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| In [[mathematics]], '''nonlinear programming''' ('''NLP''') is the process of solving an [[optimization problem]] defined by a [[Simultaneous equations|system]] of [[equation|equalities]] and [[inequalities]], collectively termed [[Constraint (mathematics)|constraints]], over a set of unknown [[real variable]]s, along with an objective [[function (mathematics)|function]] to be maximized or minimized, where some of the constraints or the objective function are [[nonlinear]].<ref>{{cite book
| | == 「シープクリフの町警察署、町の警察リーは羊を結んだ == |
| | last = Bertsekas
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| | first = Dimitri P.
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| | authorlink = Dimitri P. Bertsekas
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| | title = Nonlinear Programming
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| | edition = Second
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| | publisher = Athena Scientific
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| | year = 1999
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| | location = Cambridge, MA.
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| | isbn = 1-886529-00-0
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| }}</ref> It is the sub-field of [[Mathematical optimization]] that deals with problems that are not linear.
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| ==Applicability== | | 、笑う、これはそれがあまりにも若年特別なプレー [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード 人気バッグ]<br><br>「私自身、兄弟に行くためにあなたの目を引く、そして、あなたのアイデンティティ [http://www.dmwai.com/webalizer/kate-spade-9.html バッグ ケイトスペード]。彼に言った、「文の太陽Tianming轟音 [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード バッグ ショルダー]。<br><br>「シープクリフの町警察署、町の警察リーは羊を結んだ。 '<br><br>「郷警察、高兵士。 [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード アウトレット バッグ] '<br><br>「郷警察、陳Dajun。 '<br><br>数人の新聞名、本当に特大、聞いてdumbfoundingフラットGuodongは、これらの人々が高く、遠くの山から場所へ天皇、物品の言葉を理解していなかった余裕でしょう責めることはできないが、彼はすぐに再度沈んだと考えられている、だから、本当に既存のカウンターパートを奪還したい、人員を失っ小さく、同省は大きなを失う可能性があります。<br>「一つ屋根の下、顔を与え、Sunのチームは、生活が困難なものを作る [http://www.dmwai.com/webalizer/kate-spade-8.html ハンドバッグ ケイトスペード]。」：ソフト、ゆっくりとまっすぐ銃を、置く彼の口調を<br><br><br>は「私はデンに3チームを入れて今日の額装、クソ誰かが私に顔を与えることはできません？右、あなたが戻って、ああ、あなたが波紋以上のことを、この甥に直面与える......フラットGuodong、私は今、正式に犯罪やギャンブルを拘束していますあなたは...... '太陽Tianmingは、胸を膨らん歩い |
| A typical non[[Convex_optimization|convex]] problem is that of optimising transportation costs by selection from a set of transportion methods, one or more of which exhibit [[Economy of scale|economies of scale]], with various connectivities and capacity constraints. An example would be petroleum product transport given a selection or combination of pipeline, rail tanker, road tanker, river barge, or coastal tankship. Owing to economic batch size the cost functions may have discontinuities in addition to smooth changes.
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.1555666.net/home.php?mod=space&uid=13832&do=blog&quickforward=1&id=377414 http://www.1555666.net/home.php?mod=space&uid=13832&do=blog&quickforward=1&id=377414]</li> |
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| | <li>[http://wnbsxdpf.org.cn/plus/feedback.php?aid=27 http://wnbsxdpf.org.cn/plus/feedback.php?aid=27]</li> |
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| | <li>[http://www.xintai.org.cn/plus/feedback.php?aid=3 http://www.xintai.org.cn/plus/feedback.php?aid=3]</li> |
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| | </ul> |
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| Modern engineering practice involves much numerical optimization. Except in certain narrow but important cases such as passive electronic circuits, engineering problems are non-linear, and they are usually very complicated.
| | == 彼の怪我はなく、彼自身の傷を == |
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| In experimental science, some simple data analysis (such as fitting a spectrum with a sum of peaks of known location and shape but unknown magnitude) can be done with linear methods, but in general these problems, also, are non-linear. Typically, one has a theoretical model of the system under study with variable parameters in it and a model the experiment or experiments, which may also have unknown parameters. One tries to find a best fit numerically. In this case one often wants a measure of the precision of the result, as well as the best fit itself.
