nonlinear programming

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nonlinear programming

[′nän‚lin·ē·ər ′prō‚gram·iŋ]
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 ?
However, it may be difficult to solve neutrosophic nonlinear optimization models in indeterminate nonlinear optimization design problems, such as multiple-bar truss structure designs under neutrosophic number environments, by the Karush-Kuhn-Tucker (KKT) necessary conditions.
By building large scale nonlinear optimization tools and using them to assimilate electrophysiological data, we will develop a method for automatically finding the network parameters that accurately reproduce biological motor sequences and their adaptation to multiple physiological inputs.
In each control cycle, sequential quadratic programming (SQP) algorithm is used to deal with the constrained nonlinear optimization problem, realizing rolling optimal control.
In the primal-dual method, a nonlinear optimization problem (primal problem) can be converted into another nonlinear optimization problem (dual problem).
The problem of optimal distribution of reactive power is a nonlinear optimization problem with multiple constraints.
This paper presents a reconstruction technique in which nonlinear optimization is used in combination with an impact model to quickly and efficiently find a solution to a given set of parameters and conditions to reconstruct a collision.
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.
Velasquez, "R-chaosoptimiser: an optimiser for unconstrained global nonlinear optimization written in R language for statistical computing", Ingenieria e Investigacion, vol.
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.
Other developments included the beginnings of the development of nonlinear optimization, the application of linear optimization methods to management theory in 1951, the publication of the first linear optimization textbook in 1953, the development of dual simplex methods in 1953, and early methods for transportation problems in 1954 (Brentjes, 1994).

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