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The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
However, to apply it, the origin (all variables equal to 0) must be a feasible point. This condition is satisfied only when all the constraints (except non-negativity) are less-than constraints and with positive constant on the right-hand side. The Big M method introduces surplus and artificial variables to convert all inequalities into that form.
The bucket elimination algorithm can be adapted for constraint optimization. A given variable can be indeed removed from the problem by replacing all soft constraints containing it with a new soft constraint. The cost of this new constraint is computed assuming a maximal value for every value of the removed variable.
If the objective function is quadratic and the constraints are linear, quadratic programming techniques are used. If the objective function is a ratio of a concave and a convex function (in the maximization case) and the constraints are convex, then the problem can be transformed to a convex optimization problem using fractional programming ...
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.
A price index aggregates various combinations of base period prices (), later period prices (), base period quantities (), and later period quantities (). Price index numbers are usually defined either in terms of (actual or hypothetical) expenditures (expenditure = price * quantity) or as different weighted averages of price relatives ( p t ...
Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning). Constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints.