MISTTDFPM — Julia Implementation

Modified Inertial Spectral Three-Term Derivative-Free Projection Method for solving nonlinear equations with convex constraints.

Problem: Find x in C such that F(x) = 0, where C is a closed convex subset of R^n and F: R^n -> R^n is continuous and monotone.

The method combines inertial extrapolation, a spectral three-term conjugate-gradient-type search direction (adapted from Amini-Faramarzi), a unified Armijo-type line search framework (7 variants from Ibrahim, Alshahrani & Al-Homidan, JOTA 2026), and feasible inexact projection via the Frank-Wolfe (conditional gradient) method.

Prerequisites

Julia 1.12+ (developed on 1.12.4)
All dependencies are declared in Project.toml / Manifest.toml

Install dependencies

cd jcode/
julia --project=. -e 'using Pkg; Pkg.instantiate()'

This installs: LazySets, DataFrames, CSV, ProgressMeter, Plots, BenchmarkProfiles, Images, TestImages, Wavelets, XLSX, Colors, BenchmarkTools.

Quick Start

# From jcode/ directory
include("src/includes.jl")

# Define problem
F(x) = exp.(x) .- 1.0
C = Ball2(zeros(5), 1.0)    # unit ball in R^5

# Solve with default (tuned) parameters
solver = MISTTDFPM(F, C)
x, k, converged = solve(solver; maxiter=1000, verbose=true)

# Solve with original paper parameters
solver = MISTTDFPM(F, C; preset=:paper)
x, k, converged = solve(solver)

# Reference algorithms (same solve() interface)
x, k, converged = solve(IHZIPM(F, C))
x, k, converged = solve(STTDFPM(F, C))
x, k, converged = solve(ISTTDFPM(F, C))

# Use built-in test problems
for p in safe_all_problems(1000)
    s = MISTTDFPM(p.F, p.C; x0=p.x0)
    x, k, conv = solve(s)
    println("$(p.name): k=$k, converged=$conv")
end

Reproducing Paper Results

All paper experiments can be reproduced from the jcode/ directory. Scripts use modular ARGS dispatch: pass subset names to run specific parts, or no args to run everything.

Step 1: Verify installation

julia --project=. scripts/01_smoke_test.jl

Runs all 4 algorithms on a few problems. Should complete in under a minute with all tests passing.

Step 2: Experiment 1 — Large-scale comparison (Paper Section 5)

15 problems x 10 starting points x 9 dimensions (1,350 instances). Compares MISTTDFPM vs IHZIPM, STTDFPM, ISTTDFPM (exact & inexact).

# Full run (all 9 dimensions: 1k, 5k, 10k, 20k, 30k, 50k, 80k, 100k, 120k)
julia --project=. scripts/50_experiment1.jl

# Or run by tier
julia --project=. scripts/50_experiment1.jl small        # dims [1k, 5k, 10k]
julia --project=. scripts/50_experiment1.jl mid          # dims [20k, 30k, 50k]
julia --project=. scripts/50_experiment1.jl large        # dims [80k, 100k, 120k]

# Generate LaTeX tables and summary (after benchmarks complete)
julia --project=. scripts/50_experiment1.jl latex summary

Output: results/experiment1/

experiment1_raw.csv — per-instance results
experiment1_matrix.csv — pivot table for performance profiles
experiment1_tables.tex — LaTeX tables for the paper
exp1_*.pdf — performance profile figures

Step 3: Compressed sensing image recovery (Paper Section 6)

DWT-based scrambled Hadamard compressed sensing with image restoration.

# Full run
julia --project=. scripts/65_compressed_sensing.jl

# Or run by part
julia --project=. scripts/65_compressed_sensing.jl partA      # scaling (cameraman 64²/128²/256²)
julia --project=. scripts/65_compressed_sensing.jl partB      # generality (4 images, ratio=0.3 & 0.5)
julia --project=. scripts/65_compressed_sensing.jl partC      # visual comparison PNGs
julia --project=. scripts/65_compressed_sensing.jl latex summary

Output: results/compressed_sensing/

partA_scaling_raw.csv, partB_generality_raw.csv — numerical results
compressed_sensing_tables.tex — LaTeX tables
visual_*.png — recovered image comparisons

Optional: Parameter search & sensitivity analysis

These scripts were used during development to find optimal parameters. The winning parameters are already baked into the :default preset, so re-running them is not required for reproduction.

