This instructional school will cover topics that come under the broad title of Mathematical Programming. Some of the subjects to be covered in this course include linear programming and game theory, semidefinite programming, optimality conditions for genral nonlinear programming problems, linear complementarity problems, combinatorial optimization
and fixed points and its computations.
Name of the Speaker

Affiliation 
No. of Lectures (each of 1 ½ hrs) 
Detailed Syllabus 
Prof. R. B. Bapat [RBB] 
Indian Statistical Institute Delhi 
6 
Fixed point theorems and computation of fixed points, Sperner's lemma, graph theoretic proof, constructive proof. Brouwer's fixed point theorem, proof using Sperner's lemma, applications. No retraction theorem, hairy ball theorem, Kronecker index theorem. Kakutani's fixed point theorem, Nash equilibrium. Continuous mappings of a circle, homotopy and degree of a mapping. Tucker's lemma, BorsukUlam theorem. 
Dr. A. Chandrashekaran [ACS]

Central University of Tamil Nadu, Thiiruvarur 
6 
Convex sets and convex functions, separation theorem, theorems of alternatives(Farkas lemma and Gordans theorem). Necessary and sufficient conditions for local and global optimality of a feasible point, Weierstrass Theorem. Definitions of normal cone, cone of feasible directions and tangent cone. Optimality conditions based on these cones. Fritz John optimality conditions and KKT optimality conditions. Lagrangian duality; strong and weak duality theorems. 
Dr. I. Jeyaraman [IJ] 
National Institute of Technology, Suratkal 
6 
Complementarity problems: Problem Description, Source Problems, Examples, Existence and Uniqueness of solutions, Matrix classes: Positive definite, $P$, $Q$, $P_0$, sufficient, nondegenerate and semimonotone, Algorithms, Generalizations. 
Prof. G. S. R. Murthy [GSR] 
Indian Statistical Institute Hyderabad 
6 
Introduction to Semidefinite Programming Problem (SDP) with a historical perspective; some applications of SDP; how Linear Programming and Quadratic Programming Problems are special cases of SDP; some properties of SDP such as convexity; brief discussion on algorithms for solving SDPs 
Prof. T. Parthasarathy [TP] 
Indian Statistical Institute Chennai 
6 
Introduction to zerosum two person matrix games with some examples. Statement of Von Neumann's minimax theorem. Formulation of the game problem as a linear programming problem thereby giving an algoritm to find optimal strategies for matrix games through simplex method due to Dantzig. Statement of Nash theorem on bimatrix games and its LCP formulation. Stochastic games(if time permits) 
Prof. Sharad S Sane [SS]

Indian Institute of Technology, Mumbai 
6 
Theory of network flows, augmenting paths and FordFulkerson algorithm and its scaling improvement, EdmondsKarp polynomial time algorithm, Some applications: weighted bipartite matching problem, demandsupply network, airline scheduling, image segmentation and baseball elimination. 
References:
1. M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear programming: Theory and Algorithms, (2nd ed.)Wiley, New York, 1993.
2. T.H. Cormen, C E. Leiserson and R. L. Rivest, Introduction to algorithms, Tata McGraw Hill, 1990.
3. R. W. Cottle, J. S. Pang, R. E. Stone, The linear complementarity problem, Academic Press, Boston, 1992.
4. B. Craven and B. Mond, Linear programming with matrix variables, Linear Algebra Appl., 38, pp. 7380, 1981.
5. R. Fletcher, A nonlinear programming problem in statistics , SIAM J. Sci. Statist. Comput., 2, pp. 257267, 1981.
6. R. Fletcher, Semidefinite matrix constraints in optimization, SIAM J. Control Optim., 23, pp. 493513, 1985.
7. D.Gale, Theory of linear economic models, University of Chicago Press, 1960.
8. J Kleinberg and E. Tardos : Algorithm designs, Pearson Education Limited, 2014.
9. G.Owen, Game Theory, Academic Press, 1995.
10. Yu. A. Shashkin, Fixed Points, Universities Press (India) Ltd., 1991.
11. J.Von Neumann and O.Morgenstern, `Theory of Games and Economic Behaviour, Princeton University Press, 2007.
12. L. Vandenberghe and S. Boyd, Semidefinite Programming, Siam Review, Vol. 38, No.1, pp.4995, March 1996.
13. D.B. West, Introduction to graph theory, Prentice Hall of India, 2000.
TimeTable (with names of speakers and course associates/tutors):
Day 
Date 
Lecture 1 (9.30–11.00) 
Tea (11.00 – 11.30) 
Lecturer 2 (11.30–1.00) 
Lunch (1.00–2.30) 
Tutorial (2.30–3.30) 
Tea (3.30  4.00) 
Tutorial (4.005.00) 
Snacks 5.00  5.30 


(name of the speaker) 

(name of the speaker) 

(name of the speaker / tutor) 
Tea



Mon 
04.07.2016 
RBB 

SS 

Allotment of 2 tutors for each speaker will be done at a later point. 
Allotment of 2 tutors for each speaker will be done at a later point. 

Tues 
05.07.2016 
SS 

RBB 


Wed 
06.07.2016 
RBB 

SS 


Thu 
07.07.2016 
SS 

RBB 


Fri 
08.07.2016 
RBB 

SS 


Sat 
09.07.2016 
SS 

RBB 


SUNDAY 

Mon 
11.07.2016 
TP 

GSR 

Allotment of 2 tutors for each speaker will be done at a later point. 
Tea

Allotment of 2 tutors for each speaker will be done at a later point. 
Snacks

Tues 
12.07.2016 
GSR 

TP 


Wed 
13.07.2016 
TP 

GSR 


Thu 
14.07.2016 
GSR 

TP 


Fri 
15.07.2016 
TP 

GSR 


Sat 
16.07.2016 
GSR 

TP 


SUNDAY 

Mon 
18.07.2016 
IJ 

ACS 

Allotment of 2 tutors for each speaker will be done at a later point. 
Tea

Allotment of 2 tutors for each speaker will be done at a later point. 
Snacks

Tues 
19.07.2016 
ACS 

IJ 


Wed 
20.07.2016 
IJ 

ACS 


Thu 
21.07.2016 
ACS 

IJ 


Fri 
22.07.2016 
IJ 

ACS 


Sat 
23.07.2016 
ACS 

IJ 
