Lecture I (Days 2,3,4): Convex Analysis

2. Convex Sets Notes; Convex Sets Slides

   affine set, convex set, convex combinations and hull, convex cone, hyperplanes and halfspaces, Euclidean balls and ellipsoids, norm balls and norm cones, polyhedra, positive semidefinite cone, minimum and minimal elements, separating and supporting hyperplane theorems (and proofs), dual cones
   Assignments:
   • Derive necessary (and sufficient) conditions for positive definiteness of X in Sⁿ based on the angle between X and I.
   • Prove that K=K⁺⁺ iff K is a c.c.c.

3. Convex Functions Notes; Convex Functions Slides

   first and second order conditions, epigraph, Jensen's inequality, compositions, pointwise supremum, minimization, conjugate function
   Assignments:
   • Prove that a real valued convex function on Rⁿ is continuous.
   • Prove that a real valued convex function on Rⁿ is locally Lipshitz continuous.
   • Prove that the function G(X)=XX^T-I, from n by n matrices to n by n symmetric matrices, is K-convex with K the cone of semidefinite matrices.
   • Prove that the function g(X)=log det (X) is strictly concave on the set of n by n positive definite matrices X. Find the gradient and Hessian of g(X).

4. Convex Optimization Problems Notes ; Convex Optimization Problems Slides

   standard form, optimal points, implicit constraints, feasibility, (geometric) optimality conditions, quasiconvex optimization, fractional programs, quadratic programs, quadratically constrained quadratic programs, second-order cone programs, robust linear programming, geometric programming, semidefinite programming, eigenvalue minimization, multicriteria minimization, portfolio optimization

5. Duality Notes; Duality Slides

   Lagrangian, Lagrange dual function, Lagrange dual problem, weak and strong duality!!, constraint qualifications, KKT optimality conditions, dual SDP

Supplementary Information

• NEOS Server Stories

• Suggested Texts/References:

  • Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications

    by Aharon Ben-Tal and Arkadi Nemirovski
    MPS/SIAM Series on Optimization 2
    URL: http://www.ec-securehost.com/SIAM/MP02.html

  • Numerical Optimization

    Jorge Nocedal Stephen J. Wright
    URL: http://www.ece.northwestern.edu/~nocedal/book/num-opt.html
  • Lecture Notes for/from Boyd and Vandenberghe book

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