Many people have the knowledge of linear systems or problems that are common in the field of engineering or generally in sciences. These are usually expressed as vectors. Such systems or problems are also applicable to different forms whereby variables are separated to two subsets that are disjointed with the left-hand side being linear for every separate set. This gives optimization problems that have bilinear objective functions accompanied by one or two constraints, a form known the biliniar problem.
Generally, the bilinear functions are known to have subclasses of quadratic function as well as quadratic programming. This programming usually have several applications such as when dealing with the constrained bi-matrix games, dealing with Markovian assignments, and complementarity problems. At the same time, majority of 0-1 integer programs can be described similarly.
There are various similarities that can be noted between linear systems and bi-linear systems. For instance, both systems have some homogeneity where the right side constants identically become zero. In addition, one can always add multiples to the equations without altering their solutions. On the contrary, these problems can be further classified into two forms, that is the complete and the incomplete forms. The complete forms normally have unique solutions apart from the number of variables being equal to the number of equations.
On the other hand, for the incomplete forms, there are usually more variables than equations and the solution to such problem is indefinite and lies between some range of values. The formulation of these problems takes various forms. Nevertheless, the more common practical problems include an objective function that is bilinear, accompanied by one or more linear constraints. For the expressions that take this form, theoretical results can always be obtained.
These programming problems can also be expressed in form of concave minimization problems, due to their importance in coming up with the concave minimizations. There are two main reasons for this case. First, the bilinear programming may be applied in numerous problems in a real world. Secondly, some techniques often used in solving the bilinear problems have similarities with those techniques often applied in solving the general concave problems in minimization.
There are various scenarios in which these programming problems remain applicable. These include the representation of situations such as the ones facing bimatrix game players. Other areas of previous successful use are such as multi-commodity flow networks, multilevel assignment problems, decision-making theory, scheduling orthogonal production as well as locating of a freshly acquired facility.
In addition, optimization problems usually involving bilinear programs are also necessary when conducting petroleum blending activities as well as water networks operations across the globe. Non-convex-bilinear constraints are mostly needed when modeling the proportions to be mixed from different streams within the petroleum blending and water network systems.
The pooling problem as well make use of these forms of problems. Their application also include solving various planning problems and multi-agent coordination. Nonetheless, these generally places focus on numerous features of the Markov process that is commonly used in decision-making process.
Generally, the bilinear functions are known to have subclasses of quadratic function as well as quadratic programming. This programming usually have several applications such as when dealing with the constrained bi-matrix games, dealing with Markovian assignments, and complementarity problems. At the same time, majority of 0-1 integer programs can be described similarly.
There are various similarities that can be noted between linear systems and bi-linear systems. For instance, both systems have some homogeneity where the right side constants identically become zero. In addition, one can always add multiples to the equations without altering their solutions. On the contrary, these problems can be further classified into two forms, that is the complete and the incomplete forms. The complete forms normally have unique solutions apart from the number of variables being equal to the number of equations.
On the other hand, for the incomplete forms, there are usually more variables than equations and the solution to such problem is indefinite and lies between some range of values. The formulation of these problems takes various forms. Nevertheless, the more common practical problems include an objective function that is bilinear, accompanied by one or more linear constraints. For the expressions that take this form, theoretical results can always be obtained.
These programming problems can also be expressed in form of concave minimization problems, due to their importance in coming up with the concave minimizations. There are two main reasons for this case. First, the bilinear programming may be applied in numerous problems in a real world. Secondly, some techniques often used in solving the bilinear problems have similarities with those techniques often applied in solving the general concave problems in minimization.
There are various scenarios in which these programming problems remain applicable. These include the representation of situations such as the ones facing bimatrix game players. Other areas of previous successful use are such as multi-commodity flow networks, multilevel assignment problems, decision-making theory, scheduling orthogonal production as well as locating of a freshly acquired facility.
In addition, optimization problems usually involving bilinear programs are also necessary when conducting petroleum blending activities as well as water networks operations across the globe. Non-convex-bilinear constraints are mostly needed when modeling the proportions to be mixed from different streams within the petroleum blending and water network systems.
The pooling problem as well make use of these forms of problems. Their application also include solving various planning problems and multi-agent coordination. Nonetheless, these generally places focus on numerous features of the Markov process that is commonly used in decision-making process.
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