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In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]
where denotes the vector (x 1, x 2). In this example, the first line defines the function to be minimized (called the objective function , loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint.
For a 2-D Cartesian (trilateration) situation, these restrictions take one of two equivalent forms: The allowable interior angle at P between lines P-C1 and P-C2: The ideal is a right angle, which occurs at distances from the baseline of one-half or less of the baseline length; maximum allowable deviations from the ideal 90 degrees may be ...
Also, let Q = (x 1, y 1) be any point on this line and n the vector (a, b) starting at point Q. The vector n is perpendicular to the line, and the distance d from point P to the line is equal to the length of the orthogonal projection of on n. The length of this projection is given by:
More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain , codomain and graph ( ) = {(,) ():}. Similarly, one can define a right-restriction or range restriction R B . {\displaystyle R\triangleright B.}
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Letting μ = e −2λ, the problem becomes finding the maximum of μ 5 −2μ 4 +μ 3. Differentiating, the μ has to satisfy 5μ 4 −8μ 3 +3μ 2 = 0. This equation has roots 0, 0.6, and 1. As μ is actually e −2λ, it has to be greater than zero but less than one.
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