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Classical Optimization
Published in Albert G. Holzman, Mathematical Programming, 2020
A point at which the first derivative is zero is called a stationary point. Therefore, we have proven that a necessary condition for x0 to be a local maximum or minimum is that x0 be a stationary point. Since Theorem 7 provides only a necessary condition, a stationary point may be either a maximum, a minimum, or neither. The following obvious examples illustrate this: f1(x) = 4x2 has a minimumf2(x) = -3x2 has a maximumf3(x) = 7x3 has neither a maximum nor a minimum
Optimum Design
Published in William H. Middendorf, Richard H. Engelmann, Design of Devices and Systems, 2017
William H. Middendorf, Richard H. Engelmann
Elementary calculus demonstrates that if the first derivative is zero, the function is at a maximum or a minimum, or at least at a stationary point. That is, dC/dx = 0 at each of the five points designated in Figure 12.2. The word “extremum” is used to refer to maximum and minimum points, and the optimum design is usually at one of the extrema. A stationary point is usually of no significance. To determine which of the extrema has been located or whether a stationary point has been found, one method is to evaluate the second derivative of the function. If the second derivative is positive, the extremum is at the least a relative minimum. If the second derivative is negative, the extremum is a maximum, but again it may be only a relative maximum, as shown in the figure. If the second derivative is zero, a stationary point has been located.
Steps in the Finite Element Method
Published in Chandrakant S. Desai, Tribikram Kundu, Introductory Finite Element Method, 2017
Chandrakant S. Desai, Tribikram Kundu
Extremum points are also known as stationary points because the function value is stationary at these points. Since the slope dfdx=0 at stationary points the first variation of the function δf=dfdx⋅Δx must be zero at those points.
A tutorial introduction to reinforcement learning
Published in SICE Journal of Control, Measurement, and System Integration, 2023
Now we give a brief introduction to SA. Suppose is some function and d can be any integer. The objective of SA is to find a solution to the equation , when only noisy measurements of are available. The SA method was introduced in Ref. [21], where the objective was to find a solution to a scalar equation , where . The extension to the case where d>1 was first proposed in Ref. [22]. The problem of finding a fixed point of a map can be formulated as the above problem with . If it is desired to find a stationary point of a function , then we simply set . Thus the above problem formulation is quite versatile. More details are given at the start of Section 4.2.
Intelligent step-length adjustment for adaptive path-following in nonlinear structural mechanics based on group method of data handling neural network
Published in Mechanics of Advanced Materials and Structures, 2022
Ali Maghami, Seyed Mahmoud Hosseini
As shown in Figure 6, the critical points on the path could be considered as the stationary and inflection points. A stationary point is a point on the path where the function’s derivative is zero. While, for the multivariable case, it is a point where the gradient of λ is equal to zero, as follows: and the gradient of λ is where, n is the number of degrees of freedom, and the vector index is moved in front of each quantity to reduce clutter. To compute the garadient of λ, we obtain the first derivative of the displacement vector with respect to the load factor as follows: where, for point i on the equilibrium path, can be computed by: we can rewrite (38) according to (7) as: for a point on the equilibrium path, is equal to zero (see Figure 1). Thus, relation (39) yields: where, is computed during the path-following through the relation (8). Therefore, the gradient of λ would be:
Analysis of screening decisions in inventory models with imperfect quality items
Published in International Journal of Production Research, 2021
Zsuzsanna Hauck, Boualem Rabta, Gerald Reiner
However, even though the above conditions are necessary, they are not sufficient to ensure the optimality of a solution. If the cost function is convex, then KKT conditions become sufficient given that the constraint is linear. However, the convexity of the objective function holds only for particular forms of the screening cost and defect detection functions. In the general case (see below), the cost function is not convex. A stationary point, in this case, can be a local minimum, a local maximum, or a saddle point. It is also possible that multiple local minima exist that are determined by checking the second-order optimality condition. If the above KKT system of equations leads to a small number of solutions, then the global minimum can be determined by examining the values of the objective function at those local minima.