.
+///
+/// Interior point methods could offer a further alternative, for example, the one in:
+///
+/// * Valkonen T. - _A method for weighted projections to the positive definite
+/// cone_, <https://doi.org/10.1080/02331934.2014.929680>.
+///
+/// This function returns the number of iterations taken.
+pub fn quadratic_nonneg<F, I>(
+ method : InnerMethod,
+ mA : &DMatrix<F::MixedType>,
+ g : &DVector<F::MixedType>,
+ //c_ : F,
+ λ : F,
+ x : &mut DVector<F::MixedType>,
+ τ : F,
+ iterator : I
+) -> usize
+where F : Float + ToNalgebraRealField,
+ I : AlgIteratorFactory<F>
+{
+
+ match method {
+ InnerMethod::FB =>
+ quadratic_nonneg_fb(mA, g, λ, x, τ, iterator),
+ InnerMethod::SSN =>
+ quadratic_nonneg_ssn(mA, g, λ, x, τ, iterator).unwrap_or_else(|e| {
+ println!("{}", format!("{e}. Using FB fallback.").red());
+ let ins = InnerSettings::<F>::default();
+ quadratic_nonneg_fb(mA, g, λ, x, τ, ins.iterator_options)
+ })
+ }
+}