Introduction to Optimization for Engineers – ELEC 400O | MECH 410Q | MECH 540G

Winter Term, 2026–27. Instructors: Alberto Padoan and Dominic Liao-McPherson.
Links: Website | Piazza | Canvas 400O, Canvas 41OQ, Canvas 540G.

Lectures

Time: Every Tuesday and Thursday, 11:00 – 12:30.
Room: UBCV | Gateway Health Building (GWHB) | Floor 4 | Room 4290

Tutorials

Time: Every Wednesday, 15:00 – 16:00.
Room: UBCV | Hugh Dempster Pavilion (DMP) | Floor 3 | Room 310

Office Hours

Primarily via Piazza. Alternatively, after lectures by appointment via written email.

Credits

Units: 3 (ELEC 400O, MECH 410Q) and 4 (MECH 540G). Letter grade.

Course Description

This course is an introduction to optimization for engineers, with an emphasis on algorithms: how they are derived, why they converge, and how they behave on real problems. Topics include:

Mathematical background: linear algebra, conditioning, and multivariable calculus
Formulating and reformulating engineering problems as optimization problems
Optimality conditions, unconstrained and constrained, and the Karush–Kuhn–Tucker (KKT) conditions
Duality and sensitivity
Algorithms: gradient descent, line search, Newton and damped Newton, projected gradient, interior-point methods, and sequential quadratic programming
Trajectory optimization as a closing application

Learning Objectives

Students completing this course should be able to:

Recognise the class an optimization problem belongs to, and formulate an engineering problem into standard form
State the optimality conditions appropriate to a problem, and use them to certify or reject a candidate solution
Derive the KKT conditions and interpret multipliers as sensitivities
Derive the main algorithms of the course and explain the assumptions each one needs
Implement gradient descent, Newton's method, and a projection-based method, and diagnose their failure modes
Explain how conditioning and numerical error affect the behaviour of an optimization algorithm
Use an off-the-shelf solver competently and interpret what it reports

Course Schedule

Mathematical foundations

Introduction: what optimization is for, logistics
Sets, vectors, matrices, norms
Eigenvalues, spectral decomposition, the SVD, conditioning
Multivariable calculus, Taylor expansion, Newton's method for rootfinding

Unconstrained optimization: theory

Standard form, feasible sets, minimisers; LP, QP and conic programs
Modelling and problem manipulation, with engineering examples
First-order necessary conditions
Second-order necessary and sufficient conditions
Convexity and global optimality

Unconstrained optimization: algorithms

Gradient descent and its convergence
Line search, backtracking, convergence rates
Newton's method and damped Newton
Projected gradient descent

Midterm 1 – Thursday, October 22, 2026 (tentative), in person (during regular class hours).

Constrained optimization: theory

Feasible-set geometry: active sets, tangent and normal cones
Constraint qualifications and the KKT conditions
Second-order sufficiency and sensitivity
Duality

Constrained optimization: algorithms

Penalties, barriers, and interior-point methods
Projected Newton

Midterm 2 – Thursday, November 26, 2026 (tentative), in person (during regular class hours).

Bonus topics

Sequential quadratic programming
Trajectory optimization

No classes: 30 September (National Day for Truth and Reconciliation), and 10-11 November (midterm break and Remembrance Day).

Material & References

Course notes and problem sets are distributed through Canvas. New material is posted before each lecture or shortly afterwards.

J. Nocedal and S. J. Wright, Numerical Optimization. 2nd ed., Springer, 2006.
S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. (optional)
D. P. Bertsekas, Nonlinear Programming. 3rd ed., Athena Scientific, 2016. (optional)

Prerequisites

Linear algebra and multivariable calculus at the level of a second-year engineering course.
Familiarity with programming (needed for both tutorials and final project).
No prior exposure to optimization is assumed.

Assessment

ELEC 400-O and MECH 410-Q

35% – Midterm 1
35% – Midterm 2
30% – Final project

MECH 540-G

30% – Midterm 1
30% – Midterm 2
30% – Final project
10% – Research presentation (last 2 weeks of lectures)

Note: Use of generative AI is permitted as a tool. However, all submitted work must reflect the student’s own understanding. Work primarily generated by AI will receive a grade of zero.

Final Project

A short research-style report and a runnable Python notebook, taking one method from the course further than the lectures do. Main idea: identify a problem for which standard formulations or off-the-shelf algorithms fall short, develop a formulation or algorithm based on material from the course, and demonstrate its effectiveness through numerical experiments, theoretical analysis, or both. Groups of two are preferred.

Benchmark problems may be suggested by the instructors, but original topics are especially welcome.

Timeline

Topic proposal – 20 October, one paragraph
Project talks – From 18 November, during the tutorial hour
Report and notebook – due 4 December, 16:00

Late policy

Deadlines are firm. Late work or missed assessments will not be graded, consistent with the Academic Calendar on Grading Practices.

Disclaimers

Lectures and course materials, including presentations, tests, outlines, and similar materials, are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

This is not an official course webpage from UBC; it is maintained personally by the instructors. This being the inaugural run, please anticipate occasional hiccups. Thank you for your flexibility as we refine the experience.

Feedback

If you have suggestions or found the material useful, we would be happy to hear from you. Please use:
alberto [DOT] padoan [@] ubc [DOT] ca
dliaomcp [@] mech [DOT] ubc [DOT] ca