Non-Linear Programming

Non-linear programming (NLP) is a branch of mathematical optimization dealing with problems in which the objective function or the constraints, or both, are non-linear. This field is a subset of mathematical programming focusing on optimizing a system by adjusting its parameters to the best possible values under given constraints.

In the realm of mathematics, optimization problems are formalized as a means to determine the extrema — maxima, minima, or stationary points — of a function. Non-linear programming emerges when some of these functions or constraints deviate from linearity. Contrasted with linear programming, where all relationships must be linear, NLP allows for more complex, realistic modeling of many real-world scenarios.

Mathematical Formulation

A typical non-linear programming problem can be expressed as:

[ \text{Minimize or Maximize } f(x) ]

subject to:

[ g_i(x) \leq c_i, \quad i = 1, \ldots, m ]

[ h_j(x) = b_j, \quad j = 1, \ldots, p ]

where ( f(x) ) is the objective function, ( g_i(x) ) and ( h_j(x) ) represent the inequality and equality constraints respectively, and the functions ( f, g_i, h_j ) are non-linear.

Techniques and Methods

Convex Optimization

Special cases arise when the objective function is convex for minimization problems or concave for maximization problems, combined with a convex constraint set. Methods from convex optimization are particularly effective in solving these issues, leveraging the properties of convexity to ensure global optima.

Quadratic Programming

When the objective function is quadratic and the constraints are linear, the problems are termed quadratic programming. These problems are a subset of NLP and have their specialized techniques, which simplify problem-solving in many engineering and economic applications.

Applications

Non-linear programming finds applications across numerous domains, including:

Economics: Modeling non-linear cost functions or utility functions.
Engineering: Designing systems and processes with non-linear characteristics.
Machine Learning: Training models with non-linear decision boundaries.
Experimental Sciences: Analyzing data where systems are represented by non-linear models.

Historical Context and Contributions

One notable contributor to the field of non-linear programming was Albert W. Tucker, whose work laid foundational concepts in optimization theory and game theory, influencing related fields extensively.