Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficiency and Convergence Insights in Large-Scale Optimization Using the Improved Inexact-Newton-Smart Algorithm and Interior-Point Framework

Neda Bagheri Renani, Maryam Jaefarzadeh, Daniel Sevcovic

TL;DR

The paper addresses solving large-scale nonlinear optimization problems by comparing the Improved Inexact-Newton-Smart (INS) algorithm against a primal–dual Interior-Point Method (IPM). It analyzes short-step and long-step IPMs, inexact Newton directions, and INS enhancements including Hessian regularization, quasi-Newton updates, and preconditioning, across extensive synthetic benchmarks. On 100 benchmarks with up to $n=10^6$ variables, IPM achieved an average of $68.11$ iterations and $0.13$ s, versus INS at $100.00$ iterations and $0.23$ s, with IPM reaching the primary termination criterion in $100 ext{%}$ of runs and INS in $32 ext{%}$; INS under tuned parameters narrowed the gap, highlighting adaptive potential. The results advocate a robust IPM baseline and a tunable INS as a complementary option, with future work focusing on hybrid INS–IPM strategies that integrate Newton-type flexibility with barrier-based robustness.$

Abstract

We present a head-to-head evaluation of the Improved Inexact--Newton--Smart (INS) algorithm against a primal--dual interior-point framework for large-scale nonlinear optimization. On extensive synthetic benchmarks, the interior-point method converges with roughly one third fewer iterations and about one half the computation time relative to INS, while attaining marginally higher accuracy and meeting all primary stopping conditions. By contrast, INS succeeds in fewer cases under default settings but benefits markedly from moderate regularization and step-length control; in tuned regimes its iteration count and runtime decrease substantially, narrowing yet not closing the gap. A sensitivity study indicates that interior-point performance remains stable across parameter changes, whereas INS is more affected by step length and regularization choice. Collectively, the evidence positions the interior-point method as a reliable baseline and INS as a configurable alternative when problem structure favors adaptive regularization.

Efficiency and Convergence Insights in Large-Scale Optimization Using the Improved Inexact-Newton-Smart Algorithm and Interior-Point Framework

TL;DR

The paper addresses solving large-scale nonlinear optimization problems by comparing the Improved Inexact-Newton-Smart (INS) algorithm against a primal–dual Interior-Point Method (IPM). It analyzes short-step and long-step IPMs, inexact Newton directions, and INS enhancements including Hessian regularization, quasi-Newton updates, and preconditioning, across extensive synthetic benchmarks. On 100 benchmarks with up to variables, IPM achieved an average of iterations and s, versus INS at iterations and s, with IPM reaching the primary termination criterion in of runs and INS in ; INS under tuned parameters narrowed the gap, highlighting adaptive potential. The results advocate a robust IPM baseline and a tunable INS as a complementary option, with future work focusing on hybrid INS–IPM strategies that integrate Newton-type flexibility with barrier-based robustness.$

Abstract

We present a head-to-head evaluation of the Improved Inexact--Newton--Smart (INS) algorithm against a primal--dual interior-point framework for large-scale nonlinear optimization. On extensive synthetic benchmarks, the interior-point method converges with roughly one third fewer iterations and about one half the computation time relative to INS, while attaining marginally higher accuracy and meeting all primary stopping conditions. By contrast, INS succeeds in fewer cases under default settings but benefits markedly from moderate regularization and step-length control; in tuned regimes its iteration count and runtime decrease substantially, narrowing yet not closing the gap. A sensitivity study indicates that interior-point performance remains stable across parameter changes, whereas INS is more affected by step length and regularization choice. Collectively, the evidence positions the interior-point method as a reliable baseline and INS as a configurable alternative when problem structure favors adaptive regularization.

Paper Structure

This paper contains 24 sections, 3 theorems, 46 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Interior-Point Method (IPM) Background
  3. Analysis of the Short-Step Method in IPM
  4. Local Model of Complementarity
  5. Theoretical Implications
  6. Role of the inexactness level delta and its impact on convergence.
  7. Analysis of the Long-Step Method in IPM
  8. Rationale for $\tau\in(0,1)$ and its effect.
  9. Inexact-Newton-Smart Test (INS) Method
  10. Newton System with Regularization
  11. Step length and stopping
  12. Inexactness and contraction
  13. Equality-Constrained Newton Phase (ECNP)
  14. Inexact linear solves and forcing condition
  15. Merit function and backtracking
  16. ...and 9 more sections

Key Result

lemma 3.1

Assume that the sequence $\{\mu_k\}_{k \geq 0}$ of nonnegative numbers $\mu_k \geq 0$ satisfies the inequality where $\omega \in (0,1)$, $C>0$ are constants. If $\mu_0 < \omega/C$, then As a consequence, the sequence $\{\mu_k\}_{k \geq 0}$ converges to zero at an exponential rate. Furthermore, the number of iterations $k$ needed to achieve a given precision goal $0 \leq \mu_k < \varepsilon$ is

Figures (2)

  • Figure 1: Comparison of the optimal value of the objective function
  • Figure 2: Comparison of the number of iterations

Theorems & Definitions (6)

  • lemma 3.1
  • proof
  • lemma 3.2
  • proof
  • theorem 3.3
  • proof