# OAT sensitivity analysis
julia --project=. scripts/10_sensitivity_analysis.jl

# Scale-aware LHS parameter search (3-phase)
julia --project=. scripts/20_parameter_search.jl

# CS-specific parameter search
julia --project=. scripts/25_cs_parameter_search.jl

Algorithms

MISTTDFPM (our method — Algorithm 1)

Step	Operation
0	Initialize x_0 in C
1	Stop if ‖F(x_k)‖ < ε
2	Inertial: w_k = (1 - λ_k) x_k + λ_k w_{k-1}
3	Direction: d_k = -θ_k F(w_k) + β_k d_{k-1} + ξ_k p_k
4	Line search: find α_k via Armijo condition
5	Projection: x_{k+1} = P_C[w_k - ζ_k F(z_k), x_k, θ_k]

Reference algorithms

Algorithm	Description	Projection
`IHZIPM`	Improved HZ two-term CG direction	Inexact (Frank-Wolfe)
`STTDFPM`	Spectral three-term, no inertia	Exact (default) or inexact
`ISTTDFPM`	Inertial STTDFPM	Exact (default) or inexact

All four share the same solve(solver; maxiter, verbose, timeout, ...) interface.

Presets

Named parameter sets for MISTTDFPM(F, C; preset=...):

Preset	Description
`:default`	Best tuned parameters from LHS search (used in paper)
`:paper`	Original conservative paper parameters
`:no_inertia`	Default with inertia disabled (λ=0)
`:tuned`	Alias for `:default`
`:v2`	Previous defaults (before scale-aware search)
`:scale_small`	Best for n ∈ {1k, 5k, 10k}
`:scale_large`	Best for n ∈ {75k, 100k, 120k}
`:cs`	CS / image restoration — clipped LS
`:cs_constant`	CS / image restoration — constant LS

User kwargs always override the preset: MISTTDFPM(F, C; preset=:paper, λ=0.5).

Line Search Types

Seven Armijo-type variants from the unified framework (JOTA 2026), selected via ls_type:

`ls_type`	γ_k formula
`:constant`	1
`:residual`	‖F(z_k)‖
`:normalized`	‖F(z_k)‖ / (1 + ‖F(z_k)‖)
`:blend`	τ + (1-τ)‖F(z_k)‖
`:capped`	min(1, ‖F(z_k)‖)
`:clipped`	clamp(‖F(z_k)‖, η₁, η₂) (default)
`:adaptive`	clipped + stabilized initial step

Additionally, EMA history smoothing (β_smooth < 1) can be applied to any variant.

Key Parameters

Parameter	Default	Description
`ε`	`1e-6`	Stopping tolerance
`η`	`1.419`	Initial line search step
`ρ`	`0.460`	Line search contraction ∈ (0,1)
`σ`	`0.1`	Armijo parameter
`ls_type`	`:clipped`	Line search variant
`η₁_ls`	`0.085`	Clipping lower bound
`η₂_ls`	`0.881`	Clipping upper bound
`β_smooth`	`0.712`	EMA smoothing (1.0 = off)
`λ`	`0.200`	Inertial parameter bound
`α_min`	`0.740`	Spectral parameter lower bound
`α_max`	`10.0`	Spectral parameter upper bound
`κ`	`6.967`	Parameter for μ_k
`ξ̄`	`0.393`	Adaptive parameter bound ∈ (0,1)
`p_choice`	`:F_w`	Direction p_k: `:F_w`, `:F_w_prev`, `:d_prev`, `:y`
`γ_proj`	`1.782`	Projection step scaling
`θ̄`	`0.04`	Inexact projection bound

See CLAUDE.md for the full parameter reference.

Test Problems

15 problems used in paper experiments, available via safe_all_problems(n):

Problem	Constraint
ExponentialI	Ball, Box
ExponentialIII	Box
SmoothSine	Box
NonsmoothSine	Triangle
ModifiedNonsmoothSine (2 variants)	Triangle
PolynomialI	Box
Logarithmic	Box
NonsmoothLogarithmic	Box
ModifiedTrigI	Box
ARWHEAD	Ball
ENGVAL1	Ball
PENALTY1	Ball
DIXON3DQ	Ball

Compressed sensing problems: dwt_cs_problem(...) for DWT-based image recovery.

safe_all_problems(n)           # 15 problems at dimension n
all_problems(n)                # all 16 (includes ModifiedTridiagonal-Triangle)
ball_problems(n)               # ball-constrained subset
general_constraint_problems(n) # BallBox + BoxLinear constraints (no closed-form projection)

