Implement Line Search with Negative Curvature Detection for MINRES Based on Liu et al. (2022)

src/minres.jl

@@ -18,13 +18,15 @@
 # Dominique Orban, <[email protected]>
 # Brussels, Belgium, June 2015.
 # Montréal, August 2015.
+#
+# Liu, Yang & Roosta, Fred. (2022). A Newton-MR algorithm with complexity guarantees for nonconvex smooth unconstrained optimization. 10.48550/arXiv.2208.07095. 
 
 export minres, minres!
 
 """
     (x, stats) = minres(A, b::AbstractVector{FC};
                         M=I, ldiv::Bool=false, window::Int=5,
-                        λ::T=zero(T), atol::T=√eps(T),
+                        linesearch::Bool=false, λ::T=zero(T), atol::T=√eps(T),
                         rtol::T=√eps(T), etol::T=√eps(T),
                         conlim::T=1/√eps(T), itmax::Int=0,
                         timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
@@ -68,6 +70,7 @@
 * `M`: linear operator that models a Hermitian positive-definite matrix of size `n` used for centered preconditioning;
 * `ldiv`: define whether the preconditioner uses `ldiv!` or `mul!`;
 * `window`: number of iterations used to accumulate a lower bound on the error;
+* `linesearch`: if `true`, indicate that the solution is to be used in an inexact Newton method with linesearch. If negative curvature is detected at iteration k > 0, the solution of iteration k-1 is returned. If negative curvature is detected at iteration 0, the right-hand side is returned (i.e., the negative gradient);
 * `λ`: regularization parameter;
 * `atol`: absolute stopping tolerance based on the residual norm;
 * `rtol`: relative stopping tolerance based on the residual norm;
@@ -88,6 +91,8 @@
 #### Reference
 
 * C. C. Paige and M. A. Saunders, [*Solution of Sparse Indefinite Systems of Linear Equations*](https://doi.org/10.1137/0712047), SIAM Journal on Numerical Analysis, 12(4), pp. 617--629, 1975.
+
+* Liu, Yang & Roosta, Fred. (2022). A Newton-MR algorithm with complexity guarantees for nonconvex smooth unconstrained optimization. 10.48550/arXiv.2208.07095. 
 """
 function minres end
 
@@ -108,6 +113,7 @@
 
 def_kwargs_minres = (:(; M = I                     ),
                      :(; ldiv::Bool = false        ),
+                     :(; linesearch::Bool = false  ),
                      :(; λ::T = zero(T)            ),
                      :(; atol::T = √eps(T)         ),
                      :(; rtol::T = √eps(T)         ),
@@ -124,7 +130,7 @@
 
 args_minres = (:A, :b)
 optargs_minres = (:x0,)
-kwargs_minres = (:M, :ldiv, :λ, :atol, :rtol, :etol, :conlim, :itmax, :timemax, :verbose, :history, :callback, :iostream)
+kwargs_minres = (:M, :ldiv, :linesearch ,:λ, :atol, :rtol, :etol, :conlim, :itmax, :timemax, :verbose, :history, :callback, :iostream)
 
 @eval begin
   function minres!(solver :: MinresSolver{T,FC,S}, $(def_args_minres...); $(def_kwargs_minres...)) where {T <: AbstractFloat, FC <: FloatOrComplex{T}, S <: AbstractVector{FC}}
@@ -173,18 +179,36 @@
     # β₁ M v₁ = b.
     kcopy!(n, r2, r1)  # r2 ← r1
     MisI || mulorldiv!(v, M, r1, ldiv)
+
+    rNorm =  knorm_elliptic(n, r2, r1)  # = ‖r‖
+    history && push!(rNorms, rNorm)
+
+    if rNorm == 0
+      stats.niter = 0
+      stats.solved, stats.inconsistent = true, false
+      stats.timer = start_time |> ktimer
+      stats.status = "x is a zero-residual solution"
+      history && push!(rNorms, zero(T))
+      history && push!(ArNorms, zero(T))
+      history && push!(Aconds, zero(T))
+      warm_start && kaxpy!(n, one(FC), Δx, x)
+      solver.warm_start = false
+      return solver
+    end
+
     β₁ = kdotr(m, r1, v)
     β₁ < 0 && error("Preconditioner is not positive definite")
     if β₁ == 0
       stats.niter = 0
       stats.solved, stats.inconsistent = true, false
       stats.timer = start_time |> ktimer
-      stats.status = "x is a zero-residual solution"
+      stats.status = "b is a zero-curvature directions"
       history && push!(rNorms, β₁)
       history && push!(ArNorms, zero(T))
       history && push!(Aconds, zero(T))
       warm_start && kaxpy!(n, one(FC), Δx, x)
       solver.warm_start = false
+      linesearch && kcopy!(n, x, b) # x ← b
       return solver
     end
     β₁ = sqrt(β₁)
@@ -275,6 +299,21 @@
       ArNorm = ϕbar * root  # = ‖Aᴴrₖ₋₁‖
       history && push!(ArNorms, ArNorm)
 
+      # Check if -ve curvature encountered.
+      if linesearch
+        if cs * γbar ≥ 0 # or eps(T)
+          (verbose > 0) && @printf(iostream, "nonpositive curvature detected:  cs * γbar = %e\n", cs * γbar)
+          stats.solved = true
+          stats.niter = iter
+          stats.inconsistent = false
+          stats.timer = start_time |> ktimer
+          stats.status = "nonpositive curvature"
+          iter == 1 && kcopy!(n, x, b)  # x ← b # we start  at iteration 1
+          solver.warm_start = false
+          return solver
+        end
+      end
+
       # Compute the next plane rotation.
       γ = sqrt(γbar * γbar + β * β)
       γ = max(γ, ϵM)

test/test_minres.jl

@@ -69,6 +69,27 @@
       @test(resid ≤ minres_tol * norm(A) * norm(x))
       @test(stats.solved)
 
+      # Test linesearch
+      A, b = symmetric_indefinite(FC=FC)
+      x, stats = minres(A, b, linesearch=true)
+      @test stats.status == "nonpositive curvature"
+
+      # Test Linesearch which would stop on the first call since A is negative definite
+      A, b = symmetric_indefinite(FC=FC; shift = 5)
+      x, stats = minres(A, b, linesearch=true)
+      @test stats.status == "nonpositive curvature"
+      @test stats.niter == 1 # in Minres they add 1 to the number of iterations first step
+      @test all(x .== b)
+      @test stats.solved == true
+
+      # Test when b^TAb=0 and linesearch is true
+      A, b = system_zero_quad(FC=FC)
+      x, stats = minres(A, b, linesearch=true)
+      @test stats.status == "nonpositive curvature"
+      @test all(x .== b)
+      @test stats.solved == true
+
+
       # test callback function
       solver = MinresSolver(A, b)
       storage_vec = similar(b, size(A, 1))