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| {{refimprove|date=May 2008}}
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| '''Bilevel optimization''' is a special kind of [[optimization]] where one problem is embedded (nested) within another. The outer optimization task is commonly referred to as the upper-level optimization task, and the inner optimization task is commonly referred to as the lower-level optimization task. These problems involve two kinds of variables, referred to as the upper-level variables and the lower-level variables.
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| == Mathematical formulation of the problem ==
| | A central defender (CD) won't be too happy about playing at right back (RB) unless he already has experience of playing there. Provide from tv broadcast online from atdhe, justin tv or ustream, Myp2p or sopcast. Donovan's ringing endorsement typifies the sentiments of experienced players and coaches alike. The calendar 2011 consists of costly decoration and that charm their watchers in just one go, easily become impressive wall decoration pieces for our home, which we feel like watching every day. On the other hand, France coach Raymond Domenech will take just his squad of 23 players and nobody else to a FIFA World Cup training camp later this month. <br><br>With world-class content we'll help you coach, consistent with your beliefs and philosophy or play with confidence. The beauty of these FIFA 09 football game tricks are, they are simple enough to master, plus you get to do a little showing off on the field, which makes scoring against your rival that much more satisfying. It may seem too theoretical or unnecessary, while we should spend some time idealizing what you want. While high hopes were placed on Brazil for winning the tournament, instead they placed in fourth, with Germany grabbing the top prize. Ellis Park, also close to the city centre, is in the process of a R2billion upgrade that will include new upper tiers behind each goal. <br><br><br><br>By pressing the arrow keys you can change the corner taker. Unlike everybody else, in Acapulco the National squad was there to get in tune early for the most important soccer competition in the planet. What happens in cases like this you simply make use of the 59th Minute Strategy in combination with the Ultimate Database and simply scoop up easy money. The decent TV player would greatly facilitate your World Cup life. Group stage of FIFA world cup 2010 is now over and the tournament has unfolded some dramatic surprises. <br><br><br><br>Pressing the (default) "C" button will bring one of your players close to the corner and a quick short pass can give him a good place for a cross. This way you will avoid making mistakes that will cost you a lot of coins that you will have bought. If you are a Windows user, Blaze - Video HDTV Player is your best option. The sixteen qualifiers were completed by Mexico and the USA from Central and North America, Japan and South Korea from Asia, and Ghana from Africa. This is a sum equivalent to four hundred dollars each for every man, woman and child in this nation of two hundred million. <br><br>Shortly enough, it will be your opening to overcome your enemies, after you've creamed the nuts and bolts of FIFA. Opening kickoff for this pivotal clash is scheduled for today at 4 p. In FUT 13, the traditionalmoney makingmethods are either now ineffective or have too much competition. The particular FIFA age 14 computer game claims a long time of enjoyable along with your footballer friends. Gambling on packs to try and play the market in order to make coins is risky- packs dont always contain high value items and the whole process takes a lot of time. <br><br>ESPN TV and CBC-Canada TV will both get their USA vs. In FIFA 15, the stadiums are animated and they are filled from the emotion that only football can make. #links# It held in South Africa for the first time, this is the first African countries to host the it, suggesting the cause of [http://browse.deviantart.com/?q=African+Football African Football] gradually into the peak. Striker and captain Asamoah Gyan is amongst the top [http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=scorers&gs_l=news scorers] of FIFA world cup 2010. The purpose of the V is to nurture, support and train the whole Qnet Network of 6 Million plus.<br><br>Here is more info in regards to [https://www.facebook.com/fifa15cheatsandhack Fifa 15 Hack] stop by our own website. |
| A general formulation of bilevel optimization problem can be written as follows: | |
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| <math> \min\limits_{x \in X, y \in Y}\;\; F(x,y) </math>
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| subject to:
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| <math> G_i(x,y) \leq 0</math> , for <math>i \in \{ 1,2,\ldots ,I \}</math>;
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| <math> y \in \arg \min \limits_{z \in Y} \{ f(x,z) : g_{j}(x,z) \leq 0, j \in \{ 1,2,\ldots,J \} \}</math>
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| where
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| :<math> F,f: R^{n_x} \times R^{n_y} \to R</math>
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| :<math> G_i,g_j: R^{n_x} \times R^{n_y} \to R</math>
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| :<math> X \subseteq R^{n_x}</math>
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| :<math> Y \subseteq R^{n_y}.</math>
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| In the above formulation, <math>F</math> represents the upper-level objective function and <math>f</math> represents the lower-level objective function. Similarly <math>x</math> represents the upper-level decision vector and <math>y</math> represents the lower-level decision vector. <math>G_i</math> and <math>g_j</math> represent the inequality constraint functions at the upper and lower levels respectively. Equality constraints may also be present in a bilevel program, but they have been omitted for brevity.
