Who, What, Where and When

• Time (tentative): Tuesday, 6:00-9:00 pm
• Place: RUTCOR Bldg, Room 139
• Organizational Meeting: Tuesday, Sep 2, 6:00 pm, RUTCOR, Room 139
  In this meeting we'll fix the class time. If you cannot attend, please e-mail your preferences to:
• Instructor: Adi Ben-Israel
  • E-mail: bisrael@rutcor.rutgers.edu
  • Office: 113 RUTCOR Bldg, Busch Campus
  • Telephone: 445-5631 Fax: 445-5472
• Text: R. Tyrrell Rockafellar and Roger J.-B. Wets , Variational Analysis , Springer-Verlag, 1997 (to appear)
• Description: This course is an introduction to convex & nonsmooth analysis, with emphasis on topics that are used in optimization theory and algorithms.
• Prerequisites: Linear algebra and advanced calculus. Examples and applications are mostly from linear/nonlinear programming; knowledge of these would be helpful (but is not formally required).
• Intended for: Graduate students in operations research, mathematics, engineering, economics and related areas.

Syllabus

1. Review.
2. Convex functions of one real variable. Basic properties. Inequalities. Continuity. 1st and 2nd order differentiation. Conjugate functions.
3. Introduction to optimization algorithms. Generalities. Descent and steepest descent directions. 1st order methods. Newton methods. Conjugate direction methods.
4. Convex sets, cones and polyhedra. Generalities. Calculus of convex sets. Projections associated with convex sets and cones. Separation theorems. The Minkowski-Farkas lemma. Conical approximations of convex sets: Tangent and normal cones.
5. Convex functions of several variables. Basic properties. Functional operations preserving convexity. Local and global behavior of convex functions. 1st and 2nd order differentiation. Calculus of conjugate functions.
6. Sublinearity and support functions. Basic properties. Support functions of closed convex polyhedra. The isomorphism between closed convex sets and closed sublinear functions. Norms, dual norms and polarity. Calculus with support functions.
7. Subdifferentials of finite convex functions. Alternative definitions and basic properties. Calculus with subdifferentials. The subdifferential as a multifunction: Monotonicity and continuity properties.
8. Constrained convex minimization problems. Abstract optimality conditions. Exact penalty. Optimality conditions involving the tangent and normal cones. The Lagrangian, Lagrange multipliers, and their economic interpretation. Saddle points and minmax. Karush-Kuhn-Tucker conditions. Constraint qualifications. Duality theory of convex programming.
9. Introduction to nonsmooth analysis. The clarke generalized derivative. Nonsmooth calculus and optimization problems.

Course Work

References

• F.H. Clarke, Optimization and Nonsmooth Analysis (2nd edition), J. Wiley, 1989
• J-P Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, I: Fundamentals, Springer-Verlag, 1993.
• J-P Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, II: Advanced Theory and Bundle Methods, Springer-Verlag, 1993.
• R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970.
• V.M. Tikhomirov, Convex Analysis, Part I of Vol. 14 Analysis II of the Encyclopedia of Mathematical Sciences (R.V. Gamkrelidze, Ed.), Springer, 1990.