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Industrial Mathematics Final Year Topic: Convergence of Gradient Descent on Quadratic Objectives with Controlled Conditioning

This Industrial Mathematics final year project connects an iteration's convergence behaviour with the spectrum of a stated quadratic objective.

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Why choose this project topic?

A focused study of gradient conditioning gives you a specific question in numerical optimisation. It connects an iteration's convergence behaviour with the spectrum of a stated quadratic objective. The bounded comparison creates room to explain how your evidence supports an interpretation and where the method has limits.

How do condition number and step-size choice affect gradient-descent convergence on positive-definite quadratics?

Agree the apparatus or dataset, comparison range and feasible measurement schedule for gradient conditioning with your supervisor.

Proposed project objectives

  1. 01Define the materials, variables and comparison conditions for gradient conditioning.
  2. 02Derive the iteration matrix and a valid step range.
  3. 03Evaluate the measurements or model outputs in relation to this question: How do condition number and step-size choice affect gradient-descent convergence on positive-definite quadratics?

A suggested research approach

Derive the iteration matrix and a valid step range. Construct quadratics with known spectra, compare objective and iterate errors and verify the relationship between the theoretical bound and observed convergence. Agree the available resources and record uncertainty, deviations from the protocol and any observations that challenge the initial interpretation.

What you will need

  • Positive-definite quadratic examples with known spectra
  • Gradient-method convergence references
  • Reproducible iteration and error calculations

Keep your project scope clear

A quadratic convergence result does not establish global behaviour for an arbitrary non-convex objective.

Industrial Mathematics project chapter outline

Use this outline as a starting point. You can edit the chapter titles to match your department’s format during setup.

  1. Chapter 1Introduction
  2. Chapter 2Literature Review
  3. Chapter 3Theory and Methodology
  4. Chapter 4Results and Applications
  5. Chapter 5Summary, Conclusion and Recommendations

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