nonmonotone PANOC

Author: aldma

src/algorithms/panoc.jl

@@ -32,6 +32,7 @@ See also: [`PANOC`](@ref).
 - `minimum_gamma=1e-7`: lower bound to `gamma` in case `adaptive == true`.
 - `max_backtracks=20`: maximum number of line-search backtracks.
 - `directions=LBFGS(5)`: strategy to use to compute line-search directions.
+- `monotonicity=1`: parameter controlling the averaging scheme for nonmonotone linesearch; monotonicity ∈ (0,1], monotone scheme by default.
 
 # References
 1. Stella, Themelis, Sopasakis, Patrinos, "A simple and efficient algorithm for nonlinear model predictive control", 56th IEEE Conference on Decision and Control (2017).
@@ -49,6 +50,7 @@ Base.@kwdef struct PANOCIteration{R,Tx,Tf,TA,Tg,TLf,Tgamma,D}
     minimum_gamma::R = real(eltype(x0))(1e-7)
     max_backtracks::Int = 20
     directions::D = LBFGS(5)
+    monotonicity::R = real(eltype(x0))(1)
 end
 
 Base.IteratorSize(::Type{<:PANOCIteration}) = Base.IsInfinite()
@@ -65,6 +67,7 @@ Base.@kwdef mutable struct PANOCState{R,Tx,TAx,TH}
     g_z::R            # value of nonsmooth term (at z)
     res::Tx           # fixed-point residual at iterate (= x - z)
     H::TH             # variable metric
+    merit::R = zero(gamma)
     tau::R = zero(gamma)
     x_prev::Tx = similar(x)
     res_prev::Tx = similar(x)
@@ -108,6 +111,8 @@ function Base.iterate(iter::PANOCIteration{R}) where {R}
         res = x - z,
         H = initialize(iter.directions, x),
     )
+    # initialize merit
+    state.merit = f_model(iter, state) + state.g_z
     return state, state
 end
 
@@ -138,7 +143,7 @@ reset_direction_state!(iter::PANOCIteration, state::PANOCState) =
 function Base.iterate(iter::PANOCIteration{R,Tx,Tf}, state::PANOCState) where {R,Tx,Tf}
     f_Az, a, b, c = R(Inf), R(Inf), R(Inf), R(Inf)
 
-    f_Az_upp = if iter.adaptive == true
+    if iter.adaptive == true
         gamma_prev = state.gamma
         state.gamma, state.g_z, f_Az, f_Az_upp = backtrack_stepsize!(
             state.gamma,
@@ -160,13 +165,8 @@ function Base.iterate(iter::PANOCIteration{R,Tx,Tf}, state::PANOCState) where {R
         if state.gamma != gamma_prev
             reset_direction_state!(iter, state)
         end
-        f_Az_upp
-    else
-        f_model(iter, state)
     end
 
-    # compute FBE
-    FBE_x = f_Az_upp + state.g_z
 
     # compute direction
     set_next_direction!(iter, state)
@@ -192,6 +192,8 @@ function Base.iterate(iter::PANOCIteration{R,Tx,Tf}, state::PANOCState) where {R
     copyto!(state.z_curr, state.z)
     state.f_Ax = state.f_Ax_d
 
+    # retrieve merit and set threshold
+    FBE_x = state.merit
     sigma = iter.beta * (0.5 / state.gamma) * (1 - iter.alpha)
     tol = 10 * eps(R) * (1 + abs(FBE_x))
     threshold = FBE_x - sigma * norm(state.res)^2 + tol
@@ -250,7 +252,8 @@ function Base.iterate(iter::PANOCIteration{R,Tx,Tf}, state::PANOCState) where {R
     end
 
     update_direction_state!(iter, state)
-
+    # update merit with averaging rule
+    state.merit = (1 - iter.monotonicity) * state.merit + iter.monotonicity * FBE_x_new
     return state, state
 end
 

test/problems/test_lasso_small.jl

@@ -191,6 +191,17 @@ using ProximalAlgorithms:
 
     end
 
+    @testset "PANOC (fixed, nonmonotone)" begin
+        x0 = zeros(T, n)
+        x0_backup = copy(x0)
+        solver = ProximalAlgorithms.PANOC(tol = TOL, monotonicity=R(0.3))
+        x, it = @inferred solver(x0 = x0, f = f_autodiff, A = A, g = g, Lf = Lf)
+        @test eltype(x) == T
+        @test norm(x - x_star, Inf) <= TOL
+        @test it < 20
+        @test x0 == x0_backup
+    end
+
     @testset "PANOC (adaptive step)" begin
         x0 = zeros(T, n)
         x0_backup = copy(x0)
@@ -202,6 +213,17 @@ using ProximalAlgorithms:
         @test x0 == x0_backup
     end
 
