Module Catalogues, Xi'an Jiaotong-Liverpool University

Module Code: MTH318
Module Title: Optimisation Theory
Module Level: Level 3
Module Credits: 5.00
Semester: SEM2
Originating Department: Mathematical Sciences
Pre-requisites: MTH107ANDMTH117ANDMTH118

 Aims Many mathematical problems arising in the real world can be formulated as optimisation problems. This module provides students with the tools and training to recognise optimisation problems that arise in mathematics and related fields, presenting the basic theory and concentrating on results that are useful in computation. The module helps students understand how such problems are solved and gain experience in solving them, giving them the background required to use the methods in their future careers.
 Learning outcomes A Investigate the existence of solutions to an optimisation problem. B Discuss and analyse simple optimisation problems and set them out in standard form.C Show the theorems of the alternative and use them to derive necessary conditions for optimality.D Make use of the necessary conditions for optimality to solve the primal problem.E Recognise the important role of convexity of an objective function in optimisation.F State constraint qualifications for optima with inequality constraints and apply them to discuss constrained optimisation problems.G Formulate and solve the dual problem.
 Method of teaching and learning This module will be delivered by formal lectures and tutorials. Students are expected to attend lectures as well as tutorials. Lectures will introduce students to the academic content and practical skills which are the subject of the module, while tutorials will allow students to discuss and analyse optimisation problems and to practice relevant skills.
 Syllabus Optimisation problems, basic concepts, illustrative examples, Weierstrass theorem. Convex sets: convex hulls, closure and interior, hyperplanes; separation and support of sets; Theorem of the alternative (Farkas’s Theorem, Gordan’s Theorem); convex cone and polarity; Polyhedral sets, extreme points, extreme directions. Convex functions: definition and basic properties; subgradients; differentiable convex functions; minima and maxima of convex functions; generalisations of convex functions. Unconstrained optima (first-order conditions, second-order conditions). Fritz John and Kurush-Kuhn-Tucker Optimality: constrained optimisation problems (inequality constraints, inequality and equality constraints); second-order necessary and sufficient optimality conditions for constrained problems. Constraint qualifications: cone of tangents; other constraint qualifications; problems with inequality and equality constraints. Lagrangian duality and saddle point optimality conditions: Lagrangian dual; Duality Theorems and Saddle point optimality conditions; properties of the dual function; formulating and solving the dual problem; the primal solution; Linear and quadratic problems.
Delivery Hours
 Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total Hours/Semester 39 13 98 150

## Assessment

 Sequence Method % of Final Mark 1 Class Test 15.00 2 Coursework 10.00 3 Final Exam 75.00
 Module Catalogue generated from SITS CUT-OFF: 12/16/2019 8:15:15 AM