Topic outline

  • General

    Textbook and Optional References

    •        Stephen Boyd; Lieven Vandenberghe, Convex Optimization (Textbook!)

    •        Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications

    •        Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course

    •        J. Gallier and J. Quaintance, Algebra, Topology, Dierential Calculus, and Optimization Theory For Computer Science and Engineering

    Course Requirements and Grading


    •        4 homework assignments. Homeworks will normally be assigned on Wendsdays and will be due in 2 weeks.

    •        Final Exam. The format will be decided depending on the sitation.


    Homeworks: 60%. Exam: 40%. These weights are approximate. We reserve the right to change them later. Can be also discussed.


    Concentrates on recognizing and solving (using standard packages) convex optimization problems that arise in practice!

    •        Convex sets, functions, and optimization problems.

    •        Least-squares, linear and quadratic programs.

    •        Semidefinite programming (SDP).

    •        Minimax, extremal volume, and other problems with geometric interpretation.

    •        Optimality conditions, duality theory, theorem of alternatives.

    •        Introduction to unconstrained optimization algorithms.

    •        Introduction to interior-point methods for constrained optimization.

    •        Applications.

    Course Objectives

    •        to give the tools and training to recognize convex optimization problems that arise in electrical engineering and computer science

    •        to present the basic theory of such problems, concentrating on results that are useful in computations

    •        to give a thorough understanding of how such problems are solved, and some experience in solving them

    •        to give the background required to use the standard methods and software packages (CVX toolbox) in your own research work

    •        to give a number of examples of successful application of convex optimization techniques for solving problem in applied mathematics, computer science, statistics, electrical engineering, and science and engineering in general

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      Some general features of convex problems

      Example: Aalto Design

      What the duality theory is for?

      What we will/won't do

      How many problems are convex?

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      (Vector) Spaces 


      Systems of linear equations 

      Least squares

      Special matrix forms

      Quadratic and Hermitian forms

      Eigenvalues and eigenvectors


      Singular value decomposition

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      Subspace, affine set, convex set, convex cone

      Simple examples and properties

      Combination and hulls

      Ellipsoids, polyhedra, norm balls

      Affine and projective transformations

      Separating and supporting hyperplanes

      Generalized inequalities

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      Assignment 1: Convex Sets

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      Convex functions


      Simple examples, elementary properties

      More examples / exercises, more properties

      Jensen's inequality

      Quasiconvex, quasiconcave functions

      Log-convex and log-concave functions


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      Assignment 2: Convex Functions

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      Solutions to Problems 6 and 7 of Assignment 1 which we didn't finish in the exercise session yesterday.

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      Abstract form problem

      Standard form problem

      Convex optimization problem

      Standard form with generalized inequalities

      Mulitcriterion optimization

      Restriction and relaxation

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      Linear programming (LP) 

      Examples and applications of LP

      Linear fractional programming 

      Quadratic optimization problems (QP)

      Quadratically constrained quadratic problems (QCQP) 

      Examples and applications of QP and QCQP

      Semidefinite programming (SDP) format

      More applications

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      Assignment 3: Convex Problems 

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      The lecture will be given by a visiting lecturer from Nokia Bell labs. It is necessary for doing Assignment 3.  

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      General descent method

      Line search types

      Gradient method

      Steepest descent method

      Newton's method

      Convergence analysis

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      Brief history of Sequential Unconstrained Minimization Techniques (SUMT) & Interior point (IP) methods

      Logarithmic barrier function

      Central path

      Basic SUMT

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      Lagrange dual function

      Lagrange dual problem

      Strong duality and Slater's condition

      Karush-Kuhn-Tucker (KKT) optimality conditions
      Sensitivity analysis

      Generalized inequalities

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      Assignment 4: Optimization Algorithms