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| ==[[Stackelberg]] Games==
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| Bilevel optimization was first realized in the field of game theory by a German economist [[Heinrich Freiherr von Stackelberg]] who published Market Structure and Equilibrium (Marktform und Gleichgewicht) in 1934 that described this hierarchical problem. The strategic game described in his book came to be known as Stackelberg game that consists of a leader and a follower. The leader is commonly referred as a Stackelberg leader and the follower is commonly referred as a Stackelberg follower. In a Stackelberg game, the players of the game compete with each other, such that the leader makes the first move, and then the follower reacts optimally to the leader's action. This kind of a hierarchical game is asymmetric in nature, where the leader and the follower can not be interchanged. The leader knows ex ante that the follower observes its actions before responding in an optimal manner. Therefore, if the leader wants to optimize its objective, then it needs to anticipate the optimal response of the follower. In this setting, the leader's optimization problem contains a nested optimization task that corresponds to the follower's optimization problem. In the Stackelberg games, the upper level optimization problem is commonly referred as the leader's problem and the lower level optimization problem is commonly referred as the follower's problem.
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| == Applications ==
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| Bilevel optimization problems are commonly found in a number of real-world problems. This includes problems in the domain of [[transportation]], [[economics]], [[decision science]], [[business]], [[engineering]], [[environmental economics]] etc. Some of the practical bilevel problems studied in the literature are briefly discussed.<ref>{{cite web |url=http://bilevel.org/applications/ |title=Scope: Evolutionary Bilevel Optimization |last1= |first1= |last2= |first2= |date= |website=http://www.bilevel.org |publisher= |accessdate=6 October 2013}}</ref>
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| ===Toll setting problem===
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| In the field of transportation, bilevel optimization commonly appears in the toll-setting problem. Consider a network of highways that is operated by the government. The government wants to maximize its revenues by choosing the optimal toll setting for the highways. However, the government can maximize its revenues only by taking the highway users' problem into account. For any given tax structure the highway users solve their own optimization problem, where they minimize their traveling costs by deciding between utilizing the highways or an alternative route. Under these circumstances, the government's problem needs to be formulated as a bilevel optimization problem. The upper level consists of the governments objectives and constraints, and the lower level consists of the highway users' objectives and constraints for a given tax structure. It is noteworthy that the government will be able to identify the revenue generated by a particular tax structure only by solving the lower level problem that determines to what extent the highways are used.
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| ===Structural optimization===
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| Structural optimization problems comprise of two levels of optimization task and are commonly referred as mathematical programming problems with equilibrium constraints ([[MPEC]]). The upper level objective in such problems may involve cost minimization or weight minimization subject to bounds on displacements, stresses and contact forces. The decision variables at the upper level usually are shape of the structure, choice of materials, amount of material etc. However, for any given set of upper level variables, the state variables (displacement, stresses and contact forces) can only be figured out by solving the potential energy minimization problem that appears as an equilibrium satisfaction constraint or lower level minimization task to the upper level problem.
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| ===Defense applications===
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| Bilevel optimization has a number of applications in defense, like [[strategic offensive]] and defensive force structure design, strategic bomber force structure, and allocation of tactical aircraft to missions. The offensive entity in this case may be considered a leader and the defensive entity in this case may be considered a follower. If the leader wants to maximize the damage caused to the opponent, then it can only be achieved if takes the reactions of the follower into account. A rational follower will always react optimally to the leaders offensive. Therefore, the leader's problem appears as an upper level optimization task, and the optimal response of the follower to the leader's actions is determined by solving the lower level optimization task.
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| == Solution methodologies ==
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| {{Empty section|date=October 2013}}
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| == Evolutionary bilevel optimization ==
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| For complex bilevel problems, classical methods fail due to difficulties like [[non-linearity]], [[discreteness]], non-[[differentiability]], non-[[convexity]] etc. In such situations, evolutionary methods, though computationally demanding, could be an alternative tool to offset some of these difficulties and lead to an approximate optimal solution.
