Teaching plan for the course unit



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General information


Course unit name: Optimization

Course unit code: 572662

Academic year: 2021-2022

Coordinator: Gerardo Gomez Muntane

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S



Estimated learning time

Total number of hours 150


Face-to-face and/or online activities



-  Lecture





-  Problem-solving class




Supervised project


Independent learning




Competences to be gained during study


That the students acquire knowledge and understanding that provide them a basis or opportunity for originality in developing and / or applying ideas, often in a research context





Learning objectives


Referring to knowledge

 That the students can apply their knowledge and their ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their field of study



Teaching blocks


1. I. Unconstrained optimization and optimality conditions

*   I.1 Nonlinear programming

 I.2 Application contexts

 I.3 Characterization issue

 I.4 Computation issue

 I.5 Duality

 I.6 Unconstrained optimization

 I.7 Local minima

 I.8 Necessary conditions

 I.9 Sucient conditions for local minima

 I.10 The role of convexity

2. II. Gradient methods for unconstrained optimization

*   II.1 Quadratic unconstrained problems

 II.2 Existence of optimal solutions

 II.3 Iterative computational rethods

 II.4 Gradient methods. Motivation

 II.5 Principal gradient methods

 II.6 Choices of direction and stepsize

 II.7 Convergence

3. III. Newton and Gauss-Newton methods

*   III.1 Newtons method

 III.2 Convergence

 III.3 Variants of Newtons method

 III.4 Least squares problems

 III.5 The Gauss-Newton method

 III.6Quasi-Newton methods

4. IV. Convexity

*   IV.1 Convex sets and convex functions

 IV.2 Diㄦential properties of convex functions

 IV.3 Extrema of convex functions

 IV.4 Optimality conditions for convex programms

 IV.5 Optimization subect to bounds

 IV.6 Optimization over a simplex

 IV.7 Optimal routing

 IV.8 Projection over a convex set

5. V. Constrained optimization. Lagrange multipliers

*   V.1 Equality constrained problems

   V.1.1 Lagrange multiplier theorem

   V.1.2 Equality constrained problems. Sucientciency conditions

   V.1.3 Convexi^Lcation using augmented Lagrangians

   V.1.4 Sensitivity

 V.2 Inequality Constrained Problems

   V.2.1 Necessary and subectcient conditions

   V.2.2 Linear constraints

6. VI. Duality

*   VI.1 Convex cost. Linear constraints

 VI.2 Duality theorem

 VI.3 Linear programming duality

 VI.4 Quadratic programming duality

 VI.5 Geometrical framework for duality

 VI.6 Geometric multipliers

 VI.7 The dual problem

 VI.8 Properties of the dual function

 VI.9 Duality and G-multipliers

 VI.10 Strong duality theorem

 VI.11 Linear equality constraints

7. VII. Interior point methods

*   VII.1 Barrier and interior point methods

 VII.2 Linear programs and the logarithmic barrier

 VII.3 Path following using Newtons method

8. VIII. Penalty methods

*   VIII.1 Quadratic penalty methods

 VIII.2 Multiplier methods

9. IX. Stochastic Optimization



Teaching methods and general organization


Presentation, by the teacher, of the main ideas and results of the different thematic blocks,and resolution, by the students, of exercises



Official assessment of learning outcomes


- Resolution of exercises  (50% of the final mark)

- Final exam (50% of the final mark)

- To pass you must have 4 or more points (out of 10) in each of the two ratings


Examination-based assessment

- Final exam