+    @testset "PANOC (adaptive, nonmonotone)" begin
+        x0 = zeros(T, n)
+        x0_backup = copy(x0)
+        solver = ProximalAlgorithms.PANOC(adaptive = true, tol = TOL, monotonicity=R(0.3))
+        x, it = @inferred solver(x0 = x0, f = f_autodiff, A = A, g = g)
+        @test eltype(x) == T
+        @test norm(x - x_star, Inf) <= TOL
+        @test it < 35
+        @test x0 == x0_backup
+    end
+
     @testset "PANOCplus (fixed step)" begin
         x0 = zeros(T, n)
         x0_backup = copy(x0)

test/problems/test_lasso_small_strongly_convex.jl

@@ -188,6 +188,15 @@ using ProximalAlgorithms
         @test x0 == x0_backup
     end
 
+    @testset "PANOC (nonmonotone)" begin
+        solver = ProximalAlgorithms.PANOC(tol=TOL, monotonicity=T(0.2))
+        y, it = solver(x0=x0, f=fA_autodiff, g=g, Lf=Lf)
+        @test eltype(y) == T
+        @test norm(y - x_star, Inf) <= TOL
+        @test it < 45
+        @test x0 == x0_backup
+    end
+
     @testset "PANOCplus" begin
         solver = ProximalAlgorithms.PANOCplus(tol=TOL)
         y, it = solver(x0=x0, f=fA_autodiff, g=g, Lf=Lf)

test/problems/test_nonconvex_qp.jl

@@ -35,6 +35,16 @@ using Test
         @test x0 == x0_backup
     end
 
+    @testset "PANOC (nonmonotone)" begin
+        x0 = zeros(T, n)
+        x0_backup = copy(x0)
+        solver = ProximalAlgorithms.PANOC(tol = TOL, monotonicity=T(0.1))
+        x, it = solver(x0 = x0, f = f, g = g)
+        z = min.(upp, max.(low, x .- gamma .* (Q * x + q)))
+        @test norm(x - z, Inf) / gamma <= TOL
+        @test x0 == x0_backup
+    end
+
     @testset "PANOCplus" begin
         x0 = zeros(T, n)
         x0_backup = copy(x0)
@@ -112,6 +122,16 @@ end
             @test x0 == x0_backup
         end
 
+        @testset "PANOC (nonmonotone)" begin
+            x0 = zeros(T, n)
+            x0_backup = copy(x0)
+            solver = ProximalAlgorithms.PANOC(tol = TOL, monotonicity=T(0.1))
+            x, it = solver(x0 = x0, f = f, g = g)
+            z = min.(upp, max.(low, x .- gamma .* (Q * x + q)))
+            @test norm(x - z, Inf) / gamma <= TOL
+            @test x0 == x0_backup
+        end
+
         @testset "PANOCplus" begin
             x0 = zeros(T, n)
             x0_backup = copy(x0)

test/problems/test_sparse_logistic_small.jl

@@ -120,6 +120,17 @@ using LinearAlgebra
         @test x0 == x0_backup
     end
 
+    @testset "PANOC (adaptive, nonmonotone)" begin
+        x0 = zeros(T, n)
+        x0_backup = copy(x0)
+        solver = ProximalAlgorithms.PANOC(adaptive = true, tol = TOL, monotonicity=R(0.5))
+        x, it = solver(x0 = x0, f = f_autodiff, A = A, g = g)
+        @test eltype(x) == T
+        @test norm(x - x_star, Inf) <= 1e-4
+        @test it < 50
+        @test x0 == x0_backup
+    end
+
     @testset "PANOCplus (adaptive step)" begin
         x0 = zeros(T, n)
         x0_backup = copy(x0)