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, 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.
Usually, some similarities may be noted between the linear and the bi-linear systems. For example, both systems have homogeneity in which case the right hand side constants become zero. Additionally, you may add multiples to equations without the need to alter their solutions. At the same time, these problems can further be classified into other two forms that include the complete as well as the incomplete forms. Generally, the complete form usually have distinct solutions other than the number of the variables being the same as the number of the equations.
On the contrary, incomplete forms usually have an indefinite solution that lies in some specified range, and contain more variables compared to the number of equations. In formulating these problems, various forms can be developed. Nonetheless, a more common and practical problem includes the bilinear objective functions that are bound by some constraints that are linear. All expressions taking this form usually have a theoretical result.
Such programming problems may as well be expressed as concave minimization problems. This is because of their importance when coming up with concave minimizations. Two main reasons exist for this. To begin with, the bilinear programming can be applied to numerous problems in the real world. The second is that some of the techniques utilized when solving bilinear programs bear similarities with the techniques applied in solving general concave problems on minimization.
These programming problems may be applied in several ways. These applications are such as in models which try to represent circumstances the players of bimatrix games often face. It has also been used previously in decision making theory, locating newly acquired equipment, multi-commodity network flow, multi-level assignment issues and scheduling orthogonal production.
On the other hand, optimization issues normally connected to bilinear programs remain necessary when undertaking water network operations and even petroleum blending activities around the world. Non-convex-bilinear constraints can be required in the modeling of proportions from different streams that are to be combined in petroleum blending as well as water networking 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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Author: Matthew FoxThis author has published 1 articles so far.