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Matrix Calculus for Machine Learning
Published in Richard M. Golden, Statistical Machine Learning, 2020
The Lagrange Multiplier Theorem is often used to transform a constrained optimization problem into an unconstrained optimization problem that can be solved with a descent algorithm. For example, suppose that the goal is to minimize the restriction of a continuously differentiable function V:Ω→Rd to a smooth nonlinear hypersurface {x∈Rd:ϕ(x)=0m}. This problem can be transformed into an unconstrained nonlinear optimization problem using the Lagrange Multiplier Theorem provided that ϕ is continuously differentiable and the Jacobian of ϕ evaluated at x* has full row rank.
Simulation, Evaluation, and Optimization of Thermal Energy Systems
Published in Steven G. Penoncello, Thermal Energy Systems, 2018
Each Lagrange multiplier in an optimization problem is associated with a constraint. In Example 6.2, the Lagrange multiplier is associated with the constraint pertaining to the total length of tubing available to form the tube bundle inside the shell of the heat exchanger. Recall that the units of the Lagrange multiplier for Example 6.2 is [$/m]. This indicates that the value of the Lagrange multiplier has some relationship between the heat exchanger initial cost (objective function) and the length of tubing available (constraint). To reveal this relationship, consider Example 6.2 again with 101 m of tubing available instead of 100. Solving the optimization problem for this case results in, Ds = 0.69480 m, LHX = 1.3319 m, which results in a minimized heat exchanger cost of CHX = $4,202.20. By making an additional 1 m of tubing available, the heat exchanger cost increases by $4,202.20 – $4,181.59 = $20.61. Referring back to Example 6.2, the Lagrange multiplier was determined to be λ = −20.616 $/m. This exercise reveals the significance of the Lagrange multiplier; it indicates the change in the minimized objective function per unit change in the constraint, ∂f*∂Φk=−λk
Body-Coordinate Formulation
Published in Parviz E. Nikravesh, Planar Multibody Dynamics, 2018
where D is the system Jacobian matrix, as in Eqs. (7.51) and (7.52), and λ is an array of coefficients, known as Lagrange multipliers. The number of multipliers, that is, the dimension of array λ, is equal to the number of constraints nc (or the number of rows in D). In other words, for each constraint, there is one Lagrange multiplier. The immediate advantage of describing the array (r)h as D′λis that the number of elements in (r)h is 3 × nb (unknowns), but the number of unknown multipliers in the array λ is nc, where nc < 3 × nb; that is, we have reduced the number of unknowns in the equations of motion. With the new representation of the array of reaction forces, Eq. (7.53) is expressed as
A Tikhonov regularization parameter approach for solving Lagrange constrained optimization problems
Published in Engineering Optimization, 2018
Julio B. Clempner, Alexander S. Poznyak
The method of Lagrange multipliers is a strategy for finding the local minimum (maximum) of a function, subject to equality constraints (Poznyak, 2008). Consider the Lagrange function given by (Garcia and Zangwill, 1981; Zangwill, 1969) where the parameters α and δ are positive, the Lagrange vector-multipliers are non-negative and the components of may have any sign. Obviously, the optimization problem has a unique saddle-point on z since the optimized Equation (2) is strongly convex (Poznyak, 2008) if the parameters α and provide the condition and is strongly concave on the Lagrange multipliers for any . In view of these properties, Equation (2) has the unique saddle point (see the Kuhn–Tucker Theorem 21.13 in Poznyak [2008]) for which the following inequalities hold: for any with non-negative components and any ,