Project Structure

jcode/
├── Project.toml / Manifest.toml   # Julia dependencies (locked)
├── CLAUDE.md                      # Full development documentation
├── README.md                      # This file
├── src/
│   ├── includes.jl                # Entry point (load order)
│   ├── deps.jl                    # Package dependencies
│   ├── conditionalGradient.jl     # Inexact projection (Frank-Wolfe)
│   ├── direction.jl               # Inertial step + spectral direction
│   ├── linesearch.jl              # Armijo-type line search (7 variants)
│   ├── MISTTDFPM.jl               # Our algorithm (struct, iterator, solve)
│   ├── exact_projection.jl        # Exact projection (Ball2, Hyperrectangle, fallback)
│   ├── IHZIPM.jl                  # Reference: improved HZ CG + inexact projection
│   ├── ISTTDFPM.jl                # Reference: STTDFPM + ISTTDFPM
│   ├── problems.jl                # Test problems + constraint sets
│   └── benchmark.jl               # Multi-solver benchmarking utilities
├── scripts/
│   ├── 01_smoke_test.jl           # Quick verification of all algorithms
│   ├── 10_sensitivity_analysis.jl # OAT parameter sensitivity sweeps
│   ├── 15_cs_sensitivity.jl       # CS-specific sensitivity analysis
│   ├── 20_parameter_search.jl     # Scale-aware LHS parameter search
│   ├── 25_cs_parameter_search.jl  # CS-specific parameter search
│   ├── 50_experiment1.jl          # Experiment 1: large-scale comparison (paper)
│   ├── 55_experiment2_general.jl  # Experiment 2: general constraints
│   ├── 60_traffic_application.jl  # Traffic assignment application
│   ├── 65_compressed_sensing.jl   # Compressed sensing image recovery (paper)
│   └── 70_convergence_history.jl  # Convergence history plots
├── results/                       # Script output (CSVs, PDFs, PNGs, LaTeX)
└── oldcode/                       # Retired scripts and reference code

Scripts Reference

All scripts support modular ARGS dispatch. Run with no args for full execution, or pass subset names for partial runs.

Script	Purpose	ARGS
`01_smoke_test.jl`	Verify all algorithms work	—
`10_sensitivity_analysis.jl`	OAT parameter sweeps	`oat`, `joint`, `summary`, `plots`
`15_cs_sensitivity.jl`	CS sensitivity analysis	—
`20_parameter_search.jl`	3-phase LHS search	`small`, `mid`, `large`, `phase2`, `report`
`25_cs_parameter_search.jl`	CS parameter search	`search`, `report`
`50_experiment1.jl`	Paper Experiment 1	`small`, `mid`, `large`, `latex`, `summary`
`55_experiment2_general.jl`	General constraints	`small`, `large`, `latex`, `summary`
`60_traffic_application.jl`	Traffic assignment	—
`65_compressed_sensing.jl`	Paper CS experiment	`partA`, `partB`, `partC`, `latex`, `summary`
`70_convergence_history.jl`	Convergence plots	—

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MISTTDFPM — Julia Implementation

Prerequisites

Install dependencies

Quick Start

Reproducing Paper Results

Step 1: Verify installation

Step 2: Experiment 1 — Large-scale comparison (Paper Section 5)

Step 3: Compressed sensing image recovery (Paper Section 6)

Optional: Parameter search & sensitivity analysis

Algorithms

MISTTDFPM (our method — Algorithm 1)

Reference algorithms

Presets

Line Search Types

Key Parameters

Test Problems

Project Structure

Scripts Reference

About

Releases

Packages

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Name		Name	Last commit message	Last commit date
scripts		scripts
src		src
.gitignore		.gitignore
CLAUDE.md		CLAUDE.md
Project.toml		Project.toml
README.md		README.md
run_experiment1.bat		run_experiment1.bat

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MISTTDFPM — Julia Implementation

Prerequisites

Install dependencies

Quick Start

Reproducing Paper Results

Step 1: Verify installation

Step 2: Experiment 1 — Large-scale comparison (Paper Section 5)

Step 3: Compressed sensing image recovery (Paper Section 6)

Optional: Parameter search & sensitivity analysis

Algorithms

MISTTDFPM (our method — Algorithm 1)

Reference algorithms

Presets

Line Search Types

Key Parameters

Test Problems

Project Structure

Scripts Reference

About

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