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Alternatively, if the constraints are all equality constraints and are all linear, they can be solved for some of the variables in terms of the others, and the former can be substituted out of the objective function, leaving an unconstrained problem in a smaller number of variables.
For example, in economics the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value of the objective function (profit) due to the relaxation of a given constraint (e.g. through a change in income); in such a context is the marginal cost of the ...
For instance, from the example above in economics, if the maximal utility of two goods is achieved when the quantity of goods x and y are (−2, 5), and the utility is subject to the constraint x and y are greater than or equal to 0 (one cannot consume a negative quantity of goods) as is usually the case, then the actual solution to the problem ...
A typical non-convex problem is that of optimizing transportation costs by selection from a set of transportation methods, one or more of which exhibit 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 ...
Often in mathematical economics the KKT approach is used in theoretical models in order to obtain qualitative results. For example, [14] consider a firm that maximizes its sales revenue subject to a minimum profit constraint.
Global constraints are used [3] to simplify the modeling of constraint satisfaction problems, to extend the expressivity of constraint languages, and also to improve the constraint resolution: indeed, by considering the variables altogether, infeasible situations can be seen earlier in the solving process.
Constraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables , which is solved by constraint satisfaction methods.
In economics, many problems involve multiple objectives along with constraints on what combinations of those objectives are attainable.For example, consumer's demand for various goods is determined by the process of maximization of the utilities derived from those goods, subject to a constraint based on how much income is available to spend on those goods and on the prices of those goods.