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In mathematics, nonlinear programming (NLP) is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function is nonlinear. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...
For the socioeconomic meaning, see social inequality. ...
Partial plot of a function f. ...
Mathematical formulation of the problem The problem can be stated simply as:  to maximize some variable such as product throughput or  to minimize a cost function where  
Methods for solving the problem If the objective function f is linear and the constrained space is a polytope, the problem is a linear programming problem, which may be solved using well known linear programming solutions. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...
In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ...
If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then general methods from convex optimization can be used. In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. ...
In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ...
Look up Convex set in Wiktionary, the free dictionary. ...
Convex optimization is a subfield of mathematical optimization. ...
Several methods are available for solving nonconvex problems. One approach is to use special formulations of linear programming problems. Another method involves the use of branch and bound techniques, where the program is divided into subclasses to be solved with linear approximations that form a lower bound on the overall cost within the subdivision. With subsequent divisions, at some point an actual solution will be obtained whose cost is equal to or lower than the best lower bound obtained for any of the approximate solutions. This solution is optimal, although possibly not unique. The algorithm may also be stopped early, with the assurance that the best solution cannot be more than a certain percentage better than a solution that has been found. This is especially useful for large, difficult problems and problems with uncertain costs or values where the uncertainty can be estimated with an appropriate reliability estimation. Branch and bound is a general algorithmic method for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization. ...
The Karush-Kuhn-Tucker (KKT) conditions provide the necessary conditions for a solution to be optimal. In mathematics, the Karush-Kuhn-Tucker conditions (also known as the Kuhn-Tucker or the KKT conditions) are necessary for a solution in nonlinear programming to be optimal. ...
Examples
 The intersection of the line with the constrained space represents the solution A simple problem can be defined by the constraints :For other senses of this word, see dimension (disambiguation). ...
Image File history File links Example of a nonlinear programming solution. ...
Image File history File links Example of a nonlinear programming solution. ...
- x1 ≥ 0
- x2 ≥ 0
- x12 + x22 ≥ 1
- x12 + x22 ≤ 2
with an objective function to be maximized - f(x) = x1 + x2
where x = (x1, x2)
 The intersection of the top surface with the constrained space in the center represents the solution Another simple problem can be defined by the constraints :For other senses of this word, see dimension (disambiguation). ...
Image File history File links A nonlinear programming example. ...
Image File history File links A nonlinear programming example. ...
- x12 − x22 + x32 ≤ 2
- x12 + x22 + x32 ≤ 10
with an objective function to be maximized - f(x) = x1x2 + x2x3
where x = (x1, x2, x3)
See also In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. ...
Curve fitting is finding aaa curve which matches a series of data points and possibly other constraints. ...
Least squares or ordinary least squares (OLS) is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a best fit). It attempts to minimize the sum of the squares of the ordinate differences (called residuals) between points...
References - Avriel, Mordecai (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN 0-486-43227-0.
- Bazaraa, Mokhtar S. and Shetty, C. M. (1979). Nonlinear programming. Theory and algorithms. John Wiley & Sons. ISBN 0-471-78610-1.
- Nocedal, Jorge and Wright, Stephen J. (1999). Numerical Optimization. Springer. ISBN 0-387-98793-2.
- Bertsekas, Dimitri P. (1999). Nonlinear Programming: 2nd Edition. Athena Scientific. ISBN 1-886529-00-0.
External links - Software
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