Task 1: Fundamental Optimization concepts
Present important notions and key concepts in Optimization.
1. Topology of the real line, convex sets, convex hull
2. Convex functions, generalized convexity, optimality conditions in Rn
3. Linear and non-linear programming
4. (Un)constrained optimization - Lagrange multipliers
5. Multi-criteria optimization
Suggestion: Use definitions, proofs and illustrations using Mathematical software.
Task 2: Optimization with geometric constraints
Indicative topic: Billiard problems
1. Literature review: explore key references for the topic
2. Discuss the reflection of ball paths between two parallel lines
3. Discuss billiards on rectangular, triangular, hexagonal or elliptic tables.
4. Investigate the shortest path between two points on a billiard table with a fixed number of reflections. Conditions required for periodic orbits.
5. Illustrate your results using mathematical software.
You may solve other related problems (location, reflection, resource distribution, wifi or TV camera coverage, etc.), following the same pattern.
Task 3: Topic of choice in Mathematical Programming
Indicative topics: Linear(nonlinear) programming, heuristic techniques, etc.
This task aims to discuss key aspects involved in Mathematical Programming.
1. Review basic literature
2. Consider features of the problem
3. Develop a model and solve the problem
4. Discuss the applicability of your models
5. Suggest further improvements to the model