The present book is the culmination of honest and sincere efforts on the part of the authors to meet the requirements of the students who opt the paper of Non-Linear Programming. It is also meant for those who are appearing in various competitive examinations.

The entire subject matter has been arranged in a simple, systematic, graded, lucid and exhaustive manner to meet the various needs of all types of learners. Each chapter begins with some definitions followed by theorems with complete proofs and solved problems so that the study leads to perfect clarity and understanding. A large number of notes and remarks have been added for a better understanding of the subject. Exercises have been given in each chapter to provide the students with an opportunity of practice.


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

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