nonlinear programming

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nonlinear programming

[′nän‚lin·ē·ər ′prō‚gram·iŋ]
(mathematics)
A branch of applied mathematics concerned with finding the maximum or minimum of a function of several variables, when the variables are constrained to yield values of other functions lying in a certain range, and either the function to be maximized or minimized, or at least one of the functions whose value is constrained, is nonlinear.

Nonlinear programming

The area of applied mathematics and operations research concerned with finding the largest or smallest value of a function subject to constraints or restrictions on the variables of the function. Nonlinear programming is sometimes referred to as nonlinear optimization.

A useful example concerns a power plant that uses the water from a reservoir to cool the plant. The heated water is then piped into a lake. For efficiency, the plant should be run at the highest possible temperature consistent with safety considerations, but there are also limits on the amount of water that can be pumped through the plant, and there are ecological constraints on how much the lake temperature can be raised. The optimization problem is to maximize the temperature of the plant subject to the safety constraints, the limit on the rate at which water can be pumped into the plant, and the bound on the increase in lake temperature.

The nonlinear programming problem refers specifically to the situation in which the function to be minimized or maximized, called the objective function, and the functions that describe the constraints are nonlinear functions. Typically, the variables are continuous; this article is restricted to this case.

Researchers in nonlinear programming consider both the theoretical and practical aspects of these problems. Theoretical issues include the study of algebraic and geometric conditions that characterize a solution, as well as general notions of convexity that determine the existence and uniqueness of solutions. Among the practical questions that are addressed are the mathematical formulation of a specific problem and the development and analysis of algorithms for finding the solution of such problems.

The general nonlinear programming problem can be stated as that of minimizing a scalar-valued objective function f(x) over all vectors x satisfying a set of constraints. The constraints are in the form of general nonlinear equations and inequalities. Mathematically, the nonlinear programming problem may be expressed as below,

where x = (x1, x2, …, xn) are the variables of the problem, f is the objective function, gi( x ) are the inequality constraints, and hj( x ) are the equality constraints. This formulation is general in that the problem of maximizing f( x ) is equivalent to minimizing -f( x ) and a constraint gi( x ) ≥ 0 is equivalent to the constraint -gi( x ) ≤ 0.

Since general nonlinear equations cannot be solved in closed form, iterative methods must be used. Such methods generate a sequence of approximations, or iterates, that will converge to a solution under specified conditions. Newton's method is one of the best-known methods and is the basis for many of the fastest methods for solving the nonlinear programming problem.

References in periodicals archive ?
The third is the coupling between real-time image restoration and post-processing whereby the real-time feedback provides accurate prior information for the complicated nonlinear optimization in post-processing.
Most of them are nonlinear optimization problem when abstracting away these problems.
Zornig presents a textbook for undergraduate students in applied mathematics, engineering, economics, computation, and similar fields who want to become familiar with nonlinear optimization.
Nonlinear Optimization of Enzyme Kinetic Parameters.
These real world, nonlinear optimization problems involve millions of variables and constraints and a vast number of potential solutions.
But many of real world tasks cannot be solved by those techniques, or their solutions are not as good, as solutions of some special nonlinear optimization techniques.
When transportation cost on a given route is nonlinearly dependent on the number of the units transported, the transportation problem becomes the nonlinear optimization problem.
Self-concordant barrier functions are intensively studied to develop the barrier functions used in interior point methods for convex and nonlinear optimization.
Derivative-free optimization, a class of nonlinear optimization techniques useful when derivatives are unavailable or unreliable, have applications in fields including computer circuit design, engineering design, finance, operations research, and medicine.
Simple solutions for both direct and inverse problems of nonlinear optimization were obtained.
and CIEDE2000 ([DELTA]E') colour difference formulae using nonlinear optimization.
Algorithms with nonlinear optimization and a neural network process model or a neural network prediction model.

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