A number of individuals are familiar with linear systems or linear problems commonly used in engineering and generally in the field of sciences. These are commonly presented as vectors. Such problems or systems can be extended to other forms in which variables are partitioned to two disjointed subsets, in which case the left-hand-side is linear on each separate set. This gives rise to optimization problems having bilinear objectives together with one or more constraints called the biliniar problem.
Generally, bilinear functions are said to be composed of subclasses of quadratic functions and even quadratic programming. Such programming normally has a number of applications for example when dealing with constrained bi-matrix games, complementarity problems as well as when handling Markovian assignment problems. In addition, most of the 0-1 integer programs are able to be described in a similar way.
A number of similarities exist between the bilinear systems and the linear systems. For instance, these systems both possess some homogeneity with the constants on the right-hand side identically becoming zero. In addition, an individual can always introduce multiples of such equations to the system of equations without changing their solution. These problems can as well be classified further into two forms including the incomplete and the complete forms. The complete forms generally have some unique solution and with the number of equations and variables being the same.
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.
A pooling problem also utilizes these form of equations. Such a problem in programming also has its application in getting the solution to a number of multi-agent coordination and planning problems. Nevertheless, these usually focus on the various aspects of the Markov process of decision making.
Generally, bilinear functions are said to be composed of subclasses of quadratic functions and even quadratic programming. Such programming normally has a number of applications for example when dealing with constrained bi-matrix games, complementarity problems as well as when handling Markovian assignment problems. In addition, most of the 0-1 integer programs are able to be described in a similar way.
A number of similarities exist between the bilinear systems and the linear systems. For instance, these systems both possess some homogeneity with the constants on the right-hand side identically becoming zero. In addition, an individual can always introduce multiples of such equations to the system of equations without changing their solution. These problems can as well be classified further into two forms including the incomplete and the complete forms. The complete forms generally have some unique solution and with the number of equations and variables being the same.
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.
A pooling problem also utilizes these form of equations. Such a problem in programming also has its application in getting the solution to a number of multi-agent coordination and planning problems. Nevertheless, these usually focus on the various aspects of the Markov process of decision making.
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