| | 郡の主要なショックが割れた、母と息子の睡眠ケースを殺害した爆撃と郡は10年前に起こった、犠牲者は後に彼の妻の死を負担することができない大型トラックの中小企業経営者であり、痛み、そして亡命する [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード財布評判]。<br><br>真実はそこに浮上しそうです、、離婚しようとして挫折し、犯罪被害者にこの扇動である代わりに、傍若無人インターポールは彼を見つけ、起動すると、彼は会社役員の成功の少量を持っています [http://www.dmwai.com/webalizer/kate-spade-6.html ケイトスペード リボン バッグ]。インターポールの爆発共犯秘密ソリューションは手錠をかけられフード付きGuzhai郡を、バック護衛した後にお金が拘置所ラックに重い足枷ボディ、ほとんどの人間を着て、彼の本当の爆発の容疑者が逮捕されたさらさ強要として彼が変装した [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード 財布 新作]。<br>彼の怪我はなく、彼自身の傷を<br>、彼は有罪が、それは確かに彼自身のやっている、と彼は告白している [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード ショルダーバッグ]。<br>数多くの暗い隅、犯罪と犯罪との戦いはとてもシフトを継続していることを<br> [http://www.dmwai.com/webalizer/kate-spade-3.html 財布 kate spade]。<br>人々の<br>2種類の暗闇の中で生きている、二つの道は、ノーリターン、ノー終わりである.........<br>第89章前<br>ボリューム絶望 |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.mommd.com/cgi-bin/ubb-cgi/ultimatebb.cgi http://www.mommd.com/cgi-bin/ubb-cgi/ultimatebb.cgi]</li> |
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| | <li>[http://www.uberiku.com/yybbs/yybbs.cgi http://www.uberiku.com/yybbs/yybbs.cgi]</li> |
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| | <li>[http://yifang.n155.nicdns.cn/plus/feedback.php?aid=244 http://yifang.n155.nicdns.cn/plus/feedback.php?aid=244]</li> |
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| | </ul> |
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| == The general non-linear optimization problem (NLP) == | | == 特別な抗漏れ == |
| The problem can be stated simply as:
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| :<math>\max_{x \in X}f(x)</math> to maximize some variable such as product throughput
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| or
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| :<math>\min_{x \in X}f(x)</math> to minimize a cost function
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| where
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| :<math>f: R^n \to R</math>
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| :<math>x \in R^n</math>
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| s.t. (subject to)
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| :<math>h_i(x)=0, i \in I={1,\dots,p}</math>
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| :<math>g_j(x) \leq 0, j \in J={1,\dots,m}</math>
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| == Possible solutions ==
| | カー [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペード リボン バッグ]。実際に、彼は実際に馬を実証するものから前進来ることができると彼は釣りは間違いなく次の家を出荷するようにたくさんのお金であることを見ることができました [http://www.dmwai.com/webalizer/kate-spade-4.html ケイトスペード バッグ 新作]。<br>これらの人は二つの数字は、元市の5人に1、高速の1と頻繁に接触して監視さ<br>、反対側がロックすることはできませんが、抗麻薬に応じて、警察は彼らを扱う長年の経験を持って、時間はトランザクションが近くになければならない...正確な時間のためにそれらをフォローしている人に加えて、それらの信号を追跡する、という瞬間の前に取引されるように...、もっと良い方法があるでしょう。<br>10時半<br>、遠く陝西省の抗薬部門の、そこに潜在的なバイヤーというメッセージを送って、送り出さ [http://www.dmwai.com/webalizer/kate-spade-0.html ハンドバッグ ケイトスペード].........<br><br>「確かに、今日における取引時、。 [http://www.dmwai.com/webalizer/kate-spade-8.html kate spade 財布 ゴールド] '<br><br>レイが2車は陝西省と山西省に沿って高速で移動している、情報を送信し、姉妹ユニットを指摘し、計算時間に応じて7から始まる、オリジナルの旅の5個に5時間を必要とし、地方の国境に近接しています。<br><br>「徐副会場ああ、それは数時間後、今日は最後の戦いであることをようで、、私たちは成り行きを見守る必要があります [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペードバッグセール]。<br><br>特別な抗漏れ |
| * feasible, that is, for an optimal solution <math>x</math> subject to constraints, the objective function <math>f</math> is either maximized or minimized.
| | 相关的主题文章： |
| * unbounded, that is, for some <math>x</math> subject to constraints, the objective function <math>f</math> is either <math>\infty</math> or <math>-\infty</math>.
| | <ul> |
| * infeasible, that is, there is no solution <math>x</math> that is subject to constraints.
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| | | <li>[http://www.pfb999.org/thread-197045-1-1.html http://www.pfb999.org/thread-197045-1-1.html]</li> |
| ==Methods for solving the problem==
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| If the objective function ''f'' is linear and the constrained [[Euclidean space|space]] is a [[polytope]], the problem is a [[linear programming]] problem, which may be solved using well known linear programming solutions.
| | <li>[http://www.sori.cn/plus/feedback.php?aid=214 http://www.sori.cn/plus/feedback.php?aid=214]</li> |
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| If the objective function is [[Concave function|concave]] (maximization problem), or [[Convex function|convex]] (minimization problem) and the constraint set is [[Convex set|convex]], then the program is called convex and general methods from [[convex optimization]] can be used in most cases.
| | <li>[http://yuntour88.com/plus/feedback.php?aid=116 http://yuntour88.com/plus/feedback.php?aid=116]</li> |
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| If the objective function is a ratio of a concave and a convex function (in the maximization case) and the constraints are convex, then the problem can be transformed to a convex optimization problem using [[fractional programming]] techniques.
| | </ul> |
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| 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 convex (minimization problem) or 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 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 possible solution is within a tolerance from the best point found; such points are called ε-optimal. Terminating to ε-optimal points is typically necessary to ensure finite termination. 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.