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| ==Multi-objective bilevel optimization==
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| A bilevel optimization problem can be generalized to a multi-objective bilevel optimization problem with multiple objectives at one or both levels. A general multi-objective bilevel optimization problem can be formulated as follows:
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| <math> \min\limits_{x \in X, y \in Y}\;\; F(x,y) = ( F_{1} (x,y),F_{2} (x,y),\ldots,F_{p} (x,y) )</math>
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| subject to:
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| <math> G_i(x,y) \leq 0</math> , for <math>i \in \{ 1,2,\ldots,I \}</math>;
| |
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| <math> y \in \arg \min \limits_{z \in Y} \{ f(x,z) = ( f_1 (x,z),f_2 (x,z),\ldots,f_q (x,z) ) : g_{j}(x,z) \leq 0, j \in \{ 1,2,\ldots,J \} \}</math>
| |
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| where
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| :<math> F: R^{n_x} \times R^{n_y} \to R^{p}</math>
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| :<math> f: R^{n_x} \times R^{n_y} \to R^{q}</math>
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| :<math> G_i,g_j: R^{n_x} \times R^{n_y} \to R</math>
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| :<math> X \subseteq R^{n_x}</math>
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| :<math> Y \subseteq R^{n_y}.</math>
| |
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| In the above formulation, <math>F</math> represents the upper-level objective vector with <math>p</math> objectives and <math>f</math> represents the lower-level objective vector with <math>q</math> objectives. Similarly, <math>x</math> represents the upper-level decision vector and <math>y</math> represents the lower-level decision vector. <math>G_i</math> and <math>g_j</math> represent the inequality constraint functions at the upper and lower levels respectively. Equality constraints may also be present in a bilevel program, but they have been omitted for brevity.
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| ==References==
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| {{Reflist}}
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| == External links==
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| *[http://www.bilevel.org/ Evolutionary Bilevel Optimization]
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| *[http://glossary.computing.society.informs.org/ Mathematical Programming Glossary]
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| == See also == | |
| * [[optimization (mathematics)|optimization]]
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| [[Category:Mathematical optimization]]
| |
A central defender (CD) won't be too happy about playing at right back (RB) unless he already has experience of playing there. Provide from tv broadcast online from atdhe, justin tv or ustream, Myp2p or sopcast. Donovan's ringing endorsement typifies the sentiments of experienced players and coaches alike. The calendar 2011 consists of costly decoration and that charm their watchers in just one go, easily become impressive wall decoration pieces for our home, which we feel like watching every day. On the other hand, France coach Raymond Domenech will take just his squad of 23 players and nobody else to a FIFA World Cup training camp later this month.
With world-class content we'll help you coach, consistent with your beliefs and philosophy or play with confidence. The beauty of these FIFA 09 football game tricks are, they are simple enough to master, plus you get to do a little showing off on the field, which makes scoring against your rival that much more satisfying. It may seem too theoretical or unnecessary, while we should spend some time idealizing what you want. While high hopes were placed on Brazil for winning the tournament, instead they placed in fourth, with Germany grabbing the top prize. Ellis Park, also close to the city centre, is in the process of a R2billion upgrade that will include new upper tiers behind each goal.
By pressing the arrow keys you can change the corner taker. Unlike everybody else, in Acapulco the National squad was there to get in tune early for the most important soccer competition in the planet. What happens in cases like this you simply make use of the 59th Minute Strategy in combination with the Ultimate Database and simply scoop up easy money. The decent TV player would greatly facilitate your World Cup life. Group stage of FIFA world cup 2010 is now over and the tournament has unfolded some dramatic surprises.
Pressing the (default) "C" button will bring one of your players close to the corner and a quick short pass can give him a good place for a cross. This way you will avoid making mistakes that will cost you a lot of coins that you will have bought. If you are a Windows user, Blaze - Video HDTV Player is your best option. The sixteen qualifiers were completed by Mexico and the USA from Central and North America, Japan and South Korea from Asia, and Ghana from Africa. This is a sum equivalent to four hundred dollars each for every man, woman and child in this nation of two hundred million.
Shortly enough, it will be your opening to overcome your enemies, after you've creamed the nuts and bolts of FIFA. Opening kickoff for this pivotal clash is scheduled for today at 4 p. In FUT 13, the traditionalmoney makingmethods are either now ineffective or have too much competition. The particular FIFA age 14 computer game claims a long time of enjoyable along with your footballer friends. Gambling on packs to try and play the market in order to make coins is risky- packs dont always contain high value items and the whole process takes a lot of time.
ESPN TV and CBC-Canada TV will both get their USA vs. In FIFA 15, the stadiums are animated and they are filled from the emotion that only football can make. #links# It held in South Africa for the first time, this is the first African countries to host the it, suggesting the cause of African Football gradually into the peak. Striker and captain Asamoah Gyan is amongst the top scorers of FIFA world cup 2010. The purpose of the V is to nurture, support and train the whole Qnet Network of 6 Million plus.
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