Nonlinear Programming: Convex functions, Concave functions, Definitions and basic properties, subgradients of convex functions, Differentiable convex functions, Minima and Maxima of convex functions and concave functions. Generalizations of convex functions and their basic properties.
Unconstrained problems, Necessary and sufficient optimality criteria of first and second order. First order necessary and sufficient Fritz John conditions and Kuhn-Tucker conditions for Constrained programming problems with inequality constraints, with inequality and equality constraints. Kuhn Tucker conditions and linear programming problems.
Duality in Nonlinear Programming, Weak Duality Theorem, Wolfe’s Duality Theorem, Hanson-Huard strict converse duality theorem, Dorn’s duality theorem, strict converse duality theorem, Dorn’s Converse duality theorem, Unbounded dual theorem, theorem on no primal minimum. Duality in Quadratic Programming.
Quadratic Programming: Wolfe’s method, Beale’s method for Quadratic programming.
Linear fractional programming, method due to Charnes and Cooper. Nonlinear fractional programming, Dinkelbach’s approach.
Game theory – Two-person, Zero-sum Games with mixed strategies, graphical solution, solution by Linear Programming