Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimization Methods (Steepest Descent and Newton’s Method)

  • Onanaye, Adeniyi Samson
Keywords: Nonlinear Programming Problem, Unconstrained Optimization, Mathematical Programming, Newton's Method, Steepest Descent


Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as bisection method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. However, in this paper, mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple (n) design variables subject to no constraints are been used to overcome the above challenge. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained optimization is important for several reasons. Steepest Descent and Newton’s methods are employed in this paper to solve an optimization problem.


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Charnes .A. & Cooper W.W. (1962). ’Programming with linear fractional functional. Available at:

Bradley S.P. & Frey S.C. (1974). Fractional programming with homogeneous functions. Operations Research, 22(2), 350-357.

Dr. Binay Kuma. (2014). Advanced method for solution of constraint optimization problems. Available at:

Holt C. C, Modigliani F, Muth J. F, & Simon H. A. (1960). Planning production, inventories, and work-force, New Jersey: Prentice-Hall, Inc.

John W. Chinneck. (2012). Practical optimization: A gentle introduction. Available at:

Kuhn, H. & A. Tucker. (1950). Nonlinear programming. Available at:

Slater M. (1950). Lagrange multipliers revisited: A contribution to nonlinear programming. Available at:

White L. S. (2018). ’Mathematical programming models for determining freshman scholarship offers. Forgotten Books.

Winters P. R. (1962). Constrained inventory rules for production smoothing. Management Science, 8(4).

Van SlykeR. M. & Wets R. J. (1968). A duality theory for abstract mathematical programs with applications to optimal control theory. Available at:

How to Cite
Onanaye, Adeniyi Samson. (2020). Nonlinear Programming: Theories and Algorithms of Some Unconstrained Optimization Methods (Steepest Descent and Newton’s Method). International Journal of Engineering and Management Research, 10(2), 1-12.