STUDY OUTLINE FOR DETERMINISTIC PH.D.EXAMINATIONS
Linear Algebra and Matrix Theory
(References: [9; chapts. 1-3, 5-7], [8; chapts. 3.7], [2; chpt. 6], [1;chpt.2]
Vector Spaces
Linear independence
Subspaces, dimension
Bases, dimension
Orthogonality
Norms
Matrices
Rank
Null and range (column) spaces
Inverse and transpose
Projections
Norms, condition number
Partitioned matrices
Special Matrices
Symmetric
Positive definite (quadratic forms)
Orthogonal
Matrix Factorizations
LU
Cholesky
QR
Solving Linear systems
Gaussian elimination
Least squares
Iterative methods
Eigenvalues and Eigenvectors
Definitions
Eigenspaces
Properties for symmetric matrices
Analysis
(References: [8; chpts. 3], [1; chpt. 2], [3; chpt. 5], [6; chpt. 8,
Appendices A, B], [4; chpts. 1-3], [7])
Mappings from R^n to R^m
Continuity
Differentiability
Gradient
Hessian
Jacobian
Taylor series with error terms
Inverse and implicit function theorems
Geometry for m = 1
Hyperplanes
Polyhedra
Level sets
Descent direction
Feasible direction
Convexity
Convex sets
Convex functions
Algorithms
Convergence
Newton’s method
Optimization
Linear Programming
(References: [8; chpts. 1-10], 2; part I], 1, chpts. 1-6])
Formulations
Optimality conditions
Duality
Dual theorems
Farkas’ lemmas and equivalents
Complementary slackness
Sensitivity Analysis
Ranging of parameters
Adding constraints and variables
Changes in solution and objective value
Parametric programming
Simplex algorithm
Basis and basic feasible solutions
Tableaus, dictionaries and pivoting
Phase I
Unboundedness, infeasibility, and degeneracy
Revised simplex method
Upper – bound simplex method
Column generation
Dual simplex method
Networks
(References: [8;chpts. 13], [2; Part III], [1; chpts. 9-10], [10; chpts. 6,7]), [11,
chpts 6, [11, chpt. 9.2]
Formulations
Transportation and assignment
Transshipment
Maximum flow
Shortest path linear programs
Network simplex method
Associated graph objects (trees, cycles, etc.)
Min. cost flows
Dynamic Programs
(Reference: [5; chpts. 8-10])
Formulations
Inventory models
Production models
Optimality conditions
Recursion equations
Solution
Integer Programs
(Reference: [10; chpts. 13]), [12]
Formulations
Branch and bound algorithm: node-selection strategies
facility location models: setcovering, preprocessing
capacitated facility location, traveling salesman problem
References
1. Baxaara, M., Jarvis, J., Sherali, H., Linear Programming & Network Flows,
John Wiley & Sons, 1990
2. Chvatal, V., Linear Programming, W. H. Freeman, 1985.
3. Dennis, J., Schnabel, R., Numerical Methods for Unconstrained Optimizaiton
and Non-liner Equations, Prentice-Hall, 1983.
4. Edwards, C., Advanced Calculus of Several Variables, Academic Press,
1973.
5. Gill, P., Murray, W., and Wright, M., Practical Optimization, Academic Press,
1981.
6. Luenberger., D., Linear and Nonlinear Programming, Addison-Wesley, 1984.
7. Marsden J. and Hoffman, M., Elementary Classical Analysis, 2nd ed., W. H.
Freeman Company, 1993.
8. Murty, K., Linear Programming, John Wiley & Sons, 1983.
9. Strange, G., Linear Algebra and Its Applications, Third ed., Harcourt, Brace,
and Jovanovich, 1988.
10. Wagner, H., Principles of Operations Research, Prentice-Hall, 1969.
11. Ahuja, Magnanti, Orlin: Network Flows
12. Handouts in OR 210, Fall 2000