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| Under [[differentiability]] and [[constraint qualification]]s, the [[Karush–Kuhn–Tucker conditions|Karush–Kuhn–Tucker (KKT) conditions]] provide necessary conditions for a solution to be optimal. Under convexity, these conditions are also sufficient. If some of the functions are non-differentiable, [[subderivative|subdifferential]] versions of
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| [[Karush–Kuhn–Tucker conditions|Karush–Kuhn–Tucker (KKT) conditions]] are available.<ref>
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| {{cite book|last=[[Andrzej Piotr Ruszczyński|Ruszczyński]]|first=Andrzej|title=Nonlinear Optimization|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=2006|pages=xii+454|isbn=978-0691119151 |mr=2199043}}</ref>
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| ==Examples==
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| ===2-dimensional example===
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| [[Image:Nonlinear programming.svg|200px|thumb|right|The intersection of the line with the constrained space represents the solution]] | |
| A simple problem can be defined by the constraints
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| :''x''<sub>1</sub> ≥ 0 | |
| :''x''<sub>2</sub> ≥ 0
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| :''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> ≥ 1
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| :''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> ≤ 2
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| with an objective function to be maximized
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| :''f''('''x''') = ''x''<sub>1</sub> + ''x''<sub>2</sub>
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| where '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>). [http://apmonitor.com/online/view_pass.php?f=2d.apm Solve 2-D Problem].
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| ===3-dimensional example===
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| [[Image:Nonlinear programming 3D.svg|thumb|right|The intersection of the top surface with the constrained space in the center represents the solution]]
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| Another simple problem can be defined by the constraints
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| :''x''<sub>1</sub><sup>2</sup> − ''x''<sub>2</sub><sup>2</sup> + ''x''<sub>3</sub><sup>2</sup> ≤ 2
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| :''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> + ''x''<sub>3</sub><sup>2</sup> ≤ 10
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| with an objective function to be maximized
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| :''f''('''x''') = ''x''<sub>1</sub>''x''<sub>2</sub> + ''x''<sub>2</sub>''x''<sub>3</sub>
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| where '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>). [http://apmonitor.com/online/view_pass.php?f=3d.apm Solve 3-D Problem].
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| ==See also==
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| * [[Curve fitting]]
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| * [[Least squares]] minimization
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| * [[Linear programming]]
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| * [[nl (format)]]
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| * [[Mathematical optimization]]
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| * [[List of optimization software]]
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| *[[Werner Fenchel]]
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| ==References==
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| <references/>
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| ==Further reading==
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| * Avriel, Mordecai (2003). ''Nonlinear Programming: Analysis and Methods.'' Dover Publishing. ISBN 0-486-43227-0.
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| * Bazaraa, Mokhtar S. and Shetty, C. M. (1979). ''Nonlinear programming. Theory and algorithms.'' John Wiley & Sons. ISBN 0-471-78610-1.
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| * Bertsekas, Dimitri P. (1999). ''Nonlinear Programming: 2nd Edition.'' Athena Scientific. ISBN 1-886529-00-0.
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| * {{cite book|last1=Bonnans|first1=J. Frédéric|last2=Gilbert|first2=J. Charles|last3=Lemaréchal|first3=Claude| authorlink3=Claude Lemaréchal|last4=Sagastizábal|first4=Claudia A.|title=Numerical optimization: Theoretical and practical aspects|url=http://www.springer.com/mathematics/applications/book/978-3-540-35445-1|edition=Second revised ed. of translation of 1997 <!-- ''Optimisation numérique: Aspects théoriques et pratiques'' --> French| series=Universitext|publisher=Springer-Verlag|location=Berlin|year=2006|pages=xiv+490|isbn=3-540-35445-X|doi=10.1007/978-3-540-35447-5|mr=2265882}}
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| * {{cite book|last1=Luenberger|first1=David G.|authorlink1=David G. Luenberger|last2=Ye|first2=Yinyu|authorlink2=Yinyu Ye|title=Linear and nonlinear programming|edition=Third|series=International Series in Operations Research & Management Science|volume=116|publisher=Springer|location=New York|year=2008|pages=xiv+546|isbn=978-0-387-74502-2 | mr = 2423726}}
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| * Nocedal, Jorge and Wright, Stephen J. (1999). ''Numerical Optimization.'' Springer. ISBN 0-387-98793-2.
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| * [[Jan Brinkhuis]] and Vladimir Tikhomirov, 'Optimization: Insights and Applications', 2005, Princeton University Press
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| ==External links==
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| *[http://neos-guide.org/non-lp-faq Nonlinear programming FAQ]
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| *[http://glossary.computing.society.informs.org/ Mathematical Programming Glossary]
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| *[http://www.lionhrtpub.com/orms/surveys/nlp/nlp.html Nonlinear Programming Survey OR/MS Today]
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| *[http://apmonitor.com/wiki/index.php/Main/Background Overview of Optimization in Industry]
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| {{optimization algorithms}}
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| {{DEFAULTSORT:Nonlinear Programming}}
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| [[Category:Mathematical optimization]]
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| [[Category:Optimization algorithms and methods]]
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