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Industrial Mathematics Final Year Topic: Thresholding and Convergence in a Simple L1-Regularised Quadratic Problem

This Industrial Mathematics final year project connects a proximal update with an explicitly defined sparse-penalty objective.

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Why choose this project topic?

A focused study of proximal sparsity gives you a specific question in convex optimisation. It connects a proximal update with an explicitly defined sparse-penalty objective. The bounded comparison creates room to explain how your evidence supports an interpretation and where the method has limits.

How does the regularisation parameter affect sparsity and objective convergence in a stated L1-regularised least-squares problem?

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

Proposed project objectives

  1. 01Define the materials, variables and comparison conditions for proximal sparsity.
  2. 02Derive the relevant proximal operator and choose step sizes using a justified smooth-part bound.
  3. 03Evaluate the measurements or model outputs in relation to this question: How does the regularisation parameter affect sparsity and objective convergence in a stated L1-regularised least-squares problem?

A suggested research approach

Derive the relevant proximal operator and choose step sizes using a justified smooth-part bound. Compare solutions with optimality conditions and examine sparsity and residual changes across synthetic examples. Agree the available resources and record uncertainty, deviations from the protocol and any observations that challenge the initial interpretation.

What you will need

  • A synthetic quadratic-plus-L1 objective
  • Proximal-operator and convexity references
  • Optimality-residual and convergence calculations

Keep your project scope clear

A sparse mathematical solution does not prove that omitted variables have no real-world importance or causal